Mathematical symbolsWikipedia

Contents

1 Aristarchian symbols 11.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 Bourbaki dangerous bend symbol 22.1 Typography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.4 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

3 Degree symbol 53.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.2 Typography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.3 Encoding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.4 Lookalikes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.5 Keyboard entry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

4 Obelus 84.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84.2 In computer systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

5 Double turnstile 115.1 Meaning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

6 Equals sign 126.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

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6.2 Usage in mathematics and computer programming . . . . . . . . . . . . . . . . . . . . . . . . . . 136.2.1 Usage of several equals signs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146.2.2 Other uses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

6.3 Tone letter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146.4 Related symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

6.4.1 Approximately equal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146.4.2 Not equal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156.4.3 Identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156.4.4 Isomorphism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156.4.5 In logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156.4.6 In names . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156.4.7 Other related symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

6.5 Incorrect usage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

7 Innity symbol 187.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187.2 Usage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187.3 Symbolism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187.4 Graphic design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197.5 Encoding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

8 Integral symbol 248.1 Typography in Unicode and LaTeX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

8.1.1 Fundamental symbol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248.1.2 Extensions of the symbol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

8.2 Typography in other languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258.6 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

9 ISO 31-11 279.1 Mathematical logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279.2 Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279.3 Miscellaneous signs and symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279.4 Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279.5 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

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9.6 Exponential and logarithmic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279.7 Circular and hyperbolic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279.8 Complex numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279.9 Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279.10 Coordinate systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279.11 Vectors and tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279.12 Special functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279.13 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279.14 References and notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

10 List of mathematical abbreviations 2910.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3410.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

11 List of mathematical symbols 3611.1 Guide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3611.2 Basic symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3711.3 Symbols based on equality sign . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3711.4 Symbols that point left or right . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3711.5 Brackets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3711.6 Other non-letter symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3711.7 Letter-based symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

11.7.1 Letter modiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3711.7.2 Symbols based on Latin letters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3711.7.3 Symbols based on Hebrew or Greek letters . . . . . . . . . . . . . . . . . . . . . . . . . . 37

11.8 Variations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3711.9 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3711.10References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3811.11External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

12 List of mathematical symbols by subject 4012.1 Guide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4012.2 Set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

12.2.1 Denition symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4112.2.2 Set construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4112.2.3 Set operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4112.2.4 Set relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4112.2.5 Number sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4112.2.6 Cardinality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

12.3 Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4112.3.1 Arithmetic operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4112.3.2 Equality signs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

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12.3.3 Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4112.3.4 Divisibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4112.3.5 Intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4112.3.6 Elementary functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4112.3.7 Complex numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4112.3.8 Mathematical constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

12.4 Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4112.4.1 Sequences and series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4212.4.2 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4212.4.3 Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4212.4.4 Asymptotic behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4212.4.5 Dierential calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4212.4.6 Integral calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4212.4.7 Vector calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4212.4.8 Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4212.4.9 Functional analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

12.5 Linear algebra and geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4212.5.1 Elementary geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4212.5.2 Vectors and matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4212.5.3 Vector calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4212.5.4 Matrix calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4212.5.5 Vector spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

12.6 Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4212.6.1 Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4212.6.2 Group theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4212.6.3 Field theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4212.6.4 Ring theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

12.7 Combinatorics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4212.8 Stochastics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

12.8.1 Probability theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4212.8.2 Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

12.9 Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4312.9.1 Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4312.9.2 Quantiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4312.9.3 Deduction symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

12.10See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4312.11References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4312.12External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

13 Maplet 4513.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

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14 Mathematical operators and symbols in Unicode 4614.1 Dedicated blocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

14.1.1 Mathematical Operators block . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4714.1.2 Supplemental Mathematical Operators block . . . . . . . . . . . . . . . . . . . . . . . . . 4714.1.3 Mathematical Alphanumeric Symbols block . . . . . . . . . . . . . . . . . . . . . . . . . 4714.1.4 Letterlike Symbols block . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4714.1.5 Miscellaneous Mathematical Symbols-A block . . . . . . . . . . . . . . . . . . . . . . . . 4714.1.6 Miscellaneous Mathematical Symbols-B block . . . . . . . . . . . . . . . . . . . . . . . . 4714.1.7 Miscellaneous Technical block . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4714.1.8 Geometric Shapes block . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4814.1.9 Miscellaneous Symbols and Arrows block . . . . . . . . . . . . . . . . . . . . . . . . . . 4814.1.10 Arrows block . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4814.1.11 Supplemental Arrows-A block . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4814.1.12 Supplemental Arrows-B block . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4814.1.13 Combining Diacritical Marks for Symbols block . . . . . . . . . . . . . . . . . . . . . . . 4814.1.14 Arabic Mathematical Alphabetic Symbols block . . . . . . . . . . . . . . . . . . . . . . . 49

14.2 Characters in other blocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4914.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4914.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4914.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

15 Multiplication sign 5015.1 Uses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5115.2 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5115.3 Similar notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5115.4 In computer software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5115.5 Unicode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5215.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5215.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5215.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

16 Nabla symbol 5316.1 Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5316.2 Naval engineering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5316.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5316.4 Footnotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5416.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

17 Null sign 5617.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

18 Obelus 5718.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

vi CONTENTS

18.2 In computer systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5718.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5818.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5818.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

19 Percent sign 6019.1 Correct style . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

19.1.1 Spacing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6019.1.2 Usage in text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

19.2 Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6119.3 Usage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

19.3.1 Encodings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6219.3.2 In computers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6219.3.3 In linguistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

19.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6319.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6319.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

20 Plus and minus signs 6520.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6520.2 Plus sign . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6520.3 Minus sign . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6620.4 Use in elementary education . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6620.5 Use as a qualier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6620.6 Uses in computing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6720.7 Other uses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6720.8 Character codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

20.8.1 Alternative plus sign . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6820.9 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6820.10References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6820.11External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

21 Plus-minus sign 7021.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7021.2 Usage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

21.2.1 In mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7021.2.2 In statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7121.2.3 In chess . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

21.3 Minus-plus sign . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7121.4 Encodings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

21.4.1 Typing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7221.5 Similar characters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

CONTENTS vii

21.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7221.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

22 Symbols for zero 7422.1 Glyphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7422.2 Other . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7622.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

23 Table of mathematical symbols by introduction date 7723.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7723.2 Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

24 Therefore sign 7824.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7824.2 Example of use . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7824.3 Related signs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7824.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7924.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

25 Tilde 8125.1 Common use . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8125.2 Use by medieval scribes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8125.3 Diacritical use . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

25.3.1 Pitch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8225.3.2 Abbreviation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8225.3.3 Nasalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8225.3.4 Palatal n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8325.3.5 Tone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8425.3.6 International Phonetic Alphabet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8425.3.7 Letter extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8425.3.8 Other uses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8425.3.9 Precomposed Unicode characters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

25.4 Similar characters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8425.5 ASCII tilde (U+007E) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8425.6 Punctuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

25.6.1 Range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8525.6.2 Japanese . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

25.7 Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8625.7.1 As an unary operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8625.7.2 As a binary operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8625.7.3 As an equivalence operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8625.7.4 As an accent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

25.8 Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

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25.9 Economics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8725.10Electronics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8725.11Computing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

25.11.1 Directories and URLs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8725.11.2 Computer languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8725.11.3 Backup lenames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8825.11.4 Microsoft lenames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8925.11.5 Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8925.11.6 Other uses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

25.12Juggling notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8925.13Keyboards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8925.14See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9025.15References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9025.16External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

26 Tombstone (typography) 9126.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9126.2 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9226.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

27 Triple bar 9327.1 Uses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

27.1.1 Mathematics and philosophy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9327.1.2 Science . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9327.1.3 Web Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

27.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

28 Turnstile (symbol) 9528.1 Interpretations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9528.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9628.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9628.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

29 Up tack 9829.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9829.2 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

30 Vinculum (symbol) 9930.1 Other notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9930.2 Roman numerals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10030.3 Computer entry of the symbol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10030.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10030.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

CONTENTS ix

31 Weierstrass p 10131.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

32 X mark 10232.1 Unicode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10232.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10232.3 Footnotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10332.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10332.5 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 104

32.5.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10432.5.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10832.5.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

Chapter 1

Aristarchian symbols

TheAristarchian symbols are the asterisk and the obelus. They are named after Aristarchus of Samothrace.*[1]*[2]

1.1 See also Dagger (typography) Obelism Textual criticism Annotation Marginalia

1.2 References[1] Paul D. Wegner (2006). A student's guide to textual criticism of the Bible. InterVarsity Press. p. 194. ISBN 978-0-19-

814747-3.

[2] George Maximilian Anthony Grube (1965). The Greek and Roman critics. Hackett Publishing. p. 128. ISBN 978-0-87220-310-5.

1

Chapter 2

Bourbaki dangerous bend symbol

Certains passages sont destins prmunir le lecteur contre des erreurs graves, o il risquerait de tomber ; ces passagessont signals en marge par le signe ( tournant dangereux )Some passages are designed to forewarn the reader against serious errors, where he risks falling; these passages aresignposted in the margin with the sign (dangerous bend)Nicolas Bourbaki's description of the symbol in several textbooks*[1]The dangerous bend or caution symbol (U+2621 caution sign) was created by the Nicolas Bourbaki group of

Frenchvirages dangereuxroad sign, before 1949.

mathematicians and appears in the margins of mathematics books written by the group. It resembles a road sign thatindicates a dangerous bendin the road ahead, and is used to mark passages tricky on a rst reading or with anespecially dicult argument.*[2]Others have used variations of the symbol in their textbooks, and computer scientist Donald Knuth introduced anAmerican-style road-sign depiction in his Metafont and TeX systems, with a pair of adjacent signs indicating doublydangerous passages.*[3]*[4]*[5]*[6]

2

2.1. TYPOGRAPHY 3

2.1 Typography

Knuth'sDangerous Bendsign

In the LaTeX typesetting system, Knuth's dangerous bend symbol can be produced by rst loading the font manfnt(a font with extra symbols used in Knuth's TeX manual) with

\usepackage{manfnt}

4 CHAPTER 2. BOURBAKI DANGEROUS BEND SYMBOL

and then typing

\dbend

There are several variations given by \lhdbend, \reversedvideodbend, \textdbend, \textlhdbend, and \textreversedvideodbend.

2.2 See also Halmos box

2.3 References[1] See, for example, Thorie des ensembles, p. I-8.

[2] Steven G. Krantz (2011), The Proof Is in the Pudding: The Changing Nature of Mathematical Proof, Springer, ISBN0-387-48908-8, p. 92.

[3] Donald Ervin Knuth (1984), The TeXbook, Addison-Wesley, ISBN 0-201-13448-9.

[4] Donald Ervin Knuth (1986), The METAFONTbook, Addison-Wesley, ISBN 0-201-13445-4.

[5] George J. Tourlakis (2003), Lectures in Logic and Set Theory, Volume 2: Set Theory, Cambridge University Press, ISBN0-521-75374-0, p. xiv.

[6] Gerard P. Michon (2012), Dangerous Bend Symbol, doubled and tripled, Numericana

2.4 External links Knuth's use of the dangerous bend sign. Public domain GIF les. Latex style le to provide adangerenvironment marked by a dangerous bend sign, based on Knuth's book.

Chapter 3

Degree symbol

This article describes the symbol . For other meanings, see Degree (disambiguation) and ordinal indicator.

The degree symbol () is a typographical symbol that is used, among other things, to represent degrees of arc (e.g.in geographic coordinate systems), hours (in the medical eld), degrees of temperature, alcohol proof, or diminishedquality in musical harmony.*[1] The symbol consists of a small raised circle, historically a zero glyph.In Unicode it is encoded at U+00B0 degree sign (HTML ).

3.1 HistoryThe rst known recorded modern use of the degree symbol in mathematics is from 1569*[2] where the usage seemsto show that the symbol is a small raised zero, to match the prime symbol notation of sexagesimal subdivisions ofdegree such as minute , second , and tertia which originates as small raised Roman numerals.

3.2 TypographyIn the case of degrees of arc, the degree symbol follows the number without any intervening space.In the case of degrees of temperature, two scientic and engineering standards bodies (BIPM and the U.S. GovernmentPrinting Oce) prescribe printing temperatures with a space between the number and the degree symbol, as in10 C.*[3]*[4] However, in many works with professional typesetting, including scientic works published by theUniversity of Chicago Press or Oxford University Press, the degree symbol is printed with no spaces between thenumber, the symbol, and the Latin letters Cor Frepresenting Celsius or Fahrenheit, respectively (as in10C).*[5]*[6] This is also the practice of the University Corporation for Atmospheric Research, which operates theNational Center for Atmospheric Research.*[7] Use of the degree symbol to refer to temperatures measured in kelvins(symbol: K) was abolished in 1967 by the 13th General Conference on Weights and Measures (CGPM). Therefore,the triple point of water, for instance, is correctly written today as simply 273.16 K. The SI fundamental temperatureunit is now kelvin(note the lower case), and no longer degree Kelvin.

3.3 EncodingThe degree sign is included in Unicode as U+00B0 degree sign (HTML ).For use with Chinese characters there are also code points for U+2103 degree celsius (HTML ) andU+2109 degree fahrenheit (HTML ).The degree sign was missing from the basic 7-bit ASCII set of 1963, but in 1987 the ISO/IEC 8859 standard intro-duced it at position 0xB0 (176 decimal) in the Latin-1 variant. In 1991 the Unicode standard incorporated all of theLatin-1 code points, including the degree sign.

5

6 CHAPTER 3. DEGREE SYMBOL

The Windows Code Page 1252 was also an extension of the Latin-1 standard, so it had the degree sign at the samecode point. The code point in the older DOS Code Page 437 was 0xF8 (248 decimal).

3.4 LookalikesOther characters with similar appearance but dierent meanings include:

U+00BA masculine ordinal indicator (HTML ) (superscript letter used in abbreviatingwords; varies with the font and sometimes underlined)

U+02DA modier letter ring above (HTML ) (standalone) U+030A ring above (HTML ) (applied to a letter) U+0325 ring below (HTML ) (applied to a letter)

U+309C katakana-hiragana semi-voiced sound mark (HTML ) (standalone) U+309A combining katakana-hiragana semi-voiced sound mark (HTML ) (applied to aletter)

U+2070 superscript zero (HTML ) U+2218 ring operator (HTML )

3.5 Keyboard entrySome computer keyboard layouts, such as the QWERTZ layout as used in Germany, Austria and Switzerland, andthe AZERTY layout as used in France and Belgium, have the degree symbol available directly on a key. But thecommon keyboard layouts in English-speaking countries do not include the degree sign, which then has to be inputsome other way. The method of inputting depends on the operating system being used.On the Colemak keyboard layout, one can press AltGr+\ followed by D to insert a degree sign.With Microsoft Windows, there are several ways to make the degree symbol:

One can type Alt+248 or Alt+0176Note: 0176is dierent from 176"; Alt+176 produces the light shade () character.Note: The NumLock must be set rst; on full size keyboards, the numeric keypad must be used; on laptopswithout a numerical keypad, the virtual numeric keypad must be used.

The Character Map tool also may be used to obtain a graphical menu of symbols. The US-International English keyboard layout creates the degree symbol with AltGr+ Shift+:

InMicrosoft Oce and similar programs, there is often also an Insertmenu with an Insert Symbol or Symbol commandthat brings up a graphical palette of symbols to insert, including the degree symbol. In WordPerfect, pressing Ctrl+Wbrings up lists of special characters.In the Mac OS operating system, the degree symbol can be entered by typing Opt+ Shift+8. One can also use theMac OS character palette, which is available in many programs by selecting Special Characters from the Edit Menu,or from the Input Menu (ag) icon on the menu bar (enabled in the International section of the System Preferences).In iOS, the degree symbol is accessed by pressing and holding 0 and dragging your nger to the degree symbol. Thisprocedure is the same as entering diacritics on other characters.In LaTeX, the packages gensymb or textcomp that provides the commands \degree or \textdegree, respectively. Inthe absence of these packages one can write the degree symbol as ^{\circ} in math mode. In other words, it is writtenas the empty circle glyph \circ as a superscript.In Linux operating systems such as Ubuntu, this symbol may be entered via the Compose key followed by o, o. Somekeyboard layouts print this symbol upon pressing AltGr+ Shift+0 (once or twice, depending on specic keyboard

3.6. SEE ALSO 7

layout), and, in programs created by GTK+, one can enter Unicode characters in any text entry eld by rst pressingCtrl+ Shift+U+Unicode, regardless of keyboard layout. For the degree symbol, this is done by entering Ctrl+Shift+UB0.

3.6 See also Prime (symbol) Question mark Unicode Geometric Shapes

3.7 References[1] Chord Symbols. Retrieved 2013-12-16.[2] Cajori, Florian (1993) [1928-1929], AHistory of Mathematical Notations, Dover Publications, ISBN 0-486-67766-4[3] The International System of Units (PDF) (8th ed.), Bureau International des Poids et Mesures, 2006

[4] Style Manual (PDF) (30th ed.), United States Government Printing Oce, 2008

[5] 9.16 Abbreviations and symbols, Chicago Manual of Style (15th ed.), University of Chicago, 2010

[6] 10.52 Miscellaneous technical abbreviations, Chicago Manual of Style (15th ed.), University of Chicago, 2010

[7] UCAR, UCAR Communications Style Guide, retrieved 2007-09-01

3.8 External links Earliest Uses of Symbols from Geometry

Chapter 4

Obelus

Not to be confused with Obolus or Obelisk.

An obelus (symbol: , plural: obeluses or obeli) is a symbol consisting of a short horizontal line with a dot aboveand below. It is mainly used to represent the mathematical operation of division. It is therefore commonly called thedivision sign. Division may also be indicated by a horizontal line (fraction bar), or a slash.This symbol has also been used to represent subtraction in Northern Europe.*[1]

4.1 HistoryFor more details on this topic, see Obelism.The wordobeluscomes from , the Ancient Greek word for a sharpened stick, spit, or pointed pillar. Thisis the same root as that of the word "obelisk". Originally this sign (or a plain line) was used in ancient manuscriptsto mark passages that were suspected of being corrupted or spurious. The dagger symbol, also called an obelisk, isderived from the obelus and continues to be used for this purpose.

The obelus, invented by Aristarchus to mark suspected passages in Homer, is frequent in manuscriptsof the Gospel to mark just those sections, like the Pericope in John, which modern editors reject. Therst corrector of , probably the contemporary (copy-editor, rectier, proofreader), was atpains to enclose in brackets and mark with dots for deletion two famous passages in Luke written by theoriginal scribe which, being absent from BW 579 and the Egyptian versions, we infer were not acceptedin the text at that time dominant in Alexandria, viz. the incident of the "Bloody Sweat" in Gethsemane(Lk.xxi.43 f.) and the saying Father forgive them(Lk.xi.34).*[2]

Although previously used for subtraction, the obelus was rst used as a symbol for division in 1659 in the algebra bookTeutsche Algebra by Johann Rahn. Some think that John Pell, who edited the book, may have been responsible forthis use of the symbol. The usage of the obelus to represent subtraction continued in some parts of Europe (includingNorway and, until fairly recently, Denmark).*[1] Other symbols for division include the slash or solidus (/), and thefraction bar (the horizontal bar in a vertical fraction).

4.2 In computer systemsIn Microsoft Windows, the obelus is produced with Alt+0247 on the number pad or by pressing Alt Gr+ Shift++when an appropriate keyboard layout is in use. In Mac OS, it is produced with Option+/.On UNIX-based systems using Screen or X with a Compose key enabled, it can be produced by composing : (colon)and - (hyphen/minus), though this is locale- and setting-dependent. It may also be input by Unicode code-pointon GTK-based applications by pressing Control+ Shift+U, followed by the codepoint in hexadecimal (F7) andterminated by return.

8

4.3. SEE ALSO 9

Plus and minuses. The obelus or division sign used as a variant of the minus sign in an excerpt from an ocial Norwegiantrading statement form called Nringsoppgave 1 for the taxation year 2010.

In the Unicode character set, the obelus is known as the division signand has the code point U+00F7.*[3] InHTML, it can be encoded as or (at HTML level 3.2), or as .In LaTeX, the obelus is obtained by \div.

4.3 See also Commercial minus sign, , which visually resembles a tilted obelus Obelism

4.4 Notes[1] Cajori, Florian (1993), A history of mathematical notations (two volumes bound as one), Dover, pp. 242, 270271, ISBN

9780486677668. Reprint of 1928 edition.

[2] Burnett Hillman Streeter, The Four Gospels, London, Macmillan, 1924 The Aristarchus referred to was presumablyAristarchus of Samothrace.

[3] Korpela, Jukka (2006), Unicode Explained: Internationalize documents, programs, and web sites, O'Reilly Media, Inc., p.397, ISBN 9780596101213.

10 CHAPTER 4. OBELUS

4.5 External links Je Miller: Earliest Uses of Various Mathematical Symbols Michael Quinion: Where our arithmetic symbols come from

Chapter 5

Double turnstile

Not to be confused with .

In logic, the symbol , or j= is called the double turnstile. It is closely related to the turnstile symbol ` , whichhas a single bar across the middle. It is often read as "entails", "models", is a semantic consequence ofor isstronger than.*[1] In TeX, the turnstile symbols and j= are obtained from the commands \vDash and \modelsrespectively. In Unicode it is encoded at U+22A8 true (HTML )In LaTeX there is the turnstile package, which issues this sign in many ways, including the double turnstile, and iscapable of putting labels below or above it, in the correct places. The article A Tool for Logicians is a tutorial onusing this package.

5.1 MeaningThe double turnstile is a binary relation. It has several dierent meanings in dierent contexts:

To show semantic consequence, with a set of sentences on the left and a single sentence on the right, to denotethat if every sentence on the left is true, the sentence on the right must be true, e.g. ' . This usage isclosely related to the single-barred turnstile symbol which denotes syntactic consequence.

To show satisfaction, with a model (or truth-structure) on the left and a set of sentences on the right, to denotethat the structure is a model for (or satises) the set of sentences, e.g. A j= .

To denote a tautology, ' . which is to say that the expression ' is a semantic consequence of the empty set.

5.2 See also List of logic symbols List of mathematical symbols

5.3 References[1] Nederpelt, Rob (2004).Chapter 7: Strengthening and weakening. Logical Reasoning: A First Course (3rd revised ed.).

King's College Publications. p. 62. ISBN 0-9543006-7-X.

11

Chapter 6

Equals sign

"=" and " " redirect here. For double hyphens, see Double hyphen.For technical reasons, ":=" redirects here. For the computer programming assignment operator, see Assignment(computer science). For the denition symbol, see List of mathematical symbols#Symbols based on equality sign.For other uses, see Equals (disambiguation).The equals sign or equality sign (=) is a mathematical symbol used to indicate equality. It was invented in

A well-known equality featuring the equals sign

1557 by Robert Recorde. The equals sign is placed between two expressions that have the same value meaning theyrepresent the same mathematical object, as in an equation. It is assigned to the Unicode and ASCII character 003Din hexadecimal, 0061 in decimal.

6.1 HistoryThe etymology of the wordequalis from the Latin word "aequalis as meaninguniform,identical, or

equal, from aequus (level, even, or just).

The rst use of an equals sign, equivalent to 14x+15=71 in modern notation. From The Whetstone of Witte by Robert Recorde.

The "=" symbol that is now universally accepted by mathematics for equality was rst recorded by Welsh mathemati-cian Robert Recorde in The Whetstone of Witte (1557). The original form of the symbol was much wider than thepresent form. In his book Recorde explains his design of the Gemowe lines(meaning twin lines, from the Latingemellus*[1]):

to auoide the tedioue repetition of thee woordes : is equalle to : I will ette as I doe often in woorkeve, a paire of paralleles, or Gemowe lines of one lengthe, thus: =, bicaue noe .2. thynges, can be moare

12

6.2. USAGE IN MATHEMATICS AND COMPUTER PROGRAMMING 13

Recorde's introduction of "="

equalle.

to avoid the tedious repetition of these words:is equal to, I will set (as I do often in work use)a pair of parallels, or Gemowe lines, of one length (thus =), because no two things can be more equal.

According to Scotland's University of St Andrews History of Mathematics website:*[2]

The symbol '=' was not immediately popular. The symbol || was used by some and (or ), fromthe Latin word aequalis meaning equal, was widely used into the 1700s.

6.2 Usage in mathematics and computer programmingIn mathematics, the equals sign can be used as a simple statement of fact in a specic case (x = 2), or to createdenitions (let x = 2), conditional statements (if x = 2, then ), or to express a universal equivalence (x + 1)2 = x2+ 2x + 1.The rst important computer programming language to use the equals sign was the original version of Fortran, FOR-TRAN I, designed in 1954 and implemented in 1957. In Fortran, "=" serves as an assignment operator: X = 2 sets thevalue of X to 2. This somewhat resembles the use of "=" in a mathematical denition, but with dierent semantics:the expression following "=" is evaluated rst and may refer to a previous value of X. For example, the assignmentX = X + 2 increases the value of X by 2.A rival programming-language usage was pioneered by the original version of ALGOL, which was designed in 1958and implemented in 1960. ALGOL included a relational operator that tested for equality, allowing constructionslike if x = 2 with essentially the same meaning of "=" as the conditional usage in mathematics. The equals sign wasreserved for this usage.Both usages have remained common in dierent programming languages into the early 21st century. As well asFortran, "=" is used for assignment in such languages as C, Perl, Python, awk, and their descendants. But "=" is usedfor equality and not assignment in the Pascal family, Ada, Eiel, APL, and other languages.A few languages, such as BASIC and PL/I, have used the equals sign to mean both assignment and equality, distin-guished by context. However, in most languages where "=" has one of these meanings, a dierent character or, moreoften, a sequence of characters is used for the other meaning. Following ALGOL, most languages that use "=" forequality use ":=" for assignment, although APL, with its special character set, uses a left-pointing arrow.Fortran did not have an equality operator (it was only possible to compare an expression to zero, using the arithmeticIF statement) until FORTRAN IV was released in 1962, since when it has used the four characters ".EQ.to test forequality. The language B introduced the use of "==" with this meaning, which has been copied by its descendant Cand most later languages where "=" means assignment.

14 CHAPTER 6. EQUALS SIGN

6.2.1 Usage of several equals signsIn PHP, the triple equals sign (===) denotes identity,*[3] meaning that not only do the two expressions evaluate toequal values, they are also of the same data type. For instance, the expression 0 == false is true, but 0 === false isnot, because the number 0 is an integer value whereas false is a Boolean value.JavaScript has the same semantics for ===, referred to asequality without type coercion. However in JavaScriptthe behavior of == cannot be described by any simple consistent rules. The expression 0 == false is true, but 0 ==undened is false, even though both sides of the == act the same in Boolean context. For this reason it is recommendedto avoid the == operator in JavaScript in favor of ===.*[4]In Ruby, equality under == requires both operands to be of identical type, e.g. 0 == false is false. The === operatoris exible and may be dened arbitrarily for any given type. For example, a value of type Range is a range ofintegers, such as 1800..1899. (1800..1899) == 1844 is false, since the types are dierent (Range vs. Integer);however (1800..1899) === 1844 is true, since === on Range values meansinclusion in the range.*[5] Note thatunder these semantics, === is non-symmetric; e.g. 1844 === (1800..1899) is false, since it is interpreted to meanInteger#=== rather than Range#===.*[6]

6.2.2 Other usesThe equals sign is also used in dening attributevalue pairs, in which an attribute is assigned a value.

6.3 Tone letterThe equals sign is also used as a grammatical tone letter in the orthographies of Budu in the Congo-Kinshasa, inKrumen, Mwan and Dan in the Ivory Coast.*[7]*[8] The Unicode character used for the tone letter (U+A78A)*[9]is dierent from the mathematical symbol (U+003D).

6.4 Related symbolsSee also: Unicode mathematical operators

6.4.1 Approximately equalMain article: Approximation Unicode

Symbols used to denote items that are approximately equal include the following:*[10]

(U+2248, LaTeX \approx) (U+2243, LaTeX \simeq), a combination of and =, also used to indicate asymptotic equality (U+2245, LaTeX \cong), another combination of and =, which is also sometimes used to indicate isomorphismor congruence

(U+223C), which is also sometimes used to indicate proportionality, being related by an equivalence relation,or to indicate that a random variable is distributed according to a specic probability distribution

(U+223D), which is also used to indicate proportionality (U+2250, LaTeX \doteq), which can also be used to represent the approach of a variable to a limit (U+2252), commonly used in Japanese, Taiwanese and Korean (U+2253)

6.4. RELATED SYMBOLS 15

6.4.2 Not equal

The symbol used to denote inequation (when items are not equal) is a slashed equals sign "" (U+2260; 2260,Alt+Xin Microsoft Windows). In LaTeX, this is done with the "\neqcommand.Most programming languages, limiting themselves to the ASCII character set and typeable characters, use ~=, !=, /=,=/=, or to represent their Boolean inequality operator.

6.4.3 Identity

The triple bar symbol "" (U+2261, Latex \equiv) is often used to indicate an identity, a denition (which can alsobe represented by U+225D "" or U+2254 ""), or a congruence relation in modular arithmetic. The symbol ""can be used to express that an item corresponds to another.

6.4.4 Isomorphism

The symbol "" is often used to indicate isomorphic algebraic structures or congruent geometric gures.

6.4.5 In logic

Equality of truth values, i.e. bi-implication or logical equivalence, may be denoted by various symbols including =,~, and .

6.4.6 In names

A possibly unique case of the equals sign of European usage in a person's name, specically in a double-barreledname, was by pioneer aviator Alberto Santos=Dumont, as he is also known not only to have often used an equalssign (=) between his two surnames in place of a hyphen, but also seems to have personally preferred that practice, todisplay equal respect for his father's French ethnicity and the Brazilian ethnicity of his mother.*[11]The equals sign is sometimes used in Japanese as a separator between names.

6.4.7 Other related symbols

Additional symbols in Unicode related to the equals sign include:*[10]

(U+224C all equal to) (U+2254 colon equals) (U+2255 equals colon) (U+2256 ring in equal to) (U+2257 ring equal to) (U+2259 estimates) (U+225A equiangular to) (U+225B star equals) (U+225C delta equal to) (U+225E measured by) (U+225F questioned equal to).

16 CHAPTER 6. EQUALS SIGN

6.5 Incorrect usageThe equals sign is sometimes used incorrectly within amathematical argument to connect math steps in a non-standardway, rather than to show equality (especially by early mathematics students).For example, if one were nding the sum, step by step, of the numbers 1, 2, 3, 4, and 5, one might write

1 + 2 = 3 + 3 = 6 + 4 = 10 + 5 = 15.

Structurally, this is shorthand for

([(1 + 2 = 3) + 3 = 6] + 4 = 10) + 5 = 15,

but the notation is incorrect, because each part of the equality has a dierent value. If interpreted strictly as it says,it implies

3 = 6 = 10 = 15 = 15.

A correct version of the argument would be

1 + 2 = 3, 3 + 3 = 6, 6 + 4 = 10, 10 + 5 = 15.*[12]

6.6 See also 2 + 2 = 5 Double hyphen Equality (mathematics) Logical equality Plus and minus signs

6.7 Notes[1] See also geminus and Gemini.

[2] Robert Recorde. MacTutor History of Mathematics archive. Retrieved 19 October 2013.[3] Comparison Operators. PHP.net. Retrieved 19 October 2013.[4] Doug Crockford. JavaScript: The Good Parts. YouTube. Retrieved 19 October 2013.[5] why the lucky sti. 5.1 This One s For the Disenfranchised. why's (poignant) Guide to Ruby. Retrieved 19 October

2013.

[6] Brett Rasmussen (30 July 2009). Don't Call it Case Equality. Retrieved 19 October 2013.[7] Peter G. Constable; Lorna A. Priest (31 July 2006). Proposal to Encode Additional Orthographic and Modier Characters

(PDF). Retrieved 19 October 2013.

[8] Hartell, Rhonda L., ed. (1993). The Alphabets of Africa. Dakar: UNESCO and SIL. Retrieved 19 October 2013.

[9] Unicode Latin Extended-D code chart (PDF). Unicode.org. Retrieved 19 October 2013.[10] Mathematical Operators (PDF). Unicode.org. Retrieved 19 October 2013.[11] Gray, Carroll F. (November 2006). The 1906 Santos=Dumont No. 14bis. World War I Aeroplanes. No. 194: 4.[12] Capraro, Robert M.; Capraro, Mary Margaret; Yetkiner, Ebrar Z.; Corlu, Sencer M.; Ozel, Serkan; Ye, Sun; Kim, Hae

Gyu (2011).An International Perspective between Problem Types in Textbooks and Students' understanding of relationalequality. Mediterranean Journal for Research in Mathematics Education 10 (12): 187213. Retrieved 19 October 2013.

6.8. REFERENCES 17

6.8 References Cajori, Florian (1993). AHistory of Mathematical Notations. New York: Dover (reprint). ISBN 0-486-67766-4.

Boyer, C. B.: A History of Mathematics, 2nd ed. rev. by Uta C. Merzbach. New York: Wiley, 1989 ISBN0-471-09763-2 (1991 pbk ed. ISBN 0-471-54397-7)

6.9 External links Earliest Uses of Symbols of Relation Image of the page of The Whetstone of Witte on which the equals sign is introduced Scientic Symbols, Icons, Mathematical Symbols Robert Recorde invents the equals sign

Chapter 7

Innity symbol

The innity symbol (sometimes called the lemniscate) is a mathematical symbol representing the concept ofinnity.

7.1 HistoryThe shape of a sideways gure eight has a long pedigree; for instance, it appears in the cross of Saint Boniface,wrapped around the bars of a Latin cross.*[1] However, John Wallis is credited with introducing the innity symbolwith its mathematical meaning in 1655, in hisDe sectionibus conicis.*[2]*[1]*[3]*[4] Wallis did not explain his choiceof this symbol, but it has been conjectured to be a variant form of a Roman numeral for 1,000 (originally CI, alsoC), which was sometimes used to mean many, or of the Greek letter (omega), the last letter in the Greekalphabet.*[5]Leonhard Euler used an open variant of the symbol*[6] in order to denote absolutus innitus. Euler freelyperformed various operations on innity, such as taking its logarithm. This symbol is not used anymore, and doesnot exist in Unicode.

7.2 UsageIn mathematics, the innity symbol is used more often to represent a potential innity,*[1] rather than to represent anactually innite quantity such as the ordinal numbers and cardinal numbers (which use other notations). For instance,in the mathematical notation for summations and limits such as

1Xi=0

1

2i= lim

x!12x 12x1

= 2;

the innity sign is conventionally interpreted as meaning that the variable grows arbitrarily large (towards innity)rather than actually taking an innite value.In areas other than mathematics, the innity symbol may take on other related meanings; for instance, it has beenused in bookbinding to indicate that a book is printed on acid-free paper and will therefore be long-lasting.*[7]

7.3 SymbolismIn modern mysticism, the innity symbol has become identied with a variation of the ouroboros, an ancient imageof a snake eating its own tail that has also come to symbolize the innite, and the ouroboros is sometimes drawn ingure-eight form to reect this identication, rather than in its more traditional circular form.*[8]In the works of Vladimir Nabokov, including The Gift and Pale Fire, the gure-eight shape is used symbolically torefer to the Mbius strip and the innite, for instance in these books' descriptions of the shapes of bicycle tire tracks

18

7.4. GRAPHIC DESIGN 19

The symbol in several typefaces

and of the outlines of half-remembered people. The poem after which Pale Fire is entitled explicitly refers to themiracle of the lemniscate.*[9]

7.4 Graphic design

The well known shape and meaning of the innity symbol have made it a common typographic element of graphicdesign. For instance, the Mtis ag, used by the Canadian Mtis people in the early 19th century, is based around thissymbol.*[10] In modern commerce, corporate logos featuring this symbol have been used by, among others, Inniti,Room for PlayStation Portable, Microsoft Visual Studio, CoorsTek, and Lazy 8 Studios.

20 CHAPTER 7. INFINITY SYMBOL

John Wallis introduced the innity symbol to mathematical literature.

7.5 Encoding

The symbol is encoded in Unicode at U+221E innity (HTML &inn;) and in LaTeX as \infty: 1 .The Unicode set of symbols also includes several variant forms of the innity symbol, that are less frequently avail-able in fonts: U+29DC incomplete innity (HTML ISOtech entity ), U+29DD tie over innity(HTML ) and U+29DE innity negated with vertical bar (HTML ) in block MiscellaneousMathematical Symbols-B.*[11]

7.6. REFERENCES 21

Symbol used by Euler to denote innity

7.6 References[1] Barrow, John D. (2008),Innity: Where God Divides by Zero, Cosmic Imagery: Key Images in the History of Science,

W. W. Norton & Company, pp. 339340, ISBN 9780393061772

[2] De sectionibus conicis nova methodo expositis tractatus - John Wallis - Google Boeken. Books.google.com. Retrieved2013-12-01.

[3] Scott, Joseph Frederick (1981), The mathematical work of John Wallis, D.D., F.R.S., (1616-1703) (2 ed.), AmericanMathematical Society, p. 24, ISBN 0-8284-0314-7

[4] Martin-Lf, Per (1990),Mathematics of innity, COLOG-88 (Tallinn, 1988), Lecture Notes in Computer Science 417,Berlin: Springer, pp. 146197, doi:10.1007/3-540-52335-9_54, MR 1064143

[5] Clegg, Brian (2003), A brief history of innity: the quest to think the unthinkable, Robinson, ISBN 9781841196503

[6] See for instance Cor. 1 p. 174 in: Leonhard Euler. Variae observationes circa series innitas. Commentarii academiaescientiarum Petropolitanae 9, 1744, pp. 160-188.

[7] Zboray, Ronald J.; Zboray, Mary Saracino (2000), A handbook for the study of book history in the United States, Centerfor the Book, Library of Congress, p. 49, ISBN 9780844410159

22 CHAPTER 7. INFINITY SYMBOL

[8] O'Flaherty, Wendy Doniger (1986), Dreams, Illusion, and Other Realities, University of Chicago Press, p. 243, ISBN9780226618555. The book also features this image on its cover.

[9] Toker, Leona (1989),Nabokov: TheMystery of Literary Structures, Cornell University Press, p. 159, ISBN9780801422119

[10] Healy, Donald T.; Orenski, Peter J. (2003),Native American Flags, University ofOklahomaPress, p. 284, ISBN9780806135564

[11] Unicode chart (odf)" (PDF). Retrieved 2013-12-01.

7.6. REFERENCES 23

The innity symbol appears on several cards of the RiderWaite tarot deck

Chapter 8

Integral symbol

Not to be confused with Long s or Esh (letter).

The integral symbol:

(Unicode),Z

(LaTeX)

is used to denote integrals and antiderivatives in mathematics. The notation was introduced by the German mathe-matician Gottfried Wilhelm Leibniz towards the end of the 17th century. The symbol was based on the (long s)character, and was chosen because Leibniz thought of the integral as an innite sum of innitesimal summands.

8.1 Typography in Unicode and LaTeX

8.1.1 Fundamental symbolMain article: Integral calculus

The integral symbol is U+222B integral in Unicode*[1] and \int in LaTeX. In HTML, it is written as (hexadecimal), (decimal) and (named entity).The original IBM PC code page 437 character set included a couple of characters and (codes 244 and 245,respectively) to build the integral symbol. These were deprecated in subsequent MS-DOS code pages, but they stillremain in Unicode (U+2320 and U+2321, respectively) for compatibility.The symbol is very similar to, but not to be confused with, the symbol (called "esh").

8.1.2 Extensions of the symbolSee also: Multiple integral

Related symbols include:*[1]*[2]

8.2 Typography in other languagesIn other languages, the shape of the integral symbol diers slightly from the shape commonly seen in English-languagetextbooks. While the English integral symbol leans to the right, the German symbol (used throughout Central Europe)is upright, and the Russian variant leans to the left.

24

8.3. SEE ALSO 25

Regional variations (English, German, Russian) of the integral symbol.

Another dierence is in the placement of limits for denite integrals. Generally, in English-language books, limitsgo to the right of the integral symbol:

R T0f(t) dt .

By contrast, in German and Russian texts, limits for denite integrals are placed above and below the integral symbol,and, as a result, the notation requires larger line spacing:

TR0

f(t) dt .

8.3 See also Capital sigma notation

Capital pi notation

8.4 Notes[1] Mathematical Operators Unicode. Retrieved 2013-04-26.

[2] Supplemental Mathematical Operators Unicode. Retrieved 2013-05-05.

8.5 References Stewart, James (2003). Integrals. Single Variable Calculus: Early Transcendentals (5th edition ed.).Belmont, CA: Brooks/Cole. p. 381. ISBN 0-534-39330-6.

Zaitcev, V.; Janishewsky, A.; Berdnikov, A. (1999), Russian Typographical Traditions in MathematicalLiterature, EuroTeX'99 Proceedings Missing or empty |title= (help)

26 CHAPTER 8. INTEGRAL SYMBOL

8.6 External links Fileformat.info

Chapter 9

ISO 31-11

ISO 31-11 was the part of international standard ISO 31 that denes mathematical signs and symbols for use inphysical sciences and technology. It was superseded in 2009 by ISO 80000-2.*[1]Its denitions include the following:*[2]

9.1 Mathematical logic

9.2 Sets

9.3 Miscellaneous signs and symbols

9.4 Operations

9.5 Functions

9.6 Exponential and logarithmic functions

9.7 Circular and hyperbolic functions

9.8 Complex numbers

9.9 Matrices

9.10 Coordinate systems

9.11 Vectors and tensors

9.12 Special functions

9.13 See also Mathematical symbols

27

28 CHAPTER 9. ISO 31-11

Mathematical notation

9.14 References and notes[1] ISO 80000-2:2009. International Organization for Standardization. Retrieved 1 July 2010.[2] Thompson, Ambler; Taylor, Barry M (March 2008). Guide for the Use of the International System of Units (SI) NIST

Special Publication 811, 2008 EditionSecond Printing (PDF). Gaithersburg, MD, USA: NIST.[3] These brace or fence characters are upper level unicode characters, fairly recently established and so may not display

correctly in every browser. A close approximation of the appearance is found in the standard Latin characters: ( ), [ ], {}, < >. A more accurate glyph depiction of the mathematical angle bracket characters are found in the Chinese-Japanese-Korean (CJK) punctuation category: h; h;.

[4] If the perpendicular symbol, h;, does not display correctly, it is similar to h; (up tack: sometimes meaning orthogonalto) and it also appears similar to h; (the dentistry: symbol light up and horizontal)

Chapter 10

List of mathematical abbreviations

This article is a listing of abbreviated names of mathematical functions, function-like operators and other mathemat-ical terminology.

This list is limited to abbreviations of two or more letters. The capitalization of some of these abbreviationsis not standardized dierent authors use dierent capitalizations.This list is incomplete; you can help by expanding it.

AC Axiom of Choice.*[1] a.c. absolutely continuous. adj adjugate of a matrix. a.e. almost everywhere. Ai Airy function. AL Action limit. Alt alternating group (Alt(n) is also written as An.) A.M. arithmetic mean. arccos inverse cosine function. arccosec inverse cosecant function. (Also written as arccsc.) arccot inverse cotangent function. arccsc inverse cosecant function. (Also written as arccosec.) arcosech inverse hyperbolic cosecant function. (Also written as arcsch.) arcosh inverse hyperbolic cosine function. arcoth inverse hyperbolic cotangent function. arcsch inverse hyperbolic cosecant function. (Also written as arcosech.) arcsec inverse secant function. arcsin inverse sine function. arctan inverse tangent function. arg argument of a complex number.*[2] arg max argument of the maximum.

29

30 CHAPTER 10. LIST OF MATHEMATICAL ABBREVIATIONS

arg min argument of the minimum. arsech inverse hyperbolic secant function. arsinh inverse hyperbolic sine function. artanh inverse hyperbolic tangent function. a.s. almost surely. A.P. arithmetic progression. Aut automorphism group. bd boundary. Bi Airy function of the second kind. Bias bias of an estimator Card cardinality of a set.*[3] (Card(X) is also written #X, X or |X|.) cdf cumulative distribution function. c.f. cumulative frequency. char characteristic of a ring. Chi hyperbolic cosine integral function. Ci cosine integral function. cis cos + i sin function. Cl conjugacy class. cl topological closure. cod codomain. (Also written as codom.) codom codomain. (Also written as cod.) cok cokernel. (Also written as coker.) coker cokernel. (Also written as cok.) Cor corollary. corr correlation. cos cosine function. cosec cosecant function. (Also written as csc.) cosech hyperbolic cosecant function. (Also written as csch.) cosh hyperbolic cosine function. cot cotangent function. coth hyperbolic cotangent function. cov covariance of a pair of random variables. csc cosecant function. (Also written as cosec.) csch hyperbolic cosecant function. (Also written as cosech.) curl curl of a vector eld. (Also written as rot.) def dene or denition.

31

deg degree of a polynomial. (Also written as .) del del, a dierential operator. (Also written as r .) det determinant of a matrix or linear transformation. dim dimension of a vector space. div divergence of a vector eld. dkl decalitre DNE a solution for an expression does not exist, or is undened. Generally used with limits and integrals. dom domain of a function.*[1] (Or, more generally, a relation.) End categories of endomorphisms. Ei exponential integral function. Eqn equation. erf error function. erfc complementary error function. etr exponent of the trace. exp exponential function. (exp x is also written as e*x.) Ext Ext functor. ext exterior. FIP nite intersection property. FOL rst-order logic. Frob Frobenius endomorphism. Gal Galois group. (Also written as .) gcd greatest common divisor of two numbers. (Also written as hcf.) GF Galois eld. GL general linear group. G.M. geometric mean. glb greatest lower bound. (Also written as inf.) G.P. geometric progression. grad gradient of a scalar eld. hcf highest common factor of two numbers. (Also written as gcd.) H.M. harmonic mean. HOL higher-order logic. Hom Hom functor. hom hom-class. i if and only if. iid independent and identically distributed random variables. Im imaginary part of a complex number*[2] (which is also written = ).

32 CHAPTER 10. LIST OF MATHEMATICAL ABBREVIATIONS

im image inf inmum of a set. (Also written as glb.) int interior. ker kernel. lcm lowest common multiple of two numbers. lerp linear interpolation.*[4] lg common logarithm (log10) or binary logarithm (log2). LHS left-hand side of an equation. Li oset logarithmic integral function. li logarithmic integral function or linearly independent. lim limit of a sequence, or of a function. lim inf limit inferior. lim sup limit superior. ln natural logarithm, loge. log logarithm. (If without a subscript, this may mean either log10 or loge.) logh natural logarithm, loge.*[5] LST language of set theory. lub least upper bound.*[1] (Also written sup.) max maximum of a set. M.I. mathematical induction. min minimum of a set. mod modulo. mx matrix. NAND not-and in logic. No. number. NOR not-or in logic. NTS need to show. ob object class. ord ordinal number of a well-ordered set.*[3] pdf probability density function. pf proof. PGL projective general linear group. pmf probability mass function. Pr probability of an event. (See Probability theory. Also written as P or P .) PSL projective special linear group. QED Quod erat demonstrandum, a Latin phrase used at the end of a denitive proof.

33

QEF quod erat faciendum, a Latin phrase sometimes used at the end of a construction. ran range of a function. rank rank. (Also written as rk.) Re real part of a complex number.*[2] (Also written < .) resp respectively. RHS right-hand side of an equation. rk rank. (Also written as rank.) RMS root mean square. (Also written as rms.) rms root mean square. (Also written as RMS.) rng non-unital ring. rot rotor of a vector eld. (Also written as curl.) RTP required to prove. RV Random Variable. (or as R.V.) sec secant function. sech hyperbolic secant function. seg initial segment of.*[1] SFIP strong nite intersection property. sgn signum function. Shi hyperbolic sine integral function. Si sine integral function. sin sine function. sinc sinc function. sinh hyperbolic sine function. SL special linear group. Soln solution. sp linear span of a set of vectors. (Also written with angle brackets.) Spec spectrum of a ring. s.t. such that or so that. st standard part function. STP [it is] sucient to prove. sup supremum of a set.*[1] (Also written lub.) supp support of a function. Sym symmetric group (Sym(n) is also written as Sn.) tan tangent function. tanh hyperbolic tangent function. TFAE the following are equivalent.

34 CHAPTER 10. LIST OF MATHEMATICAL ABBREVIATIONS

Thm theorem. Tor Tor functor. Tr trace, either the eld trace, or the trace of a matrix or linear transformation. undef a function or expression is undened var variance of a random variable. W^5 which was what we wanted. Synonym of Q.E.D. walog without any loss of generality. w well-formed formula. whp with high probability. wlog without loss of generality. WMA we may assume. WO well-ordered set.*[1] wrt with respect to or with regard to. WTP want to prove. WTS want to show. XOR exclusive or in logic. ZF ZermeloFraenkel axioms of set theory.*[3] ZFC ZermeloFraenkel axioms (with the Axiom of Choice) of set theory.*[3]

10.1 References[1] Goldrei, Derek (1996), Classic Set Theory, London: Chapman and Hall, pp. 283287 (Index), ISBN 0-412-60610-0

[2] Priestley, H. A. (2003), Introduction to Complex Analysis (2nd ed.), Oxford University Press, p. 321 (Notation index),ISBN 978-0-19-852562-2

[3] Hamilton, A. G. (1982), Numbers, sets and axioms, Cambridge University Press, pp. 249251 (Index of symbols), ISBN0-521-24509-5

[4] Raymond, Eric (2003), LERP, Jargon File (version 4.4.7)[5] Jolley, L.B.W. (1961), Summation of Series, Second Revised Edition Dover Publications, INC., New York, Library of

Congress: 61-65274

10.2 See also Greek letters used in mathematics, science, and engineering ISO 31-11 Mathematical alphanumeric symbols Mathematical jargon Mathematical notation Notation in probability and statistics Physical constants

10.2. SEE ALSO 35

Roman letters used in mathematics Table of logic symbols Table of mathematical symbols Unicode mathematical operators

Chapter 11

List of mathematical symbols

This list is incomplete; you can help by expanding it.

This is a list of symbols found within all branches of mathematics to express a formula or to represent a constant.When reading the list, it is important to recognize that a mathematical concept is independent of the symbol chosento represent it. For many of the symbols below, the symbol is usually synonymous with the corresponding concept(ultimately an arbitrary choice made as a result of the cumulative history of mathematics), but in some situations adierent convention may be used. For example, depending on context, the triple bar "" may represent congruenceor a denition. Further, in mathematical logic, numerical equality is sometimes represented by "" instead of "=",with the latter representing equality of well-formed formulas. In short, convention dictates the meaning.Each symbol is shown both in HTML, whose display depends on the browser's access to an appropriate font installedon the particular device, and in TeX, as an image.

11.1 GuideThis list is organized by symbol type and is intended to facilitate nding an unfamiliar symbol by its visual appearance.For a related list organized by mathematical topic, see List of mathematical symbols by subject. That list also includesLaTeX and HTML markup and Unicode code points for each symbol.

Basic symbols: Symbols widely used in mathematics, roughly through rst-year calculus. More advancedmeanings are included with some symbols listed here.

Symbols based on equality "=": Symbols derived from or similar to the equal sign, including double-headedarrows. Not surprisingly these symbols are often associated with an equivalence relation.

Symbols that point left or right: Symbols, such as < and >, that appear to point to one side or another. Brackets: Symbols that are placed on either side of a variable or expression, such as |x|. Other non-letter symbols: Symbols that do not fall in any of the other categories. Letter-based symbols: Many mathematical symbols are based on, or closely resemble, a letter in some alpha-bet. This section includes such symbols, including symbols that resemble upside-down letters. Many lettershave conventional meanings in various branches of mathematics and physics. These are not listed here. TheSee also section, below, has several lists of such usages. Letter modiers: Symbols that can be placed on or next to any letter to modify the letter's meaning. Symbols based on Latin letters, including those symbols that resemble or contain an X Symbols based on Hebrew or Greek letters e.g. , , , , , , , , . Note: symbols resembling are grouped with Vunder Latin letters.

Variations: Usage in languages written right-to-left

36

11.2. BASIC SYMBOLS 37

11.2 Basic symbols

11.3 Symbols based on equality sign

11.4 Symbols that point left or right

11.5 Brackets

11.6 Other non-letter symbols

11.7 Letter-based symbolsIncludes upside-down letters.

11.7.1 Letter modiers

11.7.2 Symbols based on Latin letters

11.7.3 Symbols based on Hebrew or Greek letters

11.8 VariationsInmathematics written in Arabic, some symbols may be reversed tomake right-to-left writing and reading easier.*[13]

11.9 See also Greek letters used in mathematics, science, and engineering ISO 31-11 (Mathematical signs and symbols for use in physical sciences and technology) Latin letters used in mathematics List of mathematical abbreviations List of mathematical symbols by subject Mathematical alphanumeric symbols Mathematical constants and functions Mathematical notation Mathematical operators and symbols in Unicode Notation in probability and statistics Physical constants Table of logic symbols Table of mathematical symbols by introduction date Typographical conventions in mathematical formulae

38 CHAPTER 11. LIST OF MATHEMATICAL SYMBOLS

11.10 References[1] Math is Fun website.[2] Rnyai, Lajos (1998), Algoritmusok(Algorithms), TYPOTEX, ISBN 963-9132-16-0

[3] Deb, K.; Pratap, A.; Agarwal, S.; Meyarivan, T. (2002). A fast and elitist multiobjective genetic algorithm: NSGA-II. IEEE Transactions on Evolutionary Computation 6 (2): 182. doi:10.1109/4235.996017.

[4] Copi, IrvingM.; Cohen, Carl (1990) [1953],Chapter 8.3: Conditional Statements andMaterial Implication, Introductionto Logic (8th ed.), NewYork: Macmillan Publishers (United States), pp. 268269, ISBN 0-02-325035-6, LCCN 89037742

[5] Goldrei, Derek (1996), Classic Set Theory, London: Chapman and Hall, p. 4, ISBN 0-412-60610-0

[6] Nielsen, Michael A; Chuang, Isaac L (2000), Quantum Computation and Quantum Information, New York: CambridgeUniversity Press, p. 66, ISBN 0-521-63503-9, OCLC 43641333

[7] Goldrei, Derek (1996), Classic Set Theory, London: Chapman and Hall, p. 3, ISBN 0-412-60610-0

[8] Nielsen, Michael A; Chuang, Isaac L (2000), Quantum Computation and Quantum Information, New York: CambridgeUniversity Press, p. 62, ISBN 0-521-63503-9, OCLC 43641333

[9] Berman, Kenneth A; Paul, Jerome L. (2005), Algorithms: Sequential, Parallel, and Distributed, Boston: Course Technology,p. 822, ISBN 0-534-42057-5

[10] Goldrei, Derek (1996), Classic Set Theory, London: Chapman and Hall, p. 5, ISBN 0-412-60610-0

[11] Nielsen, Michael A; Chuang, Isaac L (2000), Quantum Computation and Quantum Information, New York: CambridgeUniversity Press, pp. 6970, ISBN 0-521-63503-9, OCLC 43641333

[12] Nielsen, Michael A; Chuang, Isaac L (2000), Quantum Computation and Quantum Information, New York: CambridgeUniversity Press, pp. 7172, ISBN 0-521-63503-9, OCLC 43641333

[13] M. Benatia, A. Lazrik, and K. Sami, "Arabic mathematical symbols in Unicode", 27th Internationalization and UnicodeConference, 2005.

11.11 External links The complete set of mathematics Unicode characters Je Miller: Earliest Uses of Various Mathematical Symbols Numericana: Scientic Symbols and Icons GIF and PNG Images for Math Symbols Mathematical Symbols in Unicode Using Greek and special characters from Symbol font in HTML Unicode Math Symbols - a quick form for using unicode math symbols. DeTeXify handwritten symbol recognition doodle a symbol in the box, and the program will tell you whatits name is

Some Unicode charts of mathematical operators:

Index of Unicode symbols Range 2100214F: Unicode Letterlike Symbols Range 219021FF: Unicode Arrows Range 220022FF: Unicode Mathematical Operators Range 27C027EF: Unicode Miscellaneous Mathematical SymbolsA

11.11. EXTERNAL LINKS 39

Range 298029FF: Unicode Miscellaneous Mathematical SymbolsB Range 2A002AFF: Unicode Supplementary Mathematical Operators

Some Unicode cross-references:

Short list of commonly used LaTeX symbols and Comprehensive LaTeX Symbol List MathML Characters - sorts out Unicode, HTML and MathML/TeX names on one page Unicode values and MathML names Unicode values and Postscript names from the source code for Ghostscript

Chapter 12

List of mathematical symbols by subject

This list of mathematical symbols by subject shows a selection of the most common symbols that are used inmodern mathematical notation within formulas, grouped by mathematical topic. As it is virtually impossible to listall the symbols ever used in mathematics, only those symbols which occur often in mathematics or mathematicseducation are included. Many of the characters are standardized, for example in DIN 1302 General mathematicalsymbols or DIN EN ISO 80000-2 Quantities and units Part 2: Mathematical signs for science and technology.The following list is largely limited to non-alphanumeric characters. It is divided by areas of mathematics and groupedwithin sub-regions. Some symbols have a dierent meaning depending on the context and appear accordingly severaltimes in the list. Further information on the symbols and their meaning can be found in the respective linked articles.

12.1 GuideThe following information is provided for each mathematical symbol:

Symbol: The symbol as it is represented by LaTeX. If there are several typographic variants, only one of thevariants is shown.

Usage: An exemplary use of the symbol in a formula. Letters here stand as a placeholder for numbers, variablesor complex expressions. Dierent possible applications are listed separately.

Interpretation: A short textual description of the meaning of the formula in the previous column.

Article: The Wikipedia article that discusses the meaning (semantics) of the symbol.

LaTeX: The LaTeX command that creates the icon. Characters from the ASCII character set can be useddirectly, with a few exceptions (pound sign #, backslash \, braces {}, and percent sign %). High-and low-position is indicated via the characters ^ and _ and is not explicitly specied.

HTML: The icon in HTML, if it is dened as a named mark. Non-named characters can be indicated in theform can nnnn by specifying the Unicode code point of the next column. High-and low-position can beindicated via < sup > and < sub > .

Unicode: The code point of the corresponding Unicode character. Some characters are combining and requirethe entry of additional characters. For brackets, the code points of the opening and the closing forms arespecied.

12.2 Set theory

40

12.3. ARITHMETIC 41

12.2.1 Denition symbols

12.2.2 Set construction

12.2.3 Set operations

12.2.4 Set relations

Note: The symbols and are used inconsistently and often do not exclude the equality of the two quantities.

12.2.5 Number sets

12.2.6 Cardinality

12.3 Arithmetic

12.3.1 Arithmetic operators

12.3.2 Equality signs

See also: Equals sign

12.3.3 Comparison

12.3.4 Divisibility

12.3.5 Intervals

12.3.6 Elementary functions

Note: the power function is not represented by its own icon, but by the positioning of the exponent as a superscript.

12.3.7 Complex numbers

Remark: real and complex parts of a complex number are often also denoted by Re and Im .

12.3.8 Mathematical constants

See also: mathematical constant for symbols of additional mathematical constants.

12.4 Calculus

42 CHAPTER 12. LIST OF MATHEMATICAL SYMBOLS BY SUBJECT

12.4.1 Sequences and series

12.4.2 Functions

12.4.3 Limits

12.4.4 Asymptotic behaviour

12.4.5 Dierential calculus

12.4.6 Integral calculus

See also: Extensions of the integral symbol

12.4.7 Vector calculus

12.4.8 Topology

12.4.9 Functional analysis

12.5 Linear algebra and geometry

12.5.1 Elementary geometry

12.5.2 Vectors and matrices

12.5.3 Vector calculus

12.5.4 Matrix calculus

12.5.5 Vector spaces

12.6 Algebra

12.6.1 Relations

12.6.2 Group theory

12.6.3 Field theory

12.6.4 Ring theory

12.7 Combinatorics

12.8 Stochastics

12.8.1 Probability theory

Remark: for operators there are several notational variants; instead of round brackets also square bracketsare used

12.9. LOGIC 43

12.8.2 Statistics

12.9 Logic

12.9.1 OperatorsSee also: Further symbols for binary connectives

12.9.2 Quantiers

12.9.3 Deduction symbols

12.10 See also List of mathematical symbols Greek letters used in mathematics, science, and engineering ISO 31-11 (Mathematical signs and symbols for use in physical sciences and technology) Latin letters used in mathematics List of mathematical abbreviations Mathematical alphanumeric symbols Mathematical constants and functions Mathematical notation Mathematical operators and symbols in Unicode Notation in probability and statistics Physical constants Table of logic symbols Table of mathematical symbols by introduction date Unicode block

12.1