# Mathematic Symbols

This is a list of mathematic symbols:

### Algebra Symbols

Symbol | Definition | Example |
---|---|---|

\(=\) | equal to | \(5+4=9\) |

\(\ne\) | not equal to | \(5\ne4\) |

\(\equiv\) | identical to | \(a \equiv b\) |

\(\not\equiv\) | not identical to | \(a \not\equiv b\) |

\(\sim\) | similar to | \(a \sim b\) |

\(\approx\) | approximately equal to | \(a \approx b\) |

\(>\) | greater than | \(5>4\) |

\(\gg\) | much greater than | \(50000 \gg 4\) |

\(<\) | less than | \(4<5\) |

\(\ll\) | much less than | \(4 \ll 50000\) |

\(\ge\) | greater than or equal to | \(a \ge b\) |

\(\le\) | less than or equal to | \(a \le b\) |

\(\Rightarrow\) | implies if...then | \(a+b+c=1\;\Rightarrow\;c+b+a=1\) |

\(\Leftrightarrow\) | is equivalent to if in only if (iff) | \(a=b+1\;\Leftrightarrow\;b=a+1\) |

\(\therefore\) | therefore | \( a=b\; \therefore\; b=a \) |

\( \sum\) | sigma, summation or sum of all | \( \sum_{n=4}^8 n = 4+5+6+7+8=30 \) |

\(\left(\; \right)\) | parentheses, calculate expression inside first | \( 5\times \left(5+4\right) = 45\) |

\(\left[\; \right]\) | brackets, calculate expression inside first | \( \left[ \left(5+4\right) \times \left(9+4\right) \right] = 117\) |

\(\{ \; \}\) | set, a collection | \(A = \{4, 5, 9 \} \) |

\(\lfloor x \rfloor\) | rounds number to lower integer | \(\lfloor 5.4 \rfloor = 5\) |

\(\lceil x \rceil\) | rounds number to upper integer | \(\lceil 5.4 \rceil = 6\) |

\(\left | x \right |\) | absolute value | \(\left | -5 \right | = 5 \) |

\(\left \| x \right \|\) | normal | |

\(x!\) | factorial | \(5!=5\;x\;4\;x\;3\;x\;2\;x\;1=120 \) |

\(a^n\) | power, an exponent | \(5^4 = 3125\) |

\(a\wedge b\) | caret, an exponent | \(5\wedge4 = 3125\) |

\( \sqrt a\) | Square root - \(\sqrt a \;\times\; \sqrt a = a\) | \(\sqrt 9= 3\) |

\(^3\sqrt a\) | cube root - \(^3\sqrt a \;\times\; ^3\sqrt a \;\times\; ^3\sqrt a = a\) | \(^3\sqrt 9 = 2.08008382305\) |

\(^4\sqrt a\) | 4th root - \(^4\sqrt a \;\times\; ^4\sqrt a \;\times\; ^4\sqrt a \;\times\; ^4\sqrt a = a\) | \(^4\sqrt 9 = 1.73205080757\) |

\( ^n\sqrt a\) | n-th root | \(n=5\;\), \(\;^n\sqrt 9 = 1.55184557391\) |

\(\gamma\) | Euler-Mascheroni constant | \(\gamma = 0.5772156649...\) |

\(\Phi\) | golden ratio | \(1 : 1.6180339887...\) |

\(\pi\) | \(\pi = 3.141592654... \) | \(C = \pi \cdot d = 2 \cdot \pi \cdot r\) |

### Angle and Line Symbols

Symbol | Definition | Example |
---|---|---|

\(\triangle\) | triangle | \(\triangle ABC \) |

\(\angle\) | angle | \(\angle ABC = 60^\circ\) |

\(\measuredangle\) | measured angle | \(\angle ABC = 60^\circ\) |

\(\sphericalangle\) | spherical angle | \(\angle AOC = 60^\circ\) |

\(\overleftrightarrow {AB}\) | infinite line distance | |

\(\overline {AB}\) | line segment from A to B | |

\(\overrightarrow {AB}\) | start line at point A | |

\(\overset{\frown} {AB}\) | arc from A to B | |

\(|A-B|\) | distance between A and B | \(|A-B| = 9\) |

\(\parallel\) | parallel to | \(\overline {AB} \parallel \overline {XY} \) |

\(\nparallel\) | not parallel to | \(\overline {AB} \nparallel \overline {XY} \) |

\(\perp\) | perpendicular lines | \(\overline {AB} \perp \overline {XY} \) |

### Basic Math Symbols

Symbol | Definition | Example |
---|---|---|

\(+\) | addition | \(5+4=9\) |

\(-\) | subtraction | \(5-4=1\) |

\(\mp\) | plus - minus, both plus and minus operations | \(5\mp4=9\) and \(1\) |

\(\pm\) | minus - plus, both minus and plus operations | \(5\mp4=1\) and \(9\) |

\(*\) | multiplication | \(5*4=20\) |

\(\times\) | multiplication | \(5\times4=20\) |

\(\bullet\) | multiplication | \(5\bullet4=20\) |

\(\div\) | division | \(5\div 4=1.25\) |

\(/\) | division | \(5/4=1.25\) |

\(-\) | horizontal line is for division / fraction | \(\frac {5} {4} =1.25\) |

\(.\) | decimal point | \(5.4\) |

\(=\) | equal | \(5+4=9\) |

\(\left(\; \right)\) | parentheses, calculate expression inside first | \( 5\times \left(5+4\right) = 45\) |

\(\left[\; \right]\) | brackets, calculate expression inside first | \( \left[ \left(5+4\right) \times \left(9+4\right) \right] = 117\) |

\(a^n\) | power, an exponent | \(5^4 = 3125\) |

\( \sqrt a\) | Square root - \(\sqrt a \;\times\; \sqrt a = a\) | \(\sqrt 9= 3\) |

% | percent, \(1\)% \(= 1/100 \) | \(5\)% \(\times 4 = 0.2 \) |

### Bracket Symbols

Symbol | Definition | Example |
---|---|---|

\(\left(\; \right)\) | parentheses, calculate expression inside first | \( 5\times \left(5+4\right) = 45\) |

\(\left[\; \right]\) | brackets, calculate expression inside first | \( \left[ \left(5+4\right) \times \left(9+4\right) \right] = 117\) |

\(\{ \; \}\) | set, a collection | \(a = {4, 5, 9 } \) |

\(\lfloor a \rfloor\) | rounds number to lower integer | \(\lfloor 5.4 \rfloor = 5\) |

\(\lceil a \rceil\) | rounds number to upper integer | \(\lceil 5.4 \rceil = 6\) |

\(\left | a \right |\) | absolute value | \(\left | -5 \right | = 5 \) |

\(\left \| x \right \|\) | normal |

### Equivalence Symbols

Symbol | Definition | Example |
---|---|---|

\(=\) | equal to | \(5+4=9\) |

\(\ne\) | not equal to | \(5\ne4\) |

\(\equiv\) | identical to | \(a \equiv b\) |

\(\not\equiv\) | not identical to | \(a \not\equiv b\) |

\(\overset{\underset{\mathrm{\Delta}}{}}{=}\) | delta equal to | |

\(\overset{\underset{\mathrm{def}}{}}{=}\) | equal to by defination | \(a \overset{\underset{\mathrm{def}}{}}{=} b\) |

\(\overset{\underset{\mathrm{m}}{}}{=}\) | measured by | \(a \overset{\underset{\mathrm{m}}{}}{=} b\) |

\(\overset{\underset{\mathrm{?}}{}}{=}\) | questioned equal to | \(a \overset{\underset{\mathrm{?}}{}}{=} b\) |

\(\sim\) | similar to | \(a \sim b\) |

\(\approx\) | approximately equal to | \(a \approx b\) |

\(\cong\) | congruent, equivalent in size and shape | \(\triangle ABC \cong \triangle XYZ\) |

\(\ncong\) | not equivalent in size and shape | \(\triangle ABC \ncong \triangle XYZ\) |

\(:=\) | is defined to be | \(a := \{2, 4, 6, 8 \}\;\) means \(\;a\;\) is defined to be set \(\;\{2, 4, 6, 8 \} \) |

\(\therefore\) | therefore | \( a=b\; \therefore\; b=a \) |

\(\because\) | because | \( a=b\; \because\; b=a \) |

\(>\) | greater than | \(5>4\) |

\(\gg\) | much greater than | \(50000 \gg 4\) |

\(<\) | less than | \(4<5\) |

\(\ll\) | much less than | \(4 \ll 50000\) |

\(\ge\) | greater than or equal to | \(a\ge b\) |

\(\le\) | less than or equal to | \(a\le b\) |

\(\geqq\) | greater than over equal to | \(a\geqq b\) |

\(\leqq\) | less than over equal to | \(a\leqq b\) |

\(\gneqq\) | greater than but not equal to | \(a\gneqq b\) |

\(\lneqq\) | less than but not equal to | \(a\lneqq b\) |

\(\Rightarrow\) | implies if then - \(\; a \Rightarrow b\;\) means if \(\;a\;\) is true then \(\;b\;\) is also true, if \(\;a\;\) is false then nothing is said about \(\;b \) | \( a = 3 \Rightarrow a3 = 9\;\) is true, but \(\;a3 = 9 \Rightarrow a = 3\;\) is in general false since \(\;a\;\) could be \(\;−3\) |

\(\rightarrow\) | same as above | same as above |

\(\Leftrightarrow\) | if and only if - \(\;a \Leftrightarrow b\;\) means \(\;a\;\) is true if \(\;b\;\) is true and \(\;a\;\) is false if \(\;b\;\) is false | \(a + 2 = b - 5 \Leftrightarrow a = b - 7\) |

\(\leftrightarrow\) | same as above | same as above |

### Geometry Symbols

Symbol | Definition | Example |
---|---|---|

\(\triangle\) | triangle | \(\triangle ABC \) |

\(\bigcirc\) | circle | |

\(\odot A\) | circle with center A | |

\(\angle\) | angle | \(\angle ABC = 60^\circ\) |

\(\measuredangle\) | measured angle | \(\measuredangle ABC = 60^\circ\) |

\(\sphericalangle\) | spherical angle | \(\sphericalangle AOC = 60^\circ\) |

\(^\circ\) | degree | 1 circle \(= 360^\circ\) |

' | arcminute | \(1^\circ = 60^\prime\) |

" | arcsecond | \(1'=60^{\prime\prime}\) |

\(r \;or\; rad\) | radiant, \(1 \;rad = 180^\circ /\pi \;\) and \(\;1^\circ = \pi / 180 \;rads\) | \(360^\circ = 2\pi\; rad\) or about \(57.2958^\circ\) |

\(g \;or\; grad\) | gradian, four hundredth (1/400) of a full circle | \( 360^\circ = 400\; grad\) |

\(\overleftrightarrow {AB}\) | infinite line distance | |

\(\overline {AB}\) | line segment from endpoint A to B | |

\(\overrightarrow{AB}\) | start line at point A | |

\(\overset{\frown}{AB}\) | arc with endpoints A and B | |

\(\overset{\frown}{ABC}\) | arc with endpoints A and C | |

m\(\overset{\frown}{AB}\) | measure arc with endpoints A and B | |

\(|A-B|\) | distance between points A and B | \(|A-B| = 9\) |

\(\parallel\) | parallel to | \(\overline {AB} \parallel \overline {XY} \) |

\(\nparallel\) | not parallel to | \(\overline {AB} \nparallel \overline {XY} \) |

\(\perp\) | perpendicular lines | \(\overline {AB} \perp \overline {XY} \) |

\(\sim\) | similarity to | \(\triangle ABC \sim \triangle XYZ\) |

\(\cong\) | congruent, equivalent in size and shape | \(\triangle ABC \cong \triangle XYZ\) |

\(\ncong\) | is not congruent to | \(\triangle ABC \ncong \triangle XYZ\) |

\(\therefore\) | therefore | \( a=b\; \therefore\; b=a \) |

\(\pi\) | \(\pi = 3.141592654... \) | \(C = \pi \cdot d = 2 \cdot \pi \cdot r\) |

### Set Symbols

Symbol | Definition | Example |
---|---|---|

\(\{ \; \}\) | set, a collection | \(A= \{ 1, 2, 3, 4 \}\) , \(B= \{ 3, 4, 5, 6 \} \) |

\(\varnothing\) | empty set | \(A=\{ \varnothing\} \) |

\(\cap\) | intersection, belonging to set A or B | \(A\cap B =\{3, 4\}\) |

\(\cup\) | union, belonging to set A or B | \(A\cup B =\{1, 2, 3, 4, 5, 6\}\) |

\(\subset\) | strict subset, A is subset of B | \(\{3, 4\} \subset \{3, 4, 5, 6\}\) |

\(\subseteq\) | subset, A subset of B, A included in B | \(\{3, 4\} \subseteq \{3, 4\}\) |

\(\nsubseteq\) | not subset, A not subset of B | \(\{6, 7\} \nsubseteq \{3, 4, 5, 6\}\) |

\(\supset\) | strict superset, A superset of B, B not equal to A | \(\{3, 4, 5, 6\} \supset \{3, 4\}\) |

\(\supseteq\) | superset, A subset of B, A includes B | \(\{3, 4, 5, 6\} \supseteq \{3, 4, 5, 6\}\) |

\(\nsupseteq\) | not superset, A not superset of B | \(\{3, 4, 5, 6\} \nsupseteq \{6, 7\}\) |

\(\in\) | belongs to | \(B=\{3, 4, 5, 6\}\) , \(3\in B\) |

\(\notin\) | does not belong to | \(B=\{3, 4, 5, 6\}\) , \(1\notin B\) |

= | equality, both sets the same A=B | \(\{3, 4, 5, 6\} = \{3, 4, 5, 6\}\) |

\(-\) | relative complement, belongs to B but not A | \(A-B = \{5, 6\}\) |

\(\ominus\) | symmetric difference, belongs to A or B gut no matches | \(A \ominus B = \{1, 2, 5, 6\}\) |

\(|\;|\) | cardinality, element of set B | \(|B|=\{3\}\) |