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Presenting Global Design The Cocoon

  •  {the design structure cocoon is a required within our environment to,
    (clean and purify the air we breath)-(within automotive and industrial equipment worldwide)
  • This will remove the following components,
  • {carbon monoxide (parts per billion)},
  • {lead (micrograms per cubic meter)},
  • {nitrogin dioxide (parts per billion)},
  • {ground level (ozone per million)},
  • {sulfer dioxide (parts per billion)}, in addition to poisonous incinerations there could be additional component groups to examine, in relation to long term health conditions example: cancer..
  • The global and environmental design is including space,{our sky},  the over all design will gather atmospheric burnt carbons.
  • Example usages: possibly to use to snuff out the earth internal chambers of active carbon minerals, without changing characteristics..
  • This process should  be included for years to come, also with surrounding locations to inhabit in Space, {this design will perfectly fit our global environment and planetary environments}.
  • In addition the concord jet design should replace our current projectiles, {our public transportation in the air is not as stable as it could be, with the concord jet design in place}. generated with  the cocoon powered design it could enter space and back to earths atmosphere and land safely.
  • The cocoon powered design in the floor of automobiles will ventelate and gather Co2 particles and carbons from the air}, a very big forward move to air quality control on this planet, currently with seven billion people using fuel..
  •  A-Z Tek Operations is also proud to present a global hub design that is motionless, it will operate exactly with the rotation of the earths movement – ground lens units will work with gigs and satellite control. controling global operations,communication, ectra…
  • This will also including a dome control units (1) north, poll (1) south poll  (6) equator, rim
    {its exterior will look like the dome on the u.s.s.r. clock design in there country.
  • The network will be applied threw reflective imagery – {Black White Filter In The Sky}-{ Imatating gray clouds }.
  • The Nuro Net Operations –  will include the light spectrum reflections directl A global colour pallet in every shad known to us, from the sun, {the Whirley gig will always be generating power, so there for, a back lit projection will always be possible in all most all ranges
  • Slag components of our metal groups, for imagery projection ., again each Whirley Gig will have this installed in the Nuro design group, “{to control our global whether  and expand our progection caricteristics.(inbound & outbound)}”.
  • I am currently looking on the USGS website looking at required maintenance of the global tremors and our global environment.
  • {Global Environment Is One Of My Main Concerns (The Design Is focused on It)
  • Generating my entire design groupings will definitely help to ensure global security and safety for us all long term.
  • Founder and Director., A-Z Tek Operations Inc.
    Current Family Name, Mr. Glenn Allan Garner. Sin: 456 118 553,
  • Original Family Name, Mr. Glenn Allan Ham. Born January 16, 1957,
  • {Capricorn, Book Of Genesis 116 The Book Of Revelations},{Listed On This Web
  • Previous Founder and Director owner D & G Operations.,{with operations within the transportation industry in Ontario Canada}
  • Your 50% partnership with myself is open and has been conducted and reviewed threw B.W.C.
  • Sending Love For Her And His Majesties In The United King Dome, And The Global Community.
  • With Special Thanks To Her Majesty Queen Elizabeth The Second, Sir King George III, And There Beautiful Grand Children. From Mr. Glenn Allan Ham. Warmest Regards To Her Majesties Palace.
  • Revisions To The Start Up Of The Global Contract Required:
    The contracts will hopefully be global.
  • The suggested return for investment will be one tenth of one percent for investors. the 360 degree partnership will also be one tenth of one present with a guarantee of a 50% reinvestment clause, to reinvest in fresh design ventures. current, new and old structured ventures. Thus continuing the expansion into infinity with business operators\ and global operations within a firm hold on advancement in the coming future.

Thank You All

Bring Forward My Design Structure Threw B.W.C. Communications.

This site is my overview of our future designs and global structured requirements.

Thank You, Again;

                           Sincerely Yours:

Mr. Glenn Allan Garner.

A-Z Tek Operations Inc.

Presenting Global Design The Cocoon And Beyond 

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P.S. I Viewed Aviation Flight Operation & Testing,  Including Strategies In Design & Tactical. It’s Very Impressive!

We Could All Take Our Atmospheric Public Crafts & War Crafts Into Space?,

Hopefully – At Best Tactical Use ages Only.

Essential  For The Security, Of Our Solar System, Planets, Eco System!

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Hopefully To Be Asserted Faction Ably!

To Generate Our Way Into The Future For Centuries To Come!

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Null (SQL)

From Wikipedia, the free encyclopedia

The Greek lowercase omega (ω)character is used to represent Null indatabase theory.

Null is a special marker used in Structured Query Language (SQL) to indicate that a data value does not exist in the database. Introduced by the creator of the relational database model, E. F. Codd, SQL Null serves to fulfill the requirement that all true relational database management systems (RDBMS)support a representation of “missing information and inapplicable information”. Codd also introduced the use of the lowercase Greek omega (ω) symbol to represent Null in database theory. NULL is also an SQL reserved keyword used to identify the Null special marker.

For people new to the subject, a good way to remember what null means is to remember that in terms of information, “lack of a value” is not the same thing as “a value of zero”; similarly, “lack of an answer” is not the same thing as “an answer of no”. For example, consider the question “How many books does Juan own?” The answer may be “zero” (we know that he owns none) or “null” (we do not know how many he owns, or doesn’t own). In a database table, the column reporting this answer would start out with a value of null, and it would not be updated with “zero” until we have ascertained that Juan owns no books.

SQL null is a state (unknown) and not a value. This usage is quite different from most programming languages, where null means not assigned to a particular instance.


E. F. Codd mentioned nulls as a method of representing missing data in the relational model in a 1975 paper in the FDT Bulletin of ACMSIGMOD. Codd’s paper that is most commonly cited in relation with the semantics of Null (as adopted in SQL) is his 1979 paper in the ACM Transactions on Database Systems, in which he also introduced hisRelational Model/Tasmania, although much of the other proposals from the latter paper have remained obscure. Section 2.3 of his 1979 paper details the semantics of Null propagation in arithmetic operations as well as comparisons employing a ternary (three-valued) logic when comparing to nulls; it also details the treatment of Nulls on other set operations (the latter issue still controversial today). In database theory circles, the original proposal of Codd (1975, 1979) is now referred to as “Krokk tables”.[1] Codd later reinforced his requirement that all RDBMSs support Null to indicate missing data in a 1985 two-part article published in ComputerWorld magazine.[2][3]

The 1986 SQL standard basically adopted Codd’s proposal after an implementation prototype in IBM System R. Although Don Chamberlin recognized nulls (alongside duplicate rows) as one of the most controversial features of SQL, he defended the design of Nulls in SQL invoking the pragmatic arguments that it was the least expensive form of system support for missing information, saving the programmer from many duplicative application-level checks (see semipredicate problem) while at the same time providing the database designer with the option not to use Nulls if he so desires; for example, in order to avoid well known anomalies (discussed in the semantics section of this article). Chamberlin also argued that besides providing some missing-value functionality, practical experience with Nulls also led to other language features which rely on Nulls, like certain grouping constructs and outer joins. Finally, he argued that in practice Nulls also end up being used as a quick way to patch an existing schema when it needs to evolve beyond its original intent, coding not for missing but rather for inapplicable information; for example, a database that quickly needs to support electric cars while having a miles-per-gallon column.[4]

Codd indicated in his 1990 book The Relational Model for Database Management, Version 2 that the single Null mandated by the SQL standard was inadequate, and should be replaced by two separate Null-type markers to indicate the reason why data is missing. In Codd’s book, these two Null-type markers are referred to as ‘A-Values’ and ‘I-Values’, representing ‘Missing But Applicable’ and ‘Missing But Inapplicable’, respectively.[5] Codd’s recommendation would have required SQL’s logic system be expanded to accommodate a four-valued logic system. Because of this additional complexity, the idea of multiple Null-type values has not gained widespread acceptance in the database practitioners’ domain. It remains an active field of research though, with numerous papers still being published.


Null has been the focus of controversy and a source of debate because of its associated three-valued logic (3VL), special requirements for its use in SQL joins, and the special handling required by aggregate functions and SQL grouping operators. Computer science professor Ron van der Meyden summarized the various issues as: “The inconsistencies in the SQL standard mean that it is not possible to ascribe any intuitive logical semantics to the treatment of nulls in SQL.”[1] Although various proposals have been made for resolving these issues, the complexity of the alternatives has prevented their widespread adoption.

Null propagation[edit]

Arithmetic operations[edit]

Because Null is not a data value, but a marker for an unknown value, using mathematical operators on Null results in an unknown value, which is represented by Null.[6] In the following example, multiplying 10 by Null results in Null:

10 * NULL          -- Result is NULL

This can lead to unanticipated results. For instance, when an attempt is made to divide Null by zero, platforms may return Null instead of throwing an expected “data exception – division by zero”.[6] Though this behavior is not defined by the ISO SQL standard many DBMS vendors treat this operation similarly. For instance, the Oracle, PostgreSQL, MySQL Server, and Microsoft SQL Server platforms all return a Null result for the following:

NULL / 0

String concatenation[edit]

String concatenation operations, which are common in SQL, also result in Null when one of the operands is Null.[7] The following example demonstrates the Null result returned by using Null with the SQL || string concatenation operator.

'Fish ' || NULL || 'Chips'   -- Result is NULL

This is not true for all database implementations. In an Oracle RDBMS for example NULL and the empty string are considered the same thing and therefore ‘Fish ‘ || NULL || ‘Chips’ results in ‘Fish Chips’.

Comparisons with NULL and the three-valued logic (3VL)[edit]

Further information: Three-valued logic

Since Null is not a member of any data domain, it is not considered a “value”, but rather a marker (or placeholder) indicating the absence of value. Because of this, comparisons with Null can never result in either True or False, but always in a third logical result, Unknown.[8] The logical result of the expression below, which compares the value 10 to Null, is Unknown:

SELECT 10 = NULL       -- Results in Unknown

However, certain operations on Null can return values if the value of Null is not relevant to the outcome of the operation. Consider the following example:

SELECT NULL OR TRUE   -- Results in True

In this case, the fact that the value on the left of OR is unknowable is irrelevant, because the outcome of the OR operation would be True regardless of the value on the left.

SQL implements three logical results, so SQL implementations must provide for a specialized three-valued logic (3VL). The rules governing SQL three-valued logic are shown in the tables below (p and q represent logical states)”[9] The truth tables SQL uses for AND, OR, and NOT correspond to a common fragment of the Kleene and Łukasiewicz three-valued logic (which differ in their definition of implication, however SQL defines no such operation).[10]

p q p OR q p AND q p = q
True True True True True
True False True False False
True Unknown True Unknown Unknown
False True True False False
False False False False True
False Unknown Unknown False Unknown
Unknown True True Unknown Unknown
Unknown False Unknown False Unknown
Unknown Unknown Unknown Unknown Unknown
p NOT p
True False
False True
Unknown Unknown

Effect of Unknown in WHERE clauses[edit]

SQL three-valued logic is encountered in Data Manipulation Language (DML) in comparison predicates of DML statements and queries. The WHERE clause causes the DML statement to act on only those rows for which the predicate evaluates to True. Rows for which the predicate evaluates to either False or Unknown are not acted on by INSERT,UPDATE, or DELETE DML statements, and are discarded by SELECT queries. Interpreting Unknown and False as the same logical result is a common error encountered while dealing with Nulls.[9] The following simple example demonstrates this fallacy:


The example query above logically always returns zero rows because the comparison of the i column with Null always returns Unknown, even for those rows where i is Null. The Unknown result causes the SELECT statement to summarily discard each and every row. (However, in practice, some SQL tools will retrieve rows using a comparison with Null.)

Null-specific and 3VL-specific comparison predicates[edit]

Basic SQL comparison operators always return Unknown when comparing anything with Null, so the SQL standard provides for two special Null-specific comparison predicates. The IS NULL and IS NOT NULL predicates (which use a postfix syntax) test whether data is, or is not, Null.[11]

The SQL standard contains an extension F571 “Truth value tests” that introduces three additional logical unary operators (six in fact, if we count their negation, which is part of their syntax), also using postfix notation. They have the following truth tables:[12]

p true false unknown
p IS TRUE true false false
p IS NOT TRUE false true true
p IS FALSE false true false
p IS NOT FALSE true false true
p IS UNKNOWN false false true
p IS NOT UNKNOWN true true false

The F571 extension is orthogonal to the presence of the boolean datatype in SQL (discussed later in this article) and, despite syntactic similarities, F571 does not introduce boolean or three-valued literals in the language. The F571 extension was actually present in SQL92,[13] well before the boolean datatype was introduced to the standard in 1999. The F571 extension is implemented by few systems however; PostgreSQL is one of those implementing it.

The addition of IS UNKNOWN to the other operators of SQL’s three-valued logic makes the SQL three-valued logic functionally complete,[14] meaning its logical operators can express (in combination) any conceivable three-valued logical function.

On systems which don’t support the F571 extension, it is possible to emulate IS UNKNOWN p by going over every argument that could make the expression p Unknown and test those arguments with IS NULL or other NULL-specific functions, although this may be more cumbersome.

Law of the excluded fourth (in WHERE clauses)[edit]

In SQL’s three-valued logic the law of the excluded middle, p OR NOT p, no longer evaluates to true for all p. More precisely, in SQL’s three-valued logic p OR NOT p is unknown precisely when p is unknown and true otherwise. Because direct comparisons with Null result in the unknown logical value, the following query

SELECT * FROM stuff WHERE ( x = 10 ) OR NOT ( x = 10 );

is not equivalent in SQL with

SELECT * FROM stuff;

if the column x contains any Nulls; in that case the second query would return some rows the first one does not return, namely all those in which x is Null. In classical two-valued logic, the law of the excluded middle would allow the simplification of the WHERE clause predicate, in fact its elimination. Attempting to apply the law of the excluded middle to SQL’s 3VL is effectively a false dichotomy. The second query is actually equivalent with:

SELECT * FROM stuff;
-- is (because of 3VL) equivalent to:
SELECT * FROM stuff WHERE ( x = 10 ) OR NOT ( x = 10 ) OR x IS NULL;

Thus, to correctly simplify the first statement in SQL requires that we return all rows in which x is not null.


In view of the above, observe that for SQL’s WHERE clause a tautology similar to the law of excluded middle can be written. Assuming the IS UNKNOWN operator is present, pOR (NOT p) OR (p IS UNKNOWN) is true for every predicate p. Among logicians, this is called law of excluded fourth.

There are some SQL expressions in which it is less obvious where the false dilemma occurs, for example:


produces no rows because IN is translates to an iterated version of equality over the argument set and 1<>NULL is Unknown, just a as 1=NULL is Unknown. (The CAST in this example is needed only in some SQL implementations like PostgreSQL, which would reject it with a type checking error otherwise. In many systems plain SELECT NULL works in the subquery.) The missing case above is of course:


Effect of Null and Unknown in other constructs[edit]


Joins evaluate using the same comparison rules as for WHERE clauses. Therefore, care must be taken when using nullable columns in SQL join criteria. In particular a table containing any nulls is not equal with a natural self-join of itself, meaning that whereas R \bowtie R = R is true for any relation R in relational algebra, a SQL self-join will exclude all rows having a null value anywhere.[15] An example of this behavior is given in the section analyzing the missing-value semantics of Nulls.

The SQL COALESCE function or CASE expressions can be used to “simulate” Null equality in join criteria, and the IS NULL and IS NOT NULL predicates can be used in the join criteria as well. The following predicate tests for equality of the values A and B and treats Nulls as being equal.


CASE expressions[edit]

SQL provides two flavours of conditional expressions. One is called “simple CASE” and operates like a switch statement. The other is called a “searched CASE” in the standard, and operates like an if…elseif.

The simple CASE expressions use implicit equality comparisons which operate under the same rules as the DML WHERE clause rules for Null. Thus, a simple CASE expressioncannot check for the existence of Null directly. A check for Null in a simple CASE expression always results in Unknown, as in the following:

SELECT CASE i WHEN NULL THEN 'Is Null'  -- This will never be returned
              WHEN    0 THEN 'Is Zero'  -- This will be returned when i = 0
              WHEN    1 THEN 'Is One'   -- This will be returned when i = 1

Because the expression i = NULL evaluates to Unknown no matter what value column i contains (even if it contains Null), the string 'Is Null' will never be returned.

On the other hand, a “searched” CASE expression can use predicates like IS NULL and IS NOT NULL in its conditions. The following example shows how to use a searchedCASE expression to properly check for Null:

SELECT CASE WHEN i IS NULL THEN 'Null Result'  -- This will be returned when i is NULL
            WHEN     i = 0 THEN 'Zero'         -- This will be returned when i = 0
            WHEN     i = 1 THEN 'One'          -- This will be returned when i = 1

In the searched CASE expression, the string 'Null Result' is returned for all rows in which i is Null.

Oracle’s dialect of SQL provides a built-in function DECODE which can be used instead of the simple CASE expressions and considers two nulls equal.

SELECT DECODE(i, NULL, 'Null Result', 0, 'Zero', 1, 'One') FROM t;

Finally, all these constructs return a NULL if no match is found; they have a default ELSE NULL clause.

IF statements in procedural extensions[edit]

SQL/PSM (SQL Persistent Stored Modules) defines procedural extensions for SQL, such as the IF statement. However, the major SQL vendors have historically included their own proprietary procedural extensions. Procedural extensions for looping and comparisons operate under Null comparison rules similar to those for DML statements and queries. The following code fragment, in ISO SQL standard format, demonstrates the use of Null 3VL in an IF statement.

      SELECT 'Result is True'
      SELECT 'Result is False'
      SELECT 'Result is Unknown';

The IF statement performs actions only for those comparisons that evaluate to True. For statements that evaluate to False or Unknown, the IF statement passes control to the ELSEIF clause, and finally to the ELSE clause. The result of the code above will always be the message 'Result is Unknown' since the comparisons with Null always evaluate to Unknown.

Analysis of SQL Null missing-value semantics[edit]

The groundbreaking work of T. Imielinski and W. Lipski (1984) provided a framework in which to evaluate the intended semantics of various proposals to implement missing-value semantics. This section roughly follows chapter 19 the “Alice” textbook.[16] A similar presentation appears in the review of Ron van der Meyden, §10.4.[1]

In selections and projections: weak representation[edit]

Constructs representing missing information, such as Codd tables, are actually intended to represent a set of relations, one for each possible instantiation of their parameters; in the case of Codd tables, this means replacement of Nulls with some concrete value. For example,

the Codd table

Name Age
George 43
Harriet NULL
Charles 56
may represent the relation

Name Age
George 43
Harriet 22
Charles 56
or equally well the relation

Name Age
George 43
Harriet 37
Charles 56

A construct (such as a Codd table) is said to be a strong representation system (of missing information) if any answer to a query made on the construct can be particularized to obtain an answer for any corresponding query on the relations it represents, which are seen as models of the construct. More precisely, if q is a query formula in the relational algebra (of “pure” relations) and if \bar{q} is its lifting to a construct intended to represent missing information, a strong representation has the property that for any query q and (table) construct T, \bar{q} lifts all the answers to the construct, i.e.:

\mathop{\mathrm{Models}}(\bar{q}(T)) = \{ q(R)\,| R \in \mathop{\mathrm{Models}}(T) \}

(The above has to hold for queries taking any number of tables as arguments, but the restriction to one table suffices for this discussion.) Clearly Codd tables do not have this strong property if selections and projections are considered as part of the query language. For example, all the answers to


should include the possibility that a relation like EmpH22 may exist. However Codd tables cannot represent the disjunction “result with possibly 0 or 1 rows”. A device, mostly of theoretical interest, called conditional table (or c-table) can however represent such an answer:

Name Age condition
Harriet ω1 ω1 = 22

where the condition column is interpreted as the row doesn’t exist if the condition is false. It turns out that because the formulas in the condition column of a c-table can be arbitrary propositional logic formulas, an algorithm for the problem whether a c-table represents some concrete relation has a co-NP-complete complexity, thus is of little practical value.

A weaker notion of representation is therefore desirable. Imielinski and Lipski introduced the notion of weak representation, which essentially allows (lifted) queries over a construct to return a representation only for sure information, i.e. if it’s valid for all “possible world” instantiations (models) of the construct. Concretely, a construct is a weak representation system if

 \bigcap\mathop{\mathrm{Models}}(\bar{q}(T)) = \bigcap \{ q(R)\,| R \in \mathop{\mathrm{Models}}(T) \}

The right-hand side of the above equation is the sure information, i.e. information which can be certainly extracted from the database regardless of what values are used to replace Nulls in the database. In the example we considered above, it’s easy to see that the intersection of all possible models (i.e. the sure information) of the query selecting WHERE Age = 22 is actually empty because, for instance, the (unlifted) query returns no rows for the relation EmpH37. More generally, it was shown by Imielinski and Lipski that Codd tables are a weak representation system if the query language is restricted to projections, selections (and renaming of columns). However, as soon as we add either joins or unions to the query language, even this weak property is lost, as evidenced in the next section.

If joins or unions are considered: not even weak representation[edit]

Let us consider the following query over the same Codd table Emp from the previous section:

SELECT Name FROM Emp WHERE Age <> 22;

Whatever concrete value one would choose for the NULL age of Harriet, the above query will return the full column of names of any model of Emp, but when the (lifted) query is run on Emp itself, Harriet will always be missing, i.e. we have:

Query result on Emp:
Query result on any model of Emp:

Thus when unions are added to the query language, Codd tables are not even a weak representation system of missing information, meaning that queries over them don’t even report all sure information. It’s important to note here that semantics of UNION on Nulls, which are discussed in a later section, did not even come into play in this query. The “forgetful” nature of the two sub-queries was all that it took to guarantee that some sure information went unreported when the above query was run on the Codd table Emp.

For natural joins, the example needed to show that sure information may be unreported by some query is slightly more complicated. Consider the table

F1 F2 F3
11 NULL 13
21 NULL 23
31 32 33

and the query

  (SELECT F2, F3 FROM J) AS F23;
Query result on J:
F1 F3
31 33
Query result on any model of J:
F1 F3
11 13
21 23
31 33

The intuition for what happens above is that the Codd tables representing the projections in the subqueries lose track of the fact that the Null values in the columns F12.F2 and F23.F2 are actually copies of the originals in the table J. This observation suggests that a relatively simple improvement of Codd tables (which works correctly for this example) would be to use Skolem constants (meaning Skolem functions which are also constant functions), say ω12 and ω22 instead of a single NULL symbol. Such an approach, called v-tables or Naive tables, is computationally less expensive that the c-tables discussed above. However it is still not a complete solution for incomplete information in the sense that v-tables are only a weak representation for queries not using any negations in selection (and not using any set difference either). The first example considered in this section is using a negative selection clause, WHERE Age <> 22, so it is also an example where v-tables queries would not report sure information.

Check constraints and foreign keys[edit]

The primary place in which SQL three-valued logic intersects with SQL Data Definition Language (DDL) is in the form of check constraints. A check constraint placed on a column operates under a slightly different set of rules than those for the DML WHERE clause. While a DML WHERE clause must evaluate to True for a row, a check constraint must not evaluate to False. (From a logic perspective, the designated values are True and Unknown.) This means that a check constraint will succeed if the result of the check is either True or Unknown. The following example table with a check constraint will prohibit any integer values from being inserted into column i, but will allow Null to be inserted since the result of the check will always evaluate to Unknown for Nulls.[17]

     i INTEGER,
     CONSTRAINT ck_i CHECK ( i < 0 AND i = 0 AND i > 0 ) );

Because of the change in designated values relative to the WHERE clause, from a logic perspective the law of excluded middle is a tautology for CHECK constraints, meaning CHECK (p OR NOT p) always succeeds. Furthermore, assuming Nulls are to be interpreted as existing but unknown values, some pathological CHECKs like the one above allow insertion of Nulls that could never be replaced by any non-null value.

In order to constrain a column to reject Nulls, the NOT NULL constraint can be applied, as shown in the example below. The NOT NULL constraint is semantically equivalent to acheck constraint with an IS NOT NULL predicate.


By default check constraints against foreign keys succeed if any of the fields in such keys are Null. For example, the table

( title VARCHAR(100),
  author_last VARCHAR(20),
  author_first VARCHAR(20),
FOREIGN KEY (author_last, author_first)
  REFERENCES Authors(last_name, first_name));

would allow insertion of rows where author_last or author_first are NULL irrespective of how the table Authors is defined or what it contains. More precisely, a null in any of these fields would allow any value in the other one, even on that is not found in Authors table. For example, if Authors contained only (‘Doe’, ‘John’), then (‘Smith’, NULL) would satisfy the foreign key constraint. SQL-92 added two extra options for narrowing down the matches in such cases. If MATCH PARTIAL is added after the REFERENCES declaration then any non-null must match the foreign key, e. g. (‘Doe’, NULL) would still match, but (‘Smith’, NULL) would not. Finally, if MATCH FULL is added then (‘Smith’, NULL) would not match the constraint either, but (NULL, NULL) would still match it.

Outer joins[edit]

Example SQL outer join query with Null placeholders in the result set. The Null markers are represented by the word NULL in place of data in the results. Results are from Microsoft SQL Server, as shown in SQL Server Management Studio.

SQL outer joins, including left outer joins, right outer joins, and full outer joins, automatically produce Nulls as placeholders for missing values in related tables. For left outer joins, for instance, Nulls are produced in place of rows missing from the table appearing on the right-hand side of the LEFT OUTER JOIN operator. The following simple example uses two tables to demonstrate Null placeholder production in a left outer join.

The first table (Employee) contains employee ID numbers and names, while the second table (PhoneNumber) contains related employee ID numbers and phone numbers, as shown below.

ID LastName FirstName
1 Johnson Joe
2 Lewis Larry
3 Thompson Thomas
4 Patterson Patricia
ID Number
1 555-2323
3 555-9876

The following sample SQL query performs a left outer join on these two tables.

SELECT e.ID, e.LastName, e.FirstName, pn.Number
FROM Employee e
LEFT OUTER JOIN PhoneNumber pn
ON e.ID = pn.ID;

The result set generated by this query demonstrates how SQL uses Null as a placeholder for values missing from the right-hand (PhoneNumber) table, as shown below.

Query result
ID LastName FirstName Number
1 Johnson Joe 555-2323
2 Lewis Larry NULL
3 Thompson Thomas 555-9876
4 Patterson Patricia NULL

Aggregate functions[edit]

SQL defines aggregate functions to simplify server-side aggregate calculations on data. Except for the COUNT(*) function, all aggregate functions perform a Null-elimination step, so that Null values are not included in the final result of the calculation.[18]

Note that the elimination of Null values is not equivalent to replacing those values with zero. For example, in the following table, AVG(i) (the average of the values of i) will give a different result from that of AVG(j):

i j
150 150
200 200
250 250

Here AVG(i) is 200 (the average of 150, 200, and 250), while AVG(j) is 150 (the average of 150, 200, 250, and 0). A well-known side effect of this is that in SQL AVG(z) is not equivalent with SUM(z)/COUNT(*).[4]

When two nulls are equal: grouping, sorting, and some set operations[edit]

Because SQL:2003 defines all Null markers as being unequal to one another, a special definition was required in order to group Nulls together when performing certain operations. SQL defines “any two values that are equal to one another, or any two Nulls”, as “not distinct”.[19] This definition of not distinct allows SQL to group and sort Nulls when the GROUP BY clause (and other keywords that perform grouping) are used.

Other SQL operations, clauses, and keywords use “not distinct” in their treatment of Nulls. These include the following:

  • PARTITION BY clause of ranking and windowing functions like ROW_NUMBER
  • UNION, INTERSECT, and EXCEPT operator, which treat NULLs as the same for row comparison/elimination purposes
  • DISTINCT keyword used in SELECT queries

The principle that Nulls aren’t equal to each other (but rather that the result is Unknown) is effectively violated in the SQL specification for the UNION operator, which does identify nulls with each other.[1] Consequently, some set operations in SQL, like union or difference, may produce results not representing sure information, unlike operations involving explicit comparisons with NULL (e.g. those in a WHERE clause discussed above). In Codd’s 1979 proposal (which was basically adopted by SQL92) this semantic inconsistency is rationalized by arguing that removal of duplicates in set operations happens “at a lower level of detail than equality testing in the evaluation of retrieval operations.”[10]

The SQL standard does not explicitly define a default sort order for Nulls. Instead, on conforming systems, Nulls can be sorted before or after all data values by using the NULLS FIRST or NULLS LAST clauses of the ORDER BY list, respectively. Not all DBMS vendors implement this functionality, however. Vendors who do not implement this functionality may specify different treatments for Null sorting in the DBMS.[17]

Effect on index operation[edit]

Some SQL products do not index keys containing NULL values. For instance, PostgreSQL versions prior to 8.3 did not, with the documentation for a B-tree index stating that[20]

B-trees can handle equality and range queries on data that can be sorted into some ordering. In particular, the PostgreSQL query planner will consider using a B-tree index whenever an indexed column is involved in a comparison using one of these operators: < ≤ = ≥ >

Constructs equivalent to combinations of these operators, such as BETWEEN and IN, can also be implemented with a B-tree index search. (But note that IS NULL is not equivalent to = and is not indexable.)

In cases where the index enforces uniqueness, NULL values are excluded from the index and uniqueness is not enforced between NULL values. Again, quoting from thePostgreSQL documentation:[21]

When an index is declared unique, multiple table rows with equal indexed values will not be allowed. Null values are not considered equal. A multicolumn unique index will only reject cases where all of the indexed columns are equal in two rows.

This is consistent with the SQL:2003-defined behavior of scalar Null comparisons.

Another method of indexing Nulls involves handling them as not distinct in accordance with the SQL:2003-defined behavior. For example, Microsoft SQL Server documentation states the following:[22]

For indexing purposes, NULL values compare as equal. Therefore, a unique index, or UNIQUE constraint, cannot be created if the key values are NULL in more than one row. Select columns that are defined as NOT NULL when columns for a unique index or unique constraint are chosen.

Both of these indexing strategies are consistent with the SQL:2003-defined behavior of Nulls. Because indexing methodologies are not explicitly defined by the SQL:2003 standard, indexing strategies for Nulls are left entirely to the vendors to design and implement.

Null-handling functions[edit]

SQL defines two functions to explicitly handle Nulls: NULLIF and COALESCE. Both functions are abbreviations for searched CASE expressions.[23]


The NULLIF function accepts two parameters. If the first parameter is equal to the second parameter, NULLIF returns Null. Otherwise, the value of the first parameter is returned.

NULLIF(value1, value2)

Thus, NULLIF is an abbreviation for the following CASE expression:

CASE WHEN value1 = value2 THEN NULL ELSE value1 END


The COALESCE function accepts a list of parameters, returning the first non-Null value from the list:

COALESCE(value1, value2, value3, ...)

COALESCE is defined as shorthand for the following SQL CASE expression:

     WHEN value2 IS NOT NULL THEN value2
     WHEN value3 IS NOT NULL THEN value3

Some SQL DBMSs implement vendor-specific functions similar to COALESCE. Some systems (e.g. Transact-SQL) implement an ISNULL function, or other similar functions that are functionally similar to COALESCE. (See Is functions for more on the IS functions in Transact-SQL.)


The Oracle NVL function accepts two parameters. It returns the first non-NULL parameter or NULL if all parameters are NULL.

A COALESCE expression can be converted into an equivalent NVL expression thus:

COALESCE ( val1, ... , val{n} )

turns into:

NVL( val1 , NVL( val2 , NVL( val3 ,  , NVL ( val{n-1} , val{n} )  )))

A use case of this function is to replace in an expression a NULL value by a fixed value like in NVL(SALARY, 0) which says, ‘if SALARY contains a NULL value, replace it with 0′.

There is, however, one notable exception. In most implementations, COALESCE evaluates its parameters until it reaches the first non-NULL one, while NVL evaluates all of its parameters. This is important for several reasons. A parameter after the first non-NULL parameter could be a function, which could either be computationally expensive, invalid, or could create unexpected side effects.

Data typing of Null and Unknown[edit]

The NULL literal is untyped in SQL, meaning that it is not designated as an integer, character, or any other specific data type.[24] Because of this, it is sometimes mandatory (or desirable) to explicitly convert Nulls to a specific data type. For instance, if overloaded functions are supported by the RDBMS, SQL might not be able to automatically resolve to the correct function without knowing the data types of all parameters, including those for which Null is passed.

Conversion from the NULL literal to a Null of a specific type is possible using the CAST introduced in SQL-92. For example:


represents an integer which has the Null value.

The actual typing of Unknown (distinct or not from NULL itself) varies between SQL implementations. For example, the following


parses and executes successfully in some environments (e.g. SQLite or PostgreSQL) which unify a NULL boolean with Unknown but fails to parse in others (e.g. in SQL Server Compact). MySQL behaves similarly to PostgreSQL in this regard (with the minor exception that MySQL regards TRUE and FALSE as no different from the ordinary integers 1 and 0). PostgreSQL additionally implements a IS UNKNOWN predicate, which can be used to test whether a three-value logical outcome is Unknown, although this is merely syntactic sugar.

BOOLEAN data type[edit]

The ISO SQL:1999 standard introduced the BOOLEAN data type to SQL, however it’s still just an optional, non-core feature, coded T031.[25]

When restricted by a NOT NULL constraint, the SQL BOOLEAN works like the Boolean type from other languages. Unrestricted however, the BOOLEAN datatype, despite its name, can hold the truth values TRUE, FALSE, and UNKNOWN, all of which are defined as boolean literals according to the standard. The standard also asserts that NULL and UNKNOWN “may be used interchangeably to mean exactly the same thing”.[26][27]

The Boolean type has been subject of criticism, particularly because of the mandated behavior of the UNKNOWN literal, which is never equal to itself because of the identification with NULL.[28]

As discussed above, in the PostgreSQL implementation of SQL, the null value is used to represent all UNKNOWN results, including the UNKNOWN BOOLEAN. PostgreSQL does not implement the UNKNOWN literal (although it does implement the IS UNKNOWN operator, which is an orthogonal feature.) Most other major vendors do not support the Boolean type (as defined in T031) as of 2012.[29] The procedural part of Oracle’s PL/SQL supports BOOLEAN however variables; these can also be assigned NULL and the value is considered the same as UNKNOWN.[30]


Common mistakes[edit]

Misunderstanding of how Null works is the cause of a great number of errors in SQL code, both in ISO standard SQL statements and in the specific SQL dialects supported by real-world database management systems. These mistakes are usually the result of confusion between Null and either 0 (zero) or an empty string (a string value with a length of zero, represented in SQL as ''). Null is defined by the ISO SQL standard as different from both an empty string and the numerical value 0, however. While Null indicates the absence of any value, the empty string and numerical zero both represent actual values.

A classic rookie error is attempting to use the equality operator to find NULL values. Most SQL implementations will execute the following query as syntactically correct (therefore give no error message) but it never returns any rows, regardless of whether NULL values do exist in the table.

FROM sometable
WHERE num = NULL;  -- Should be "WHERE num IS NULL"

In a related, but more subtle example, a WHERE clause or conditional statement might compare a column’s value with a constant. It is often incorrectly assumed that a missing value would be “less than” or “not equal to” a constant if that field contains Null, but, in fact, such expressions return Unknown. An example is below:

FROM sometable
WHERE num <> 1;  -- Rows where num is NULL will not be returned, 
                 -- contrary to many users' expectations.

Similarly, Null values are often confused with empty strings. Consider the LENGTH function, which returns the number of characters in a string. When a Null is passed into this function, the function returns Null. This can lead to unexpected results, if users are not well versed in 3-value logic. An example is below:

FROM sometable
WHERE LENGTH(string) < 20; -- Rows where string is NULL will not be returned.

This is complicated by the fact that in some database interface programs (or even database implementations like Oracle’s), NULL is reported as an empty string, and empty strings may be incorrectly stored as NULL.


The ISO SQL implementation of Null is the subject of criticism, debate and calls for change. In The Relational Model for Database Management: Version 2, Codd suggested that the SQL implementation of Null was flawed and should be replaced by two distinct Null-type markers. The markers he proposed were to stand for “Missing but Applicable” and“Missing but Inapplicable”, known as A-values and I-values, respectively. Codd’s recommendation, if accepted, would have required the implementation of a four-valued logic in SQL.[5] Others have suggested adding additional Null-type markers to Codd’s recommendation to indicate even more reasons that a data value might be “Missing”, increasing the complexity of SQL’s logic system. At various times, proposals have also been put forth to implement multiple user-defined Null markers in SQL. Because of the complexity of the Null-handling and logic systems required to support multiple Null markers, none of these proposals have gained widespread acceptance.

Chris Date and Hugh Darwen, authors of The Third Manifesto, have suggested that the SQL Null implementation is inherently flawed and should be eliminated altogether,[31]pointing to inconsistencies and flaws in the implementation of SQL Null-handling (particularly in aggregate functions) as proof that the entire concept of Null is flawed and should be removed from the relational model.[32] Others, like author Fabian Pascal, have stated a belief that “how the function calculation should treat missing values is not governed by the relational model.”[citation needed]

Closed world assumption[edit]

Another point of conflict concerning Nulls is that they violate the closed world assumption model of relational databases by introducing an open world assumption into it.[33] The closed world assumption, as it pertains to databases, states that “Everything stated by the database, either explicitly or implicitly, is true; everything else is false.”[34] This view assumes that the knowledge of the world stored within a database is complete. Nulls, however, operate under the open world assumption, in which some items stored in the database are considered unknown, making the database’s stored knowledge of the world incomplete.

See also[edit]


  1. ^ Jump up to:a b c d Ron van der Meyden, “Logical approaches to incomplete information: a survey” in Chomicki, Jan; Saake, Gunter (Eds.) Logics for Databases and Information Systems, Kluwer Academic Publishers ISBN 978-0-7923-8129-7, p. 344; PS preprint (note: page numbering differs in preprint from the published version)
  2. Jump up^ Codd, E.F. (October 14, 1985). “Is Your Database Really Relational?”. ComputerWorld.
  3. Jump up^ Codd, E.F. (October 21, 1985). “Does Your DBMS Run By The Rules?”. ComputerWorld.
  4. ^ Jump up to:a b Don Chamberlin (1998). A Complete Guide to DB2 Universal Database. Morgan Kaufmann. pp. 28–32. ISBN 978-1-55860-482-7.
  5. ^ Jump up to:a b Codd, E.F. (1990). The Relational Model for Database Management (Version 2 ed.).Addison Wesley Publishing Company. ISBN 0-201-14192-2.
  6. ^ Jump up to:a b ISO/IEC (2003). ISO/IEC 9075-2:2003, “SQL/Foundation”. ISO/IEC. Section 6.2.6:numeric value expressions..
  7. Jump up^ ISO/IEC (2003). ISO/IEC 9075-2:2003, “SQL/Foundation”. ISO/IEC. Section 6.2.8: string value expression.
  8. Jump up^ ISO/IEC (2003). ISO/IEC 9075-1:2003, “SQL/Framework”. ISO/IEC. Section 4.4.2:The null value.
  9. ^ Jump up to:a b Coles, Michael (June 27, 2005). “Four Rules for Nulls”. SQL Server Central (Red Gate Software).
  10. ^ Jump up to:a b Hans-Joachim, K. (2003). “Null Values in Relational Databases and Sure Information Answers”. Semantics in Databases. Second International Workshop Dagstuhl Castle, Germany, January 7–12, 2001. Revised Papers. Lecture Notes in Computer Science2582. pp. 119–138. doi:10.1007/3-540-36596-6_7. ISBN 978-3-540-00957-3.
  11. Jump up^ ISO/IEC (2003). ISO/IEC 9075-2:2003, “SQL/Foundation”. ISO/IEC. Section 8.7: null predicate.
  12. Jump up^ C.J. Date (2004), An introduction to database systems, 8th ed., Pearson Education, p. 594
  13. Jump up^ Jim Melton; Jim Melton Alan R. Simon (1993). Understanding The New SQL: A Complete Guide. Morgan Kaufmann. pp. 145–147. ISBN 978-1-55860-245-8.
  14. Jump up^ C. J. Date, Relational database writings, 1991-1994, Addison-Wesley, 1995, p. 371
  15. Jump up^ C.J. Date (2004), An introduction to database systems, 8th ed., Pearson Education, p. 584
  16. Jump up^ Abiteboul, Serge; Hull, Richard B.; Vianu, Victor (1995). Foundations of Databases. Addison-Wesley. ISBN 0-201-53771-0.
  17. ^ Jump up to:a b Coles, Michael (February 26, 2007). “Null Versus Null?”. SQL Server Central (Red Gate Software).
  18. Jump up^ ISO/IEC (2003). ISO/IEC 9075-2:2003, “SQL/Foundation”. ISO/IEC. Section 4.15.4:Aggregate functions.
  19. Jump up^ ISO/IEC (2003). ISO/IEC 9075-2:2003, “SQL/Foundation”. ISO/IEC. Section distinct.
  20. Jump up^ “PostgreSQL 8.0.14 Documentation: Index Types”. PostgreSQL. Retrieved6 November 2008.
  21. Jump up^ “PostgreSQL 8.0.14 Documentation: Unique Indexes”. PostgreSQL. RetrievedNovember 6, 2008.
  22. Jump up^ “Creating Unique Indexes”. PostfreSQL. September 2007. Retrieved November 6,2008.
  23. Jump up^ ISO/IEC (2003). ISO/IEC 9075-2:2003, “SQL/Foundation”. ISO/IEC. Section 6.11: case expression.
  24. Jump up^ Jim Melton; Alan R. Simon (2002). SQL:1999: Understanding Relational Language Components. Morgan Kaufmann. p. 53. ISBN 978-1-55860-456-8.
  25. Jump up^ “ISO/IEC 9075-1:1999 SQL Standard”. ISO. 1999.
  26. Jump up^ C. Date (2011). SQL and Relational Theory: How to Write Accurate SQL Code. O’Reilly Media, Inc. p. 83. ISBN 978-1-4493-1640-2.
  27. Jump up^ ISO/IEC 9075-2:2011 §4.5
  28. Jump up^ Martyn Prigmore (2007). Introduction to Databases With Web Applications. Pearson Education Canada. p. 197. ISBN 978-0-321-26359-9.
  29. Jump up^ Troels Arvin, Survey of BOOLEAN data type implementation
  30. Jump up^ Steven Feuerstein; Bill Pribyl (2009). Oracle PL/SQL Programming. O’Reilly Media, Inc. pp. 74, 91. ISBN 978-0-596-51446-4.
  31. Jump up^ Darwen, Hugh; Chris Date. “The Third Manifesto”. Retrieved May 29, 2007.
  32. Jump up^ Darwen, Hugh. “The Askew Wall” (PDF). Retrieved May 29, 2007.
  33. Jump up^ Date, Chris (May 2005). Database in Depth: Relational Theory for Practitioners. O’Reilly Media, Inc. p. 73. ISBN 0-596-10012-4.
  34. Jump up^ Date, Chris. “Abstract: The Closed World Assumption”. Data Management Association, San Francisco Bay Area Chapter. Archived from the original on 2007-05-19. RetrievedMay 29, 2007.

Further reading[edit]

External links[edit]

wikipedia – Search Index

wikipedia – Search Nul Values


Search results

  • ::= header* header ::= name NUL value NUL name ::= notnull+ value ::= notnull* notnull ::= <01> | <02> | <03> | … | <ff> NUL = <00> Duplicate names are
    4 KB (443 words) – 17:50, 3 October 2015
  • produces a continuous stream of NUL (zero value) bytes /dev/full – produces a continuous stream of NUL (zero value) bytes when read, and returns a “disk
    22 KB (2,115 words) – 00:40, 21 September 2015
  • containing the characters and terminated with a null character (‘\0’, called NUL in ASCII). Alternative names are C string, which refers to the C programming
    8 KB (1,103 words) – 02:21, 18 June 2015
  • The null character (also null terminator), abbreviated NUL, is a control character with the value zero. It is present in many character sets, including
    6 KB (852 words) – 20:25, 19 June 2015
  • pull out of Eurovision after three years and “nul points””. Radio Prague. Retrieved 23 July 2009.  “Nul Points: Tim Moore: 9780099492979:
    29 KB (2,810 words) – 12:50, 13 October 2015
  • responsible for checking the Jan De Nul Group tax filings. Goedgezelschap was reportedly bought cars for a value of 370000 €, plus two houses. On 16 April
    2 KB (246 words) – 01:33, 14 October 2014
  • Null (redirect from Null value)
    marker and keyword in SQL Null character, the zero-valued ASCII character, also designated by NUL, often used as a terminator, separator or filler. This
    3 KB (409 words) – 02:46, 27 September 2015
  • \ / : ? * ” > < | and NUL in file and directory names across all filesystems. Unices and Linux disallow the characters / and NUL in file and directory
    70 KB (2,738 words) – 15:59, 23 October 2015
  • Manager\SubSystems Registry key are also started. Creates DOS device mappings (e.g. CON:, NUL:, AUX:, COM1:, COM2:, COM3:, COM4:, PRN:, LPT1:, LPT2:, LPT3:, and drive
    3 KB (262 words) – 13:56, 7 August 2014
  • sovereignty quickly sidelined the issues — including the Charter of Quebec Valuesand the corruption allegations against the Liberals, the latter of which
    80 KB (1,539 words) – 17:29, 15 October 2015
  • NUL movement in the 1960s. This movement was opposed to the Cobra movement and wanted objective art, art that was divorced from any emotional value.
    2 KB (155 words) – 16:00, 27 August 2015
  • the future”. Retrieved 14 June 2013.  “J.F.J. De Nul | Jan De Nul Group”. Jan De Nul Group. Retrieved 11 September 2013.  Robbins, R (2006).
    26 KB (3,425 words) – 22:59, 16 October 2015
  • ought to subsidise shipping – Admiral Atukum, NUL boss”. Daily Sub. Retrieved 2010-05-20.  Francis Ugwoke. “NUL MD Spends 68 Days Abroad to Recover Ship:
    6 KB (592 words) – 18:07, 5 June 2014
  • alphabet. The NUL character with a value of 0 is represented as ^@ (@ is the ASCII character before A). The DEL character with the value 127 is usually
    2 KB (308 words) – 06:34, 29 March 2015
  • valuesValue A”, “Value B”, “Value C”, “Value D”, “Value E”, “Value F”, “Value G”, “Value H”, “Value I”, “Value J”, “Value K”, “Value L”, and “Value
    62 KB (7,894 words) – 17:57, 13 October 2015
  • taken to be ascii spaces (decimal value 32). It is stored as a six digit octal number with leading zeroes followed by a NUL and then a space. Various implementations
    23 KB (2,839 words) – 06:50, 7 October 2015
  • character. A form of control characters were introduced in the 1870 Baudot code: NULand DEL. The 1901 Murray code added the carriage return (CR) and line feed
    23 KB (3,216 words) – 14:18, 8 September 2015
  • May 22, 2009. In orthodox UTF-8, a NUL byte(\x00) is represented by a NUL byte. […] But […] we […] want NUL bytes inside […] strings […]  Sapin
    65 KB (6,167 words) – 18:21, 23 October 2015
  • contain null (NUL) characters cannot be handled directly by C string library functions: Strings using a length code are limited to the maximum value of the length
    32 KB (4,216 words) – 14:38, 3 October 2015
  • “\x1B[0m\x1B[25;1H” is a string containing 11 characters (plus a trailing NUL to mark the end of the string) with two embedded Esc characters. To output
    43 KB (3,587 words) – 15:42, 22 October 2015

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Triangular number

From Wikipedia, the free encyclopedia

The first six triangular numbers

A triangular number or triangle number counts the objects that can form an equilateral triangle, as in the diagram on the right. The nth triangle number is the number of dots composing a triangle with n dots on a side, and is equal to the sum of the n natural numbers from 1 to n. The sequence of triangular numbers (sequence A000217 in OEIS), starting at the 0th triangular number, is:

0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210, 231, 253, 276, 300, 325, 351, 378, 406 …

The triangle numbers are given by the following explicit formulas:

 T_n= \sum_{k=1}^n k = 1+2+3+ \dotsb +n = \frac{n(n+1)}{2} = {n+1 \choose 2}

where \textstyle {n+1 \choose 2} is a binomial coefficient. It represents the number of distinct pairs that can be selected from n + 1 objects, and it is read aloud as “n plus one choose two”.

Carl Friedrich Gauss is said to have found this relationship in his early youth, by multiplying n/2 pairs of numbers in the sum by the values of each pair n+1.[1] Some find it likely that its origin goes back to the Pythagoreans 5th century BC.[2] It was not, however, until 1496 the formula was documented. This was in the book Elementa Arithmetica that contains the sequence 1 3 6 10 15 21 28 (the sums: 1, 1+2,…,1+2+3+4+5+6+7).[3][Does not mean formula was known] (See also Figurate number).
The triangular number Tn solves the “handshake problem” of counting the number of handshakes if each person in a room with n + 1 people shakes hands once with each person. In other words, the solution to the handshake problem of n people is Tn−1.[4] The function T is the additive analog of the factorial function, which is the products of integers from 1 to n.

The number of line segments between closest pairs of dots in the triangle can be represented in terms of the number of dots or with a recurrence relation:

 L_n = 3 T_{n-1}= 3{n \choose 2};~~~L_n = L_{n-1} + 3(n-1), ~L_1 = 0.

In the limit, the ratio between the two numbers, dots and line segments is

 \lim_{n\to\infty} \frac{T_n}{L_n} = \frac{1}{3}

Relations to other figurate numbers[edit]

Triangular numbers have a wide variety of relations to other figurate numbers.

Most simply, the sum of two consecutive triangular numbers is a square number, with the sum being the square of the difference between the two (and thus the difference of the two being the square root of the sum). Algebraically,

T_n + T_{n-1} = \left (\frac{n^2}{2} + \frac{n}{2}\right) + \left(\frac{\left(n-1\right)^2}{2} + \frac{n-1}{2} \right ) = \left (\frac{n^2}{2} + \frac{n}{2}\right) + \left(\frac{n^2}{2} - \frac{n}{2} \right ) = n^2 = (T_n - T_{n-1})^2.

Alternatively, the same fact can be demonstrated graphically:

6 + 10 = 16 Square number 16 as sum of two triangular numbers.svg 10 + 15 = 25 Square number 25 as sum of two triangular numbers.svg

There are infinitely many triangular numbers that are also square numbers; e.g., 1, 36. Some of them can be generated by a simple recursive formula:

S_{n+1} = 4S_n \left( 8S_n + 1\right) with S_1 = 1.

All square triangular numbers are found from the recursion

S_n = 34S_{n-1} - S_{n-2} + 2 with S_0 = 0 and S_1 = 1.

A square whose side length is a triangular number can be partitioned into squares and half-squares whose areas add to cubes.

Also, the square of the nth triangular number is the same as the sum of the cubes of the integers 1 to n.

The sum of the all triangular numbers up to the nth triangular number is the nth tetrahedral number,

 \frac {n(n+1)(n+2)} {6}.

More generally, the difference between the nth m-gonal number and the nth (m + 1)-gonal number is the (n − 1)th triangular number. For example, the sixth heptagonal number (81) minus the sixth hexagonal number (66) equals the fifth triangular number, 15. Every other triangular number is a hexagonal number. Knowing the triangular numbers, one can reckon any centered polygonal number: thenth centered k-gonal number is obtained by the formula

Ck_n = kT_{n-1}+1\

where T is a triangular number.

The positive difference of two triangular numbers is a trapezoidal number.

Other properties[edit]

Triangular numbers correspond to the first-degree case of Faulhaber’s formula.

Alternating triangular numbers (1, 6, 15, 28, …) are also hexagonal numbers.

Every even perfect number is triangular (as well as hexagonal), given by the formula

M_p 2^{p-1} = M_p (M_p + 1)/2 = T_{M_p}

where Mp is a Mersenne prime. No odd perfect numbers are known, hence all known perfect numbers are triangular.

For example, the third triangular number is (3 × 2 =) 6, the seventh is (7 × 4 =) 28, the 31st is (31 × 16 =) 496, and the 127th is (127 × 64 =) 8128.

In base 10, the digital root of a nonzero triangular number is always 1, 3, 6, or 9. Hence every triangular number is either divisible by three or has a remainder of 1 when divided by nine:

0 = 9 × 0
1 = 9 × 0 + 1
3 = 9 × 0 + 3
6 = 9 × 0 + 6
10 = 9 × 1 + 1
15 = 9 × 1 + 6
21 = 9 × 2 + 3
28 = 9 × 3 + 1
36 = 9 × 4
45 = 9 × 5
55 = 9 × 6 + 1
66 = 9 × 7 + 3
78 = 9 × 8 + 6
91 = 9 × 10 + 1

The digital root pattern for triangular numbers, repeating every nine terms, as shown above, is “1, 3, 6, 1, 6, 3, 1, 9, 9”.

The converse of the statement above is, however, not always true. For example, the digital root of 12, which is not a triangular number, is 3 and divisible by three.

If x is a triangular number, then ax + b is also a triangular number, given a is an odd square and b = (a − 1) / 8

Note that b will always be a triangular number, because 8 × Tn + 1 = (2n + 1)2, which yields all the odd squares are revealed by multiplying a triangular number by 8 and adding 1, and the process for b given a is an odd square is the inverse of this operation.

The first several pairs of this form (not counting 1x + 0) are: 9x + 1, 25x + 3, 49x + 6, 81x + 10, 121x + 15, 169x + 21, … etc. Given x is equal to Tn, these formulas yieldT3n + 1, T5n + 2, T7n + 3, T9n + 4, and so on.

The sum of the reciprocals of all the nonzero triangular numbers is:

 \!\ \sum_{n=1}^{\infty}{1 \over {{n^2 + n} \over 2}} = 2\sum_{n=1}^{\infty}{1 \over {n^2 + n}} = 2 .

This can be shown by using the basic sum of a telescoping series:

 \!\ \sum_{n=1}^{\infty}{1 \over {n(n+1)}} = 1 .

Two other interesting formulas regarding triangular numbers are:

T_{a+b} = T_a + T_b + ab\


T_{ab} = T_aT_b + T_{a-1}T_{b-1},\

both of which can easily be established either by looking at dot patterns (see above) or with some simple algebra.

In 1796, German mathematician and scientist Carl Friedrich Gauss discovered that every positive integer is representable as a sum of at most three triangular numbers, writing in his diary his famous words, “EΥΡHKA! num = Δ + Δ + Δ”. Note that this theorem does not imply that the triangular numbers are different (as in the case of 20 = 10 + 10), nor that a solution with exactly three nonzero triangular numbers must exist. This is a special case of Fermat’s Polygonal Number Theorem.

The largest triangular number of the form 2k − 1 is 4095 (see Ramanujan–Nagell equation).

Wacław Franciszek Sierpiński posed the question as to the existence of four distinct triangular numbers in geometric progression. It was conjectured by Polish mathematicianKazimierz Szymiczek to be impossible. This conjecture was proven by Fang and Chen in 2007.[5][6]


A fully connected network of n computing devices requires the presence of Tn − 1 cables or other connections; this is equivalent to the handshake problem mentioned above.

In a tournament format that uses a round-robin group stage, the number of matches that need to be played between n teams is equal to the triangular number Tn − 1. For example, a group stage with 4 teams requires 6 matches, and a group stage with 8 teams requires 28 matches. This is also equivalent to the handshake problem and fully connected network problems.

One way of calculating the depreciation of an asset is the sum-of-years’ digits method, which involves finding Tn, where n is the length in years of the asset’s useful life. Each year, the item loses (bs) × (n − y)Tn, where b is the item’s beginning value (in units of currency), s is its final salvage value, n is the total number of years the item is usable, and y the current year in the depreciation schedule. Under this method, an item with a usable life of n = 4 years would lose 4/10 of its “losable” value in the first year, 3/10 in the second, 2/10 in the third, and 1/10 in the fourth, accumulating a total depreciation of 10/10 (the whole) of the losable value.

Triangular roots and tests for triangular numbers[edit]

By analogy with the square root of x, one can define the (positive) triangular root of x as the number n such that Tn = x:[7]

n = \frac{\sqrt{8x+1}-1}{2}

which follows immediately from the quadratic formula. So an integer x is triangular if and only if 8x + 1 is a square. Equivalently, if the positive triangular root n of x is an integer, then x is the nth triangular number.[7]

See also[edit]


  1. Jump up^ Hayes, Brian. “Gauss’s Day of Reckoning”. American Scientist. Computing Science. Retrieved 2014-04-16.
  2. Jump up^ Eves, Howard. “Webpage cites AN INTRODUCTION TO THE HISTORY OF MATHEMATICS”. Mathcentral. Retrieved 28 March 2015.
  3. Jump up^ Title : Elementa arithmetica… ([Reprod.]) Autor : Jordanus Nemorarius Editor : Joanne Migmanus (Parihisii) Edition year : 1496 Language : Latin Identifiant : ark:/12148/bpt6k52595c Source : Bibliothèque nationale de France Relation : Description : Collection : French books before 1601 ; 81.9 Provenance : Available on internet since : 15/10/2007 Page:
  4. Jump up^
  5. Jump up^ Chen, Fang: Triangular numbers in geometric progression
  6. Jump up^ Fang: Nonexistence of a geometric progression that contains four triangular numbers
  7. ^ Jump up to:a b Euler, Leonhard; Lagrange, Joseph Louis (1810), Elements of Algebra 1 (2nd ed.), J. Johnson and Co., pp. 332–335

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From Wikipedia, the free encyclopedia
This article is about the term. For other uses, see World war (disambiguation).

A world war is a war involving many or most of the world’s most powerful and populous countries. World wars span multiple countries on multiple continents, with battles fought in multiple theatres.

The term is applied to the two major international conflicts that occurred during the 20th century:

Both conflicts were distinct in that they began after the Industrial Revolution and before the Atomic Age.

The term may be applied to other wars, albeit infrequently, such the War of the Spanish Succession, the Seven Years’ War, the Cold War, and the Napoleonic Wars.

Origins of the term[edit]

The term “World War” was coined speculatively in the early 20th century, some years before the First World War broke out, probably as a literal translation of the German wordWeltkrieg.[1] German writer August Wilhelm Otto Niemann had used the word in the title of his anti-British novel Der Weltkrieg: Deutsche Träume (“The World War: German Dreams”) as early as 1904, published in English as The coming conquest of England. Also, the term was used as early as 1850 by Karl Marx in The Class Struggles in France, as well as his associate Friedrich Engels.[2] Rasmus B. Anderson in 1889 describes an episode in Teutonic mythology as a world war (Swedish världskrig), justifying this description by a line in an Old Norse epic poem, Völuspá: folcvig fyrst i heimi (the first great war in the world).[3] The Oxford English Dictionary cites the first known usage in the English language to a Scottish newspaper, the People’s Journal in 1848: “A war amongst the great powers is now necessarily a world-war.”

It was recognized that the complex system of opposing alliances–the German Empire, Austria-Hungary, and the Ottoman Empire vs. the French Third Republic, the Russian Empire, and the British Empire was likely to lead to a worldwide conflict in the event of war breaking out. Due to this fact, a very minute conflict between two countries has the potential to set off a domino effect of alliances, causing mass war. The fact that the powers involved had large overseas empires virtually guaranteed that a war would be worldwide, as the colonies’ resources would be a crucial strategic factor. The same strategic considerations also ensured that the combatants would strike at each other’s colonies, thus spreading the fighting far more widely than in the pre-colonial era.

Other languages have also adopted the “World War” terminology. For instance, in French, “World War” is translated as “Guerre Mondiale”; in German, “Weltkrieg”, which, prior to the war, had been used in the more abstract meaning of a global conflict; in Italian, “World War” is translated as “Guerra Mondiale”; in Spanish, — Guerra Mundial, in Danish, “Verdenskrig” and in Russian, — {lang|ru|} (Mirovaya Voyna).

Speculative fiction authors were noting the concept of a Second World War at least as early as 1919 and 1920,[4] when Milo Hastings wrote his dystopian novel City of Endless Night. In English, the term “First World War” was used by Charles à Court Repington as a title for his memoirs, published in 1920,[5] having originally discussed the matter with a Major Johnstone of Harvard University in September 1918.[6] The term “World War I” was invented by Time magazine in its issue of June 12, 1939.[7] In that same article, the term “World War II” was first used speculatively to describe the upcoming war.[8] The first use for the actual war came in its issue of September 11, 1939.[9] One week earlier, the Danish newspaper Kristeligt Dagblad used the term on its front page, saying “The second World War broke out yesterday at 11 a.m.”[10]

Large-scale wars throughout history (in chronological order)[edit]

There have been numerous wars with battles spanning two or more continents throughout history, including:

Estimated death tolls. Log. mean calculated using simple power law.
Event Lowest
Log. mean estimate[11][not in citation given] Highest
Location From To Duration (years)
Greco-Persian Wars MainlandGreece,Thrace, Aegean Islands, Asia Minor, Cyprusand Egypt 499 BCE 449 BCE 50 years
Wars of Alexander the Great Thrace, Illyria,Greece, Asia Minor, Syria,Babylonia,Persia,Sogdiana, India 335 BCE 323 BCE 12 years
Wars of the Diadochi Macedon,Greece,Thrace,Anatolia, theLevant, Egypt,Babylonia andPersia 322 BCE 275 BCE 47 years
First Punic War 285,000+ 285,000+ 285,000+ Mediterranean Sea, Sicily,Sardinia, North Africa 264 BCE 241 BCE 23 years
Second Punic War 616,000+ 616,000+ 616,000+ Italia, Sicily,Hispania,Cisalpine Gaul,Transalpine Gaul, North Africa, Greece 218 BCE 201 BCE 17 years
Roman–Seleucid War Greece andAsia Minor 192 BCE 188 BCE 4 years
Roman–Persian Wars Mesopotamia,Syria, Southern Levant, Egypt,Transcaucasus,Atropatene,Asia Minor,Balkans 92 BCE 629 CE 721 years
First Mithridatic War Asia Minor,Achaea and theAegean Sea. 89 BCE 85 BCE 4 years
Great Roman Civil War Hispania, Italia,Graecia, Illyria,Aegyptus,Africa 49 BCE 45 BCE 4 years
Byzantine–Sassanid wars Caucasus, Asia Minor, Egypt,Levant,Mesopotamia 502 CE 628 CE 126 years
Muslim conquests Mesopotamia,Caucasus,Persia, Levant,North Africa,Anatolia, Iberia,Gaul andGreater Khorasan 622 CE 1258 CE 636 years
Arab–Byzantine wars Levant, Syria, Egypt, North Africa, Anatolia, Crete, Sicily, Southern Italy 629 CE 1050s CE ~421 years
Crusades 1,000,000[12] 1,700,000 3,000,000[13] Iberia, Near East (Anatolia,Levant,Palestine),Egypt, Holy Land 1095 CE 1291 CE 197 years
Mongol conquests 30,000,000[14] 35,000,000 40,000,000[15] Eurasia 1206 CE 1324 CE 118 years
Byzantine–Ottoman Wars Asia Minor, Balkans 1265 CE 1479 CE 214 years
European colonization of the Americas 2,000,000
14,000,000 100,000,000
Americas 1492 CE 1900 CE 408 years
Ottoman–Habsburg wars Hungary, Mediterranean, Balkans, North Africa and Malta 1526 CE 1791 CE 265 years
Eighty Years’ War The Low Countries
(worldwide colonial warfare)
1568 CE 1648 CE 80 years
Anglo-Spanish War (1585–1604) Atlantic Ocean,English Channel, Low Countries,Spain, Spanish Main, Portugal,Cornwall,Ireland,Americas,Azores andCanary islands 1585 CE 1604 CE 19 years
Dutch–Portuguese War Atlantic Ocean: Brazil, West Africa, Southern Africa; Indian Ocean: India,East Indies,Indochina; China 1602 CE 1663 CE 61 years
Thirty Years’ War 3,000,000 5,900,000 11,500,000 Europe (primarily present day Germany) 1618 CE 1648 CE 30 years
Anglo-Spanish War (1654–60) Caribbean, Spain, Canary Islands andSpanish Netherlands. 1654 CE 1660 CE 6 years
Nine Years’ War Mainland Europe, Ireland,Scotland, North America, South America, Asia 1688 CE 1697 CE 9 years

War of the Spanish Succession

Europe, North America, South America 1701 CE 1714 CE 13 years
War of the Quadruple Alliance Sicily, Sardinia,Spain,Scotland, North America 1718 CE 1720 CE 2 years
Anglo-Spanish War (1727–29) Spain and Panama 1727 CE 1729 CE 2 years

War of the Austrian Succession

Europe, North America and India 1740 CE 1748 CE 8 years

Seven Years’ War

Europe, North America, South America, Africa, Asia 1754 CE 1763 CE 9 years

French Revolutionary Wars

Europe, Egypt,Middle East,Atlantic Ocean,Caribbean,Indian Ocean 1792 CE 1802 CE 9 years

Napoleonic Wars

[citation needed]
4,900,000 7,000,000[18] Europe, Atlantic Ocean,Mediterranean Sea, North Sea,Río de la Plata,French Guiana,West Indies,Indian Ocean,North America,South Caucasus 1803 CE 1815 CE 13 years
Crimean War Sicily, Sardinia,Spain,Scotland, North America 1853 CE 1856 CE 3 years

World War I

15,000,000[19] 31,000,000 65,000,000[20] Europe, Africa, the Middle East, the Pacific Islands, Chinaand off the coast of South and North America 1914 CE 1918 CE 4 years, 3 months, 1 week
Map of participants in World War II.png

World War II

40,000,000[21] 58,000,000 85,000,000[22] Europe, Pacific,Atlantic, South-East Asia,China, Middle East,Mediterranean,North Africaand Horn of Africa, brieflyNorth andSouth America 1939 CE 1945 CE 6 years and 1 day
War on Terror 272,000[23] 585,000 1,260,000[23][24][25] Global (esp. in the Greater Middle East) 2001 CE 2015 CE 14 years

Wars matching World War I by casualty count[edit]

There were a number of wars before the 20th century with as many or more casualties than the First World War (16,563,868 – 40,000,000), including:

Estimated death tolls. Log. mean calculated using simple power law.
Event Lowest
Log. mean estimate[11] Highest
Location From To Duration (years)
Three Kingdoms 36,000,000[26] 37,000,000 40,000,000[27] China 184 CE 280 CE 96 years
Mongol conquests 30,000,000[14] 35,000,000 40,000,000[15] Eurasia 1206 CE 1324 CE 118 years
Qing dynasty conquest of the Ming dynasty 25,000,000[28] 25,000,000 25,000,000 Manchuria, China proper 1616 CE 1662 CE 47 years
Taiping Rebellion 20,000,000[29] 32,000,000 100,000,000[30][31][32] China 1851 CE 1864 CE 14 years
Conquests of Tamerlane 15,000,000[33] 17,000,000 20,000,000[33] West Asia, South Asia, Central Asia, Russia 1369 CE 1405 CE 37 years
An Lushan Rebellion 13,000,000[15] 21,000,000 36,000,000[34] China 755 CE 763 CE 9 years

Large-scale wars after 1945[edit]

Most wars listed are considered part of conflicts such as the Indochina Wars, the Conflict in Afghanistan, the Gulf Wars, the War on terror, and the Cold War.

World War I and World War II[edit]

Main articles: World War I and World War II

The World Wars of the 20th century involved almost every continent on Earth. Many of the states who fought in the First World War also fought in the Second, although not always on the same sides.

Among the causes of the World Wars most commonly are named technological progress and industrialization. The technology of communication and warfare allowed to project power world-wide, while industrialization allowed mass production of military technology. A recent research stresses a geopolitical factor of circumscription or global closure.

The circumscription theory[35] says that when political expansion reaches the last frontier of a system, warfare increases until it leaves only one player standing with all others eliminated. In this aspect, the modern world repeated the same pattern passed by most pre-modern civilizations. The world was politically divided towards the Twentieth century leaving no sovereign void.[36] Soon warfare drastically increased and the world inexorably proceeded toward unipolarity.

Classical geopoliticians and many other scholars[37] perceived the fact of “the end of space”;[38] many expected it to result in world-wide wars.[39] One of them, French sociologist George Vacher de Lapouge, envisaged in 1899: “We conclude thinking about human hecatombs which the future reserves. The struggle among the contenders for universal domination will be long and necessarily merciless.”[40]

To sum up, at the most expansive phase of world history, the space for expansion abruptly ended resulting for the first time in total closure. But this time the factor ofcircumscription, already stronger than in any previous civilization, was yet multiplied by the modern technology and industry. The long peace of La Belle Epoque was doomed, to be followed by World Wars, as one of the central figures vividly expressed:

And the first gust of wind swept across a Europe grown nervous… Let Heaven at last give free rain to the fate which could no longer be thwarted. And then the first mighty lightening flash struck the earth… and with the thunder of Heaven there mingled the roar of World War batteries… The fight for freedom has begun mightier than the earth has ever seen (Mein Kampf[41]).

The two World Wars of the 20th century indeed caused unprecedented casualties and destruction across the theaters of conflict, although there are at least three wars before the 20th century with as many or more casualties than the First World War.[42] The numbers killed in both wars combined are estimated at between 60 and 100 million people. Non-combatants (mostly civilians) suffered as badly as or worse than combatants, and the distinction between combatants and non-combatants was often blurred as belligerents of both world wars mobilized for total war. Both world wars saw war crimes. Nazi Germany was responsible for multiple genocides during the Second World War, most notably theHolocaust. The Soviet Union, Canada, and United States deported and interned minority groups within their own borders, and largely due to this conflict later, many ethnic Germans were expelled in much of Eastern Europe. Imperial Japan during the Second World War was notorious for attacking neutral nations without a declaration of war, such as the bombing of Pearl Harbor, and its brutal treatment and killing of Allied prisoners of war and the inhabitants of Asia, most notably by using them for forced labor and the Rape of Nanking where 250,000 non-combatants in the city were brutally murdered by Japanese troops. The Ottoman Empire was responsible for the death of over one million Armenians during the First World War. Advances in technology were responsible for a large amount of casualties. The First World War saw major use of chemical weaponsdespite the Hague Conventions of 1899 and 1907 outlawing the use of such weapons in warfare. The Second World War was also the first (and thus far, only) conflict in whichnuclear weapons were used, devastating the Japanese cities of Hiroshima and Nagasaki.

World War I World War II
Dead 15–20M 50–85M
Injured 9–15M 20M
Conscripts 65M 90M
Battlefield size 3M km² 17M km²

The outcome of the World Wars had a profound effect on the course of world history. The old European empires collapsed or were dismantled as a direct result of the wars’ crushing costs and in some cases the defeat of imperial powers. The United States was firmly established as the dominant global power, along with its ideological foe, the Soviet Union, in close competition. These twosuperpowers exerted political influence over most of the world’s other states for decades after the end of the Second World War (ending in the late 1980s in the Soviet Union). The modern international security, economic and diplomatic system was created in the aftermath of the wars. Institutions such as the United Nations were established to collectivize international affairs, with the explicit goal of preventing another outbreak of general war.[citation needed] The wars also greatly changed the course of daily life. Technologies developed during wartime had a profound effect on peacetime life as well–for instance, jet aircraft, penicillin, nuclear energy, and electronic computers.

Since the atomic bombings of Hiroshima and Nagasaki during the Second World War, there has been a widespread and prolonged fear of a Third World War between nuclear-armed powers.

Later world wars[edit]

See also: World War III

I know not with what weapons World War III will be fought, but World War IV will be fought with sticks and stones.

Albert Einstein (1947)[43][44]

Various former government officials, politicians and authors have attempted to apply the labels of WWIII, WWIV, and WWV to various military engagements and diplomatic stand-offs since the close of WWII, such as the Cold War or the War on Terror. Among these are former American and French government officials James Woolsey[45] and Alexandre de Marenches,[46]author Eliot Cohen[47] and Zapatista leader Subcomandante Marcos.[48] Despite their efforts, none of these wars are commonly deemed world wars.

The Second Congo War (1998–2003), which involved nine nations and led to ongoing low-level warfare despite an official peace and the first democratic elections in 2006, has often been referred to as “Africa’s World War”.[49]

World War III is generally considered a hypothetical successor to World War II and is often suggested to be nuclear, devastating in nature and likely much more violent than both WWI and WWII combined. This war is anticipated and planned for by military and civil authorities, and explored in fiction in many countries. Concepts range from purely conventional scenarios or a limited use of nuclear weapons to the destruction of the planet. World War IV is sometimes mentioned as a hypothetical successor to World War III or as a plot element in books, movies or video games.

See also[edit]


  1. Jump up^ “Online Etymology Dictionary entry for World War”. 1914-08-02. Retrieved 2012-08-24.
  2. Jump up^ Engels, Frederick. “Introduction to Borkheim”.
  3. Jump up^ Rasmus Björn Anderson (translator: Viktor Rydberg), Teutonic Mythology, vol. 1, p. 139, London: S. Sonnenschein & Co., 1889 OCLC 626839.
  4. Jump up^ Hastings, Milo (1920). City of Endless Night. Dodd, Mead. Retrieved 20 October 2014.
  5. Jump up^ Repington, Charles à Court (1920). The First World War, 1914-1918. Retrieved20 October 2014.
  6. Jump up^ “The ‘First’ World War”. QI : Quite Interesting. 2014. Retrieved 20 October 2014.
  7. Jump up^ “War Machines”. TIME. June 12, 1939. Retrieved 20 October 2014. (subscription required (help)). In World War I, for example, command of the air changed hands several times, and the command changed not only when numbers varied but when one side introduced a superior new plane which could outfight the opposing machines
  8. Jump up^ “In World War II it is possible that even nations who do not take sides may play a vital military part, for they may be invaded.”
  9. Jump up^ “Grey Friday: TIME Reports on World War II Beginning”. TIME. September 11, 1939. Retrieved 20 October 2014. World War II began last week at 5:20 a. m. (Polish time) Friday, September 1, when a German bombing plane dropped a projectile on Puck, fishing village and air base in the armpit of the Hel Peninsula.
  10. Jump up^ “Den anden Verdenskrig udbrød i Gaar Middags Kl. 11”, Kristeligt Dagblad, September 4, 1939.
  11. ^ Jump up to:a b Pinto, Carla M. A.; Lopes, A. Mendes; Machado, J. A. Tenreiro (2014). Mathematical Methods in Engineering. Netherlands: Springer. pp. 173–180. ISBN 978-94-007-7182-6.
  12. Jump up^ John Shertzer Hittell, “A Brief History of Culture” (1874) p.137: “In the two centuries of this warfare one million persons had been slain…cited by White
  13. Jump up^ Robertson, John M., “A Short History of Christianity” (1902) p.278. Cited by White
  14. ^ Jump up to:a b The Cambridge History of China: Alien regimes and border states, 907–1368, 1994, p.622, cited by White
  15. ^ Jump up to:a b c Matthew White (2011-11-07). The Great Big Book of Horrible Things: The Definitive Chronicle of History’s 100 Worst Atrocities. W. W. Norton. ISBN 978-0-393-08192-3.
  16. Jump up^ Rummel, R.J. Death by Government, Chapter 3: Pre-Twentieth Century Democide
  17. Jump up^ Stannard, David E. (1993). American Holocaust: The Conquest of the New World. New York & Oxford: Oxford University Press. p. 11. ISBN 978-0-19-508557-0. In the 1940s and 1950s conventional wisdom held that the population of the entire hemisphere in 1492 was little more than 8,000,000—with fewer than 1,000,000 people living in the region north of present-day Mexico. Today, few serious students of the subject would put the hemispheric figure at less than 75,000,000 to 100,000,000 (with approximately 8,000,000 to 12,000,000 north of Mexico).
  18. Jump up^ Charles Esdaile “Napoleon’s Wars: An International History.”
  19. Jump up^ Willmott 2003, p. 307
  20. Jump up^ 1918 Influenza: the Mother of All Pandemics, CDC
  21. Jump up^ David Wallechinsky (1996-09-01). David Wallechinskys 20th Century: History With the Boring Parts Left Out. Little Brown. ISBN 978-0-316-92056-8.
  22. Jump up^ Fink, George: Stress of War, Conflict and Disaster
  23. ^ Jump up to:a b “Human costs of war: Direct war death in Afghanistan, Iraq and Pakistan October 2001 – February 2013” (PDF). Costs of War. February 2013. Retrieved 14 June 2013.
  24. Jump up^ “Update on Iraqi Casualty Data” by Opinion Research Business. January 2008.
  25. Jump up^ “Revised Casualty Analysis. New Analysis ‘Confirms’ 1 Million+ Iraq Casualties”. January 28, 2008. Opinion Research Business. Word Viewer for.doc files.
  26. Jump up^ Robert B. Marks (2011). China: Its Environment and History (World Social Change). Rowman & Littlefield Publishers. ISBN 1442212756.
  27. Jump up^ Graziella Caselli (2005). Demography – Analysis and Synthesis: A Treatise in Population. Academic Press. ISBN 012765660X.
  28. Jump up^ Alan Macfarlane (1997-05-28). The Savage Wars of Peace: England, Japan and the Malthusian Trap. Wiley-Blackwell. ISBN 978-0-631-18117-0.
  29. Jump up^ “Taiping Rebellion – Britannica Concise”. Retrieved2013-08-23.
  30. Jump up^ “The Taiping Rebellion 1850–1871 Tai Ping Tian Guo”. Retrieved 2013-08-23.
  31. Jump up^ Livre noir du Communisme: crimes, terreur, répression, page 468
  32. Jump up^ By Train to Shanghai: A Journey on the Trans-Siberian Railway By William J. Gingles page 259
  33. ^ Jump up to:a b “Timur Lenk (1369–1405)”. Retrieved 2013-08-23.
  34. Jump up^ “Death toll figures of recorded wars in human history”.
  35. Jump up^ Originally, the theory was drawn in 1970 by Anthropologist Robert Carneiro for the Bronze Age civilizations (Carneiro’s circumscription theory). One of leading world-system theorists, Christopher Chase-Dunn, noted in 1990 that the theory is applicable for the global system. “World State Formation: Historical Processes and Emergent Necessity,” California: Institute for research on World System, working paper 1, 1990, In 2007, historian Max Ostrovsky demonstrated that the circumscription factor explains much of world history and is especially relevant for the global system—the first completely circumscribed system in history. Y = Arctg X: The Hyperbola of the World Order, (Lanham; University Press of America, 2007).
  36. Jump up^ Halford J. Mackinder, The Geographical Pivot of History, (London: J. Murray, 1904);Fredrick Jackson Turner, The Frontier in American History, (Holt, Rinchart and Winston, New York, 1920). It was a particular moment in world history when the science ofgeopolitics was born, including the theory of lebensraum (living space). Y = Arctg X: The Hyperbola of the World Order, (Lanham; University Press of America, 2007, p 126-140).
  37. Jump up^ Chinese philosopher K’ang Yu-wei and French sociologist Georges Vacher de Lapouge in the late 19th century emphasized that the expansion cannot proceed indefinitely on the definite surface of the globe. The trend is bound to culminate in a world empire. K’ang Yu-wei in 1885 predicted that the matter will be decided in the contest between Washington and Berlin. The One World Philosophy, (tr. Thompson, Lawrence G., London, 1958), pp 79-80, 85. Vacher de Lapouge in 1899 foresaw the final contest between the United States and Russia and estimated the chance of the United States higher. L’Aryen: Son Rôle Social, (Nantes: 1899), chapter “L`Avenir des Aryens.”
  38. Jump up^ Stephen Kern, Culture of Time and Space’, 1880-1918′, (Massachusetts & London: Harvard University Press, 1983).
  39. Jump up^ Ignatius Clark, Voices Prophesying War: Future Wars1763-3749, (Oxford; Oxford University Press, 1992).
  40. Jump up^ George Vacher de Lapouge, L’Aryen: Son Rôle Social, (Nantes: 1899), chapter “L`Avenir des Aryens,” translated by Max Ostrovsky in his Y = Arctg X: The Hyperbola of the World Order, (Lanham; University Press of America, 2007), p 138.
  41. Jump up^ Adolf Hitler, Mein Kampf, (tr. Ralph Manheim, London: Pimlico, 1992, p 145, 148).
  42. Jump up^ “Top 10 Causes of WWI”. The Rich Ten. Retrieved 11 June 2014.
  43. Jump up^ Calaprice, Alice (2005). “The new quotable Einstein”. Princeton University Press. p. 173.ISBN 0-691-12075-7.
  44. Jump up^ “The culture of Einstein”. MSNBC. 2005-04-19. Retrieved 2012-08-24.
  45. Jump up^ “World War IV”. 2002. Retrieved 2010-02-04. Woolsey claims victory in WWIII, start of WWIV
  46. Jump up^ “The Fourth World War: Diplomacy and Espionage….”. 1992. Retrieved 2010-02-04.Book regarding alleged WWIV
  47. Jump up^ “World War IV: Let’s call this conflict what it is.”. 2001. Retrieved 2010-02-04. Why war on terrorism should be called WWIV
  48. Jump up^ Subcomandante Marcos (2001). “The Fourth World War Has Begun”. Nepantla: Views from South (Duke University Press) 2 (3): 559–572. Retrieved 20 October 2014.
  49. Jump up^ Prunier, Gerard (2014). Africa’s World War: Congo, the Rwandan Genocide, and the Making of a Continental Catastrophe. Barnes & Noble. ISBN 9780195374209. Retrieved20 October 2014.

External links[edit]

From Wikipedia, the free encyclopedia

Global Peace Index Scores, 2014. Countries appearing with a deeper shade of green are ranked as more peaceful, countries appearing more red are ranked as less peaceful.

The Institute for Economics and Peace produces the Global Peace Index.

The Global Peace Index (GPI) is an attempt to measure the relative position of nations’ and regions’ peacefulness.[1] It is the product of the Institute for Economics and Peace (IEP) and developed in consultation with an international panel of peace experts from peace institutes and think tanks with data collected and collated by the Economist Intelligence Unit. The list was launched in May 2007 and updates have been made on an annual basis since then. It is claimed to be the first study to rank countries around the world according to their peacefulness. It ranks 162 countries, up from 121 in 2007. The study is the brainchild of Australian technology entrepreneur Steve Killelea, founder of Integrated Research, and is endorsed by individuals such as Kofi Annan, the Dalai Lama, archbishop Desmond Tutu, former Finnish President Martti Ahtisaari, Nobel laureate Muhammad Yunus, economist Jeffrey Sachs, former president of Ireland Mary Robinson, current Deputy Secretary-General of the United Nations Jan Eliasson and former US president Jimmy Carter.

The index gauges global peace using three broad themes: the level of safety and security in society, the extent of domestic and international conflict, and the degree of militarization.[2] Factors are both internal such as levels of violence and crime within the country and external such as military expenditure and wars. The GPI has been criticised by Riane Eisler for not including indicators specifically relating to violence against women and children.

The updated index is released each year at events in London, Washington DC and at the United Nations Secretariat in New York. The GPI currently indicates Iceland, Denmark, Austria, and New Zealand to be the most peaceful countries and Syria, Afghanistan, South Sudan, and Iraq to be the least peaceful.[3]

Expert panel

The expert panel for the 2014 GPI consisted of:[4]


In attempting to gauge peacefulness, the GPI investigates the extent to which countries are involved in ongoing domestic and international conflicts. It also seeks to evaluate the level of harmony or discord within a nation; ten indicators broadly assess what might be described as a safety and security in society. The assertion is that low crime rates, minimal incidences of terrorist acts and violent demonstrations, harmonious relations with neighboring countries, a stable political scene and a small proportion of the population being internally displaced or refugees can be equated with peacefulness.

Countries’ peacefulness is measured on a wide range of indicators, 22 in all (originally 24 indicators, but one was dropped[which?] in 2008, and another in 2013). A table of the indicators is below.[5] In the table, UCDP stands for the Uppsala Conflict Data Program maintained by the University of Uppsala in Sweden, EIU for The Economist Intelligence Unit, UNSCT for the United Nations Survey of Criminal Trends and Operations of Criminal Justice Systems, ICPS is the International Center for Prison Studies at King’s College London, IISS for the International Institute for Strategic Studies publication The Military Balance, SIPRI for the Stockholm International Peace Research Institute Arms Transfers Database, and BICC for the Bonn International Center for Conversion.

# Indicator Source Year(s) Coding
1 Number of external and internal conflicts fought UCDP and EIU 2004 to 2009 Total number[6]
2 Number of deaths from organised conflict (external) UCDP 2010 Total number[6]
3 Number of deaths from organised conflict (internal) IISS 2010 Total number[6]
4 Level of organised conflict (internal) EIU 2010 to 2011 Qualitative scale, ranked 1 to 5
5 Relations with neighbouring countries EIU 2010 to 2011 Qualitative scale, ranked 1 to 5
6 Level of perceived criminality in society EIU 2010 to 2011 Qualitative scale, ranked 1 to 5
7 Number of refugees and displaced persons as percentage of population UNHCR and IDMC 2009 to 2010 Refugee population by country or territory of origin, plus the number of a country’s internally displaced people (IDP’s) as a percentage of the country’s total population
8 Political instability EIU 2010 to 2011 Qualitative scale, ranked 1 to 5
9 Terrorist activity Global Terrorism Index and IEP 2009 Quantitative scale, ranked 1 to 5
10 Political terror scale Amnesty International and US State Department 2010 to 2011 Qualitative scale, ranked 1 to 5
11 Number of homicides per 100,000 people UNCTS and EIU 2005 to 2009 Intentional homicides, including infanticide and excluding minor road traffic and other petty offences
12 Level of violent crime EIU 2010 to 2011 Qualitative scale, ranked 1 to 5
13 Likelihood of violent demonstrations EIU 2010 to 2011 Qualitative scale, ranked 1 to 5
14 Number of jailed persons per 100,000 people ICPS 2010 Rate of incarcerated persons as compared to the total population of the country
15 Number of internal security officers and police per 100,000 people UNCTS and EIU 2008 to 2010 Civil police force distinct from national guards or local militia [7]
16 Military expenditure as a percentage of GDP The Military Balance and IISS 2009 to 2010 Cash outlays of central or federal government to meet costs of national armed forces, as a percentage of GDP[8]
17 Number of armed-services personnel The Military Balance and IISS 2010 All full-time active armed-services personnel
18 Volume of transfers of major conventional weapons as recipient (imports) per 100,000 people SIPRI 2009 to 2010 Imports of major conventional weapons per 100,000 people[9]
19 Volume of transfers of major conventional weapons as supplier (exports) per 100,000 people SIPRI 2009 to 2010 Exports of major conventional weapons per 100,000 people[9]
20 Financial contribution to UN peacekeeping missions United Nations Committee on Contributions and IEP 2007 to 2010 Total number
21 Nuclear and heavy weapons capability The Military Balance, IISS, SIPRI, and IEP 2009 The Military Balance, IISS; SIPRI; and IEP[10]
22 Ease of access to small arms and light weapons EIU 2010 to 2011 Qualitative scale, ranked 1 to 5

Indicators not already ranked on a 1 to 5 scale were converted by using the following formula: x=(x-Min(x))/(Max(x)-Min(x)) where Max(x) and Min(x) are the highest and lowest values for that indicator of the countries ranked in the index. The 0 to 1 scores that resulted were then converted to the 1 to 5 scale. Individual indicators were then weighted according to the expert panel’s judgment of their importance. The scores were then tabulated into two weighted sub-indices: internal peace, weighted at 60% of a country’s final score, and external peace, weighted at 40% of a country’s final score.[11] ‘Negative Peace’ which is defined as the absence of violence, or fear of violence is used as the definition of peace to create the Global Peace Index.[12] An additional aim of the GPI database is to facilitate deeper study of the concept of positive peace, or those attitudes, institutions, and structures that drive peacefulness in society.[12] The GPI also examines relationships between peace and reliable international measures, including democracy and transparency, education and material well-being. As such, it seeks to understand the relative importance of a range of potential determinants, or “drivers”, which may influence the nurturing of peaceful societies, both internally and externally.

The main findings of the Global Peace Index are:[13]

  • Peace is correlated to indicators such as income, schooling and the level of regional integration
  • Peaceful countries often shared high levels of transparency of government and low corruption
  • Small, stable countries which are part of regional blocks are most likely to get a higher ranking.

Statistical analysis was applied to discover more specific drivers of peace. Specifically, the research team looked for indicators that were included and excluded from the index that had high levels of correlation with the overall score and rank of countries. Among the statistically significant indicators that were not used in the analysis were the functionality of a country’s government, regional integration, hostility to foreigners, importance of religion in national life, corruption, freedom of the media and GDP per capita.[14]

Notably absent from the 2007 study were Belarus, Iceland, many African nations, Mongolia, North Korea, and Afghanistan. They were not included because reliable data for the 24 indicators was not available.[15] Most of these countries were included in subsequent editions of the Global Peace Index, which now ranks 162 countries worldwide.

Endorsements, criticism and response

The Index has received endorsements as a political project from a number of major international figures, including the former Secretary-General of the United Nations Kofi Annan, former President of Finland and 2008 Nobel Peace Prize laureate Martti Ahtisaari, the Dalai Lama, archbishop Desmond Tutu, Muhammad Yunus, and former United States President Jimmy Carter.[16]Steve Killelea, the Australian philanthropist who conceived the idea of the Index, argues that the Index “is a wake-up call for leaders around the globe.”[17]

The Index has been widely recognized. Professor Jeffrey Sachs, Director of the Earth Institute at Columbia University said: “The GPI continues its pioneering work in drawing the world’s attention to the massive resources we are squandering in violence and conflict. The lives and money wasted in wars, incarcerations, weapons systems, weapons trade, and more, could be directed to ending poverty, promoting education, and protecting the environment. The GPI will not only draw attention to these crucial issues, but help us understand them and to invest productively in a more peaceful world.”[18]

The Economist, in publishing the first edition of the index in 2007, admitted that, “the index will run into some flak.” Specifically, according to The Economist, the weighting of military expenditure “may seem to give heart to freeloaders: countries that enjoy peace precisely because others (often the USA) care for their defense.” The true utility of the index may lie not in its specific rankings of countries now, but in how those rankings change over time, thus tracking when and how countries become more or less peaceful.[19]

The GPI has been criticised for not including indicators specifically relating to violence against women and children. Riane Eisler, writing in the Christian Science Monitor, argued that, “to put it mildly, this blind spot makes the index very inaccurate.”[20] She mentions a number of specific cases, including Egypt, where she claims 90% of women are subject to genital mutilation and China, where, she says, “female infanticide is still a problem,” according to a 2000 UNICEF study.[20]

Global Peace Index rankings

Nations considered more peaceful have lower index scores. Countries with rankings in green are in the most peaceful 20% for that year; those in red are in the bottom 20%.[21] In 2013 researchers at the Institute for Economics and Peace harmonized the Global Peace Index database to ensure that the scores were comparable over time. Several countries have been removed since 2008 due to the fact that the compilers agreed that the GPI would include nations, but not micro-states. Now, countries covered by the GPI must either have a population of more than 1 million or a land area greater than 20,000 square kilometers.[22]

Coun  try 2015 rank[23] 2015 score[23] 2014 rank[24] 2014 score[24] 2013 rank[25] 2013 score[25] 2012 rank[26][27] 2012 score[26] 2011 rank[28][27] 2011 score[28] 2010 rank[29] 2010 score[29] 2009 rank 2009 score 2008 rank 2008 score
 Iceland 1 1.148 1 1.189 1 1.162 1 1.113 1 1.148 2 1.212 1 1.203 1 1.107
 Denmark 2 1.150 2 1.193 2 1.207 2 1.239 4 1.289 7 1.341 6 1.263 5 1.238
 Austria 3 1.198 3 1.200 4 1.250 6 1.328 6 1.337 4 1.290 3 1.240 10 1.291
 New Zealand 4 1.221 4 1.236 3 1.237 2 1.239 2 1.279 1 1.188 2 1.227 2 1.190
  Switzerland 5 1.275 5 1.258 5 1.272 10 1.349 16 1.421 18 1.424 11 1.349 4 1.234
 Finland 6 1.277 6 1.297 7 1.297 9 1.348 7 1.352 9 1.352 7 1.297 7 1.273
 Canada 7 1.287 7 1.306 8 1.306 4 1.317 8 1.355 14 1.392 9 1.324 6 1.264
 Japan 8 1.323 8 1.316 6 1.293 5 1.326 3 1.287 3 1.247 4 1.243 3 1.230
 Australia 9 1.329 15 1.414 16 1.438 22 1.494 18 1.455 19 1.467 19 1.440 16 1.421
 Czech Republic 10 1.341 11 1.381 14 1.404 13 1.396 5 1.320 12 1.360 16 1.430 18 1.435
 Portugal 11 1.344 18 1.425 18 1.467 16 1.470 17 1.453 13 1.366 15 1.426 14 1.385
 Ireland 12 1.354 13 1.384 12 1.370 6 1.328 11 1.370 6 1.337 14 1.393 11 1.310
 Sweden 13 1.360 11 1.381 9 1.319 14 1.419 13 1.401 10 1.354 5 1.253 8 1.282
 Belgium 14 1.365 9 1.354 10 1.339 11 1.376 14 1.413 17 1.400 13 1.365 13 1.368
 Slovenia 15 1.378 14 1.398 13 1.374 8 1.330 10 1.358 11 1.358 10 1.343 13 1.370
 Germany 16 1.379 17 1.423 15 1.431 15 1.424 15 1.416 16 1.398 17 1.443 15 1.406
 Norway 17 1.393 10 1.371 11 1.359 18 1.480 9 1.356 5 1.322 8 1.324 9 1.288
 Bhutan 18 1.416 16 1.422 20 1.487 19 1.481 34 1.693 36 1.665 43 1.722 19 1.440
 Poland 19 1.430 23 1.532 25 1.530 24 1.524 22 1.545 29 1.618 28 1.597 33 1.610
 Netherlands 20 1.432 20 1.475 22 1.508 28 1.606 25 1.628 27 1.610 29 1.609 28 1.555
 Spain 21 1.451 26 1.548 27 1.563 25 1.548 28 1.641 25 1.588 32 1.640 27 1.550
 Hungary 22 1.463 21 1.482 23 1.520 17 1.476 20 1.495 20 1.495 25 1.574 20 1.452
 Slovakia 23 1.478 19 1.467 33 1.622 26 1.590 23 1.576 21 1.536 23 1.532 25 1.533
 Singapore 24 1.490 25 1.545 16 1.438 23 1.521 24 1.585 30 1.624 18 1.439 21 1.465
 Mauritius 25 1.503 24 1.544 21 1.497 21 1.487
 Romania 26 1.542 35 1.677 30 1.584 32 1.627 40 1.742 45 1.749 26 1.580 26 1.537
 Croatia 27 1.550 26 1.548 28 1.571 35 1.648 37 1.699 41 1.707 40 1.669 52 1.760
 Malaysia 28 1.561 33 1.659 29 1.574 20 1.590 20 1.485 22 1.539 22 1.520 23 1.517
 Chile 29 1.563 30 1.591 31 1.589 30 1.616 38 1.710 28 1.616 20 1.468 17 1.431
 Qatar 30 1.568 22 1.491 19 1.480 12 1.395 12 1.398 15 1.394 12 1.357 30 1.561
 Botswana 31 1.597 36 1.678 32 1.598 31 1.621 35 1.695 33 1.641 31 1.634 37 1.650
 Bulgaria 32 1.607 32 1.637 35 1.663 39 1.699 53 1.845 50 1.785 46 1.738 46 1.720
 Kuwait 33 1.626 37 1.679 37 1.705 47 1.792 29 1.667 39 1.693 38 1.691 42 1.670
 Costa Rica 34 1.654 42 1.755 40 1.755 36 1.659 31 1.681 26 1.590 27 1.595 31 1.570
 Taiwan 35 1.657 28 1.558 26 1.538 27 1.602 27 1.638 35 1.664 33 1.644 40 1.660
 Italy 36 1.669 34 1.675 34 1.663 38 1.690 45 1.775 40 1.701 39 1.693 34 1.620
 Lithuania 37 1.674 46 1.797 43 1.784 43 1.741 43 1.760 42 1.713 36 1.675 48 1.670
 Estonia 38 1.677 31 1.635 38 1.710 41 1.715 47 1.798 46 1.751 41 1.720 32 1.650
 United Kingdom 39 1.685 47 1.798 44 1.787 29 1.609 26 1.631 31 1.631 41 1.710 48 1.740
 Latvia 40 1.695 39 1.745 41 1.772 45 1.774 46 1.793 54 1.827 50 1.774 45 1.700
 Laos 41 1.700 38 1.723 39 1.724 37 1.662 32 1.687 34 1.661 49 1.767 55 1.770
 South Korea 42 1.701 52 1.849 47 1.822 42 1.734 50 1.829 43 1.715 42 1.716 40 1.660
 Mongolia 43 1.706 41 1.778 64 1.921 58 1.884 57 1.880 92 2.101 88 2.060 89 2.060
 Uruguay 44 1.721 29 1.565 24 1.528 33 1.628 21 1.521 24 1.568 24 1.573 24 1.524
 France 45 1.742 48 1.808 53 1.863 40 1.710 36 1.697 32 1.636 52 1.779 50 1.750
 Indonesia 46 1.768 54 1.853 54 1.879 63 1.913 68 1.979 67 1.946 60 1.843 64 1.860
 Serbia 46 1.768 52 1.849 62 1.912 64 1.920 84 2.071 90 2.071 71 1.906 80 2.020
 Namibia 48 1.784 48 1.808 46 1.807 49 1.804 54 1.850 59 1.864 59 1.837 60 1.820
 Senegal 49 1.805 72 1.974 85 2.061 78 1.994 77 2.047 79 2.031 75 1.969 74 1.950
 United Arab Emirates 49 1.805 40 1.748 36 1.679 46 1.785 33 1.690 44 1.739 30 1.619 29 1.557
 Malawi 51 1.814 77 1.995 74 1.984 60 1.894 39 1.740 51 1.813 51 1.776 62 1.840
 Albania 52 1.821 65 1.939 69 1.961 66 1.927 63 1.912 65 1.925 70 1.890 71 1.910
 Bosnia and Herzegovina 53 1.839 61 1.902 71 1.967 65 1.923 60 1.893 60 1.873 45 1.735 69 1.900
 Ghana 54 1.840 61 1.902 58 1.899 50 1.807 42 1.752 48 1.781 55 1.795 55 1.770
 Zambia 55 1.846 44 1.791 48 1.832 51 1.830 52 1.833 51 1.813 58 1.824 52 1.760
 Vietnam 56 1.848 45 1.792 41 1.772 34 1.641 30 1.670 38 1.691 48 1.764 42 1.730
 Montenegro 57 1.854 55 1.860 73 1.976 81 2.006 89 2.113 88 2.060 80 2.005
 Timor-Leste 58 1.860 69 1.947 51 1.854
 Sierra Leone 59 1.864 66 1.942 59 1.904 52 1.855 61 1.904 53 1.818
 Argentina 60 1.865 43 1.789 60 1.907 44 1.763 55 1.852 71 1.962 61 1.846 55 1.770
 Greece 61 1.878 86 2.052 68 1.957 77 1.976 65 1.947 62 1.887 62 1.850 58 1.810
   Nepal 62 1.878 76 1.989 82 2.058 80 2.001 95 2.152 82 2.044 76 1.967
 Lesotho 63 1.891 50 1.839 50 1.840 53 1.864
 Panama 64 1.903 57 1.877 56 1.893 61 1.899 49 1.812 61 1.878 66 1.862 43 1.680
 Tanzania 64 1.903 59 1.889 55 1.887 55 1.873 56 1.858 55 1.832 53 1.782 56 1.790
 Gabon 66 1.904 68 1.945 76 1.995 75 1.972 81 2.059 74 1.981 44 1.730 50 1.750
 Madagascar 67 1.911 66 1.942 90 2.074 99 2.124 105 2.239 77 2.019 69 1.886 35 1.630
 Cyprus 68 1.924 51 1.844 49 1.840 73 1.957 71 2.013 76 2.013 34 1.64 37 1.650
 Kosovo 69 1.938 64 1.929 72 1.969
 Moldova 70 1.942 71 1.971 74 1.984 66 1.927 59 1.892 66 1.938 74 1.965 78 2.000
 Jordan 71 1.944 56 1.861 52 1.858 62 1.905 64 1.918 68 1.948 67 1.864 61 1.830
 Togo 71 1.944 80 2.003 67 1.954
 Macedonia 71 1.944 87 2.056 79 2.044 68 1.935 78 2.048 83 2.048 86 2.052 76 1.960
 Nicaragua 74 1.947 58 1.882 66 1.931 81 2.006 72 2.021 64 1.924 72 1.911 71 1.910
 Oman 74 1.947 59 1.889 45 1.806 59 1.887 41 1.743 23 1.561 21 1.438 22 1.467
 Tunisia 76 1.952 79 2.001 77 2.005 72 1.955 44 1.765 37 1.678 37 1.686 40 1.660
 Benin 77 1.958 100 2.129 104 2.156 114 2.231
 Liberia 78 1.963 84 2.014 80 2.048 101 2.131 96 2.159 99 2.148
 Georgia 79 1.973 111 2.225 139 2.511 141 2.541 134 2.558 142 2.970 134 2.842
 Mozambique 80 1.976 82 2.004 61 1.910 48 1.796 48 1.809 47 1.779 47 1.762 45 1.700
 Equatorial Guinea 81 1.987 93 2.079 89 2.072 87 2.039 75 2.041 68 1.948 56 1.808 69 1.900
 Cuba 82 1.988 75 1.986 65 1.922 70 1.951 67 1.964 72 1.964 64 1.858 64 1.860
 Burkina Faso 83 1.994 78 1.998 87 2.064 56 1.881 51 1.832 57 1.852 65 1.860 72 1.930
 Bangladesh 84 1.997 98 2.106 105 2.159 91 2.071 83 2.070 87 2.058 93 2.082 93 2.100
 Ecuador 84 1.997 85 2.042 83 2.059 85 2.028 90 2.116 101 2.185 105 2.197 100 2.170
 Morocco 86 2.002 63 1.915 57 1.897 54 1.867 58 1.887 58 1.861 63 1.856 60 1.820
 Kazakhstan 87 2.008 103 2.150 78 2.031 105 2.151 93 2.137 95 2.113 97 2.141 82 2.030
 Angola 88 2.020 102 2.143 102 2.148 95 2.105 87 2.109 86 2.057 84 2.045 100 2.170
 Paraguay 89 2.023 73 1.976 84 2.060 76 1.973 66 1.954 77 2.019 73 1.950 69 1.900
 Bolivia 90 2.025 70 1.969 86 2.062 84 2.021 76 2.045 81 2.037 82 2.041 76 1.960
 Armenia 91 2.028 97 2.097 98 2.123 115 2.238 109 2.260 113 2.266
 Guyana 92 2.029 83 2.013 70 1.962 69 1.937 88 2.112 91 2.095 94 2.082
 Peru 92 2.029 119 2.304 114 2.258 79 1.995 85 2.077 89 2.067 78 2.000 89 2.060
 United States of America 94 2.038 101 2.137 100 2.126 88 2.058 82 2.063 85 2.056 104 2.195 100
 Saudi Arabia 95 2.042 80 2.003 97 2.119 106 2.178 101 2.192 107 2.216 108 2.251 105 2.250
 Papua New Guinea 96 2.064 90 2.066 99 2.126 93 2.076 94 2.139 95 2.113 91 2.075 96 2.130
 Trinidad and Tobago 97 2.070 89 2.065 90 2.074 94 2.082 79 2.051 94 2.107 77 1.985 89 2.060
 Haiti 98 2.074 99 2.127 92 2.075 107 2.179 113 2.288 114 2.270 120 2.406 114 2.350
 Gambia 99 2.086 94 2.085 93 2.091 74 1.961 62 1.910 63 1.890
 Dominican Republic 100 2.089 95 2.093 94 2.103 90 2.068 91 2.125 93 2.103 79 2.004 83 2.040
 Swaziland 101 2.102 87 2.056 88 2.069 85 2.028 69 1.995 73 1.966
 Djibouti 102 2.113 74 1.979 63 1.917 56 1.881
 Brazil 103 2.122 91 2.073 81 2.051 83 2.017 74 2.040 83 2.048 87 2.058 93 2.100
 Algeria 104 2.131 114 2.239 119 2.284 121 2.255 129 2.423 116 2.277 110 2.276 108 2.290
 Cote d’Ivoire 105 2.133 140 2.546 151 2.732 134 2.419 128 2.417 118 2.297 113 2.320 109 2.300
 Turkmenistan 106 2.135 95 2.093 103 2.154 117 2.242 108 2.248 117 2.295 90 2.075 97 2.160
 Bahrain 107 2.142 111 2.225 95 2.109 118 2.247 123 2.398 70 1.956 57 1.815 57 1.800
 Tajikistan 108 2.152 126 2.395 118 2.282 99 2.124 103 2.225
 Jamaica 109 2.153 107 2.203 117 2.274 113 2.222 106 2.244 98 2.138 89 2.065 89 2.060
 Belarus 110 2.163 92 2.078 96 2.117 109 2.208 112 2.283 105 2.204 85 2.046 89 2.060
 Cambodia 111 2.179 106 2.201 115 2.263 108 2.207 115 2.301 111 2.252 101 2.178 95 2.120
 Uganda 111 2.179 110 2.221 106 2.180 98 2.121 96 2.159 100 2.165 103 2.188 110 2.300
 Uzbekistan 113 2.187 104 2.179 124 2.333 110 2.219 109 2.260 110 2.242 109 2.274 113 2.325
 Sri Lanka 114 2.188 105 2.197 110 2.230 103 2.145 126 2.407 133 2.621 126 2.571 120 2.450
 Republic of the Congo 115 2.196 109 2.211 107 2.183 104 2.148 98 2.165 102 2.192 99 2.161 101 2.180
 Honduras 116 2.210 117 2.281 123 2.332 129 2.339 117 2.327 125 2.395 116 2.379 116 2.370
 Guinea 117 2.214 118 2.296 116 2.272 92 2.073 92 2.126
 Guatemala 118 2.215 115 2.248 109 2.221 124 2.287 125 2.405 112 2.258 102 2.187 102 2.210
 Ethiopia 119 2.234 139 2.502 146 2.630 137 2.504 131 2.468 127 2.444 130 2.635 126 2.460
 Guinea-Bissau 120 2.235 145 2.591 132 2.431 95 2.105
 Kyrgyzstan 121 2.249 125 2.382 131 2.391 131 2.359 114 2.296
 Mauritania 122 2.262 120 2.350 122 2.326 125 2.301 130 2.425 123 2.389 117 2.388 105 2.250
 El Salvador 123 2.263 116 2.280 112 2.240 111 2.220 102 2.215 103 2.195 92 2.080 93 2.100
 People’s Republic of China 124 2.267 108 2.207 101 2.142 89 2.061 80 2.054 80 2.034 83 2.045 82 2.030
 Zimbabwe 125 2.294 148 2.662 149 2.696 140 2.538 140 2.722 135 2.678 131 2.700 134 2.380
 Thailand 126 2.303 126 2.395 130 2.378 126 2.303 107 2.247 124 2.393 119 2.399 111 2.324

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