Basic operations, relations

This table describes how to type basic operation and relation symbols, along with usage examples (including the typeset equivalent.) If the expression you need is not listed, please define your own and explain the meaning of your notation.

Expression Description Examples

+ Addition

2 + 3 = 5

- Subtraction / Negation

2 - 3 = -1

* Multiplication $2 \times 3=6$
$2\cdot 3=6$

2 * 3 = 6

Note--Multiplication is also implied by juxtaposition. $xy=x \cdot y$
$3x=3 \cdot x$

xy = x*y 3x = 3*x

/ Division $\frac{4}{2}=2$
$4 \div 2=2$

4/2 = 2

= Equal

x = 1

:= Is defined to be equal to $\pi\equiv\frac{C}{D}$
$\pi\overset{\Delta}{=}\frac{C}{D}$
$\pi :=\frac{C}{D}$

pi := C/D

~ Approximately equal $\pi \approx \frac{22}{7}$

pi ~ 22/7

<> or != Not equal $4 \ne x+2,\:$ for $\: x \ne 2$

4 <> x+2, for x <> 2
4 != x+2, for x != 2

> Greater than

4 > 2

< Less than

2 < 4

>= Greater than or equal $2+x \ge 4,\:$ for $\: x \ge 2$

2+x >= 4, for x >= 2

=< Less than or equal $4 \le x+2,\:$ for $\: 2 \le x$

4 =< x+2, for 2 =< x

+/- Plus or minus. Indicates the positive and negative. $x=\pm 1 \; \Rightarrow \; x^2=1$

x = (+/-)1 ==> x^2 =1

^ Exponentiation

2^2 = 4

sqrt() Denotes the principal (nonnegative) square root of the contents of ().

Use +/- to denote both roots. $\sqrt{4}=2$
$\pm\sqrt{4}=\pm 2$

sqrt(4) = 2 (+/-)sqrt(4) = (+/-)2

cbrt(m) Denotes the cube root of m $\sqrt[3]{8}=2$

cbrt(8) = 2

[n]root(m) Denotes the nth root of m $\sqrt[5]{32}=2$

[5]root(32) = 2

! Factorial $3\boldsymbol{!}=3 \times 2 \times 1$

3! = 3*2*1

: to (ratio) The ratio of boys to girls is 4:5

|x| |z|, mod z |v| This notation is used for several related concepts.

For a real number x it denotes the absolute value of x, where |x|=sqrt(x^2).

For a complex number z=x+iy it denotes the modulus of z (so does mod z), where |z|=sqrt(x^2+y^2).

It also denotes the length, or norm, of a vector v=<x,y>, where |v|=sqrt(x^2+y^2).

The norm of v can also be denoted by ||v||, as described in a later table. $\left\lvert-7 \right\rvert =7$

$z=2-3i, \; \left\lvert z \right\rvert =\sqrt{13}$

$\mathbf{v}=\langle -5,12 \rangle, \; \left\lvert\mathbf{v} \right\rvert =13$ .

|-7| = 7

z=2-3i, |z|=sqrt(13)

v=<-5,12>, |v|=13

arg(z) For a complex number of the form
z = r[cos(theta)+i*sin(theta)],
denotes the argument of z where arg(z)=theta. $z=2(\cos{\frac{\pi}{3}}+i\sin{\frac{\pi}{3}})$
$\arg{z}=\frac{\pi}{3}$

z=2[cos(pi/3)+i*sin(pi/3)]
arg(z) = pi/3

[[x]] or floor(x) Denotes the greatest integer of x (also called floor x), where
[[x]] = the largest integer $\le x.$ $\pmb{[[}-3.2\pmb{]]}=-4$
$\pmb{[[}3.2\pmb{]]}=3$
$\lfloor -3.2 \rfloor=-4$
$\lfloor 3.2 \rfloor=3$

[[-3.2]] = -4
[[3.2]] = 3
floor(-3.2) = -4
floor(3.2) = 3

| Divides $2\vert 4$

2|4 (2 divides 4)

==> Implies $x \ge 4 \Rightarrow x > 2$

x >= 4 ==> x > 2

<== Is implied by $x > 2 \Leftarrow x \ge 4$

x > 2 <== x >= 4

<==> or iff Implies and is implied by.

If and only if. $x=2 \iff 2x=4$

x = 2 <==> 2x = 4 x = 2 iff 2x = 4

aah@ryan-usa.com
Giganews Newsgroups

Expression	Description	Examples
`+`	Addition	`2 + 3 = 5`
`-`	Subtraction / Negation	`2 - 3 = -1`
`*`	Multiplication	$2 \times 3=6$ $2\cdot 3=6$ `2 * 3 = 6`
	Note--Multiplication is also implied by juxtaposition.	$xy=x \cdot y$ $3x=3 \cdot x$ `xy = xy 3x = 3x`
`/`	Division	$\frac{4}{2}=2$ $4 \div 2=2$ `4/2 = 2`
`=`	Equal	`x = 1`
`:=`	Is defined to be equal to	$\pi\equiv\frac{C}{D}$ $\pi\overset{\Delta}{=}\frac{C}{D}$ $\pi :=\frac{C}{D}$ `pi := C/D`
`~`	Approximately equal	$\pi \approx \frac{22}{7}$ `pi ~ 22/7`
`<> or !=`	Not equal	$4 \ne x+2,\:$ for $\: x \ne 2$ `4 <> x+2, for x <> 2` `4 != x+2, for x != 2`
`>`	Greater than	`4 > 2`
`<`	Less than	`2 < 4`
`>=`	Greater than or equal	$2+x \ge 4,\:$ for $\: x \ge 2$ `2+x >= 4, for x >= 2`
`=<`	Less than or equal	$4 \le x+2,\:$ for $\: 2 \le x$ `4 =< x+2, for 2 =< x`
`+/-`	Plus or minus. Indicates the positive and negative.	$x=\pm 1 \; \Rightarrow \; x^2=1$ `x = (+/-)1 ==> x^2 =1`
`^`	Exponentiation	`2^2 = 4`
`sqrt()`	Denotes the principal (nonnegative) square root of the contents of (). Use +/- to denote both roots.	$\sqrt{4}=2$ $\pm\sqrt{4}=\pm 2$ `sqrt(4) = 2 (+/-)sqrt(4) = (+/-)2`
`cbrt(m)`	Denotes the cube root of m	$\sqrt[3]{8}=2$ `cbrt(8) = 2`
`[n]root(m)`	Denotes the nth root of m	$\sqrt[5]{32}=2$ `[5]root(32) = 2`
`!`	Factorial	$3\boldsymbol{!}=3 \times 2 \times 1$ `3! = 321`
`:`	to (ratio)	`The ratio of boys to girls is 4:5`
`\|x\| \|z\|, mod z \|v\|`	This notation is used for several related concepts. For a real number `x` it denotes the absolute value of `x`, where `\|x\|=sqrt(x^2)`. For a complex number `z=x+iy` it denotes the modulus of `z` (so does `mod z`), where `\|z\|=sqrt(x^2+y^2)`. It also denotes the length, or norm, of a vector `v=<x,y>`, where `\|v\|=sqrt(x^2+y^2)`. The norm of `v` can also be denoted by `\|\|v\|\|`, as described in a later table.	$\left\lvert-7 \right\rvert =7$ $z=2-3i, \; \left\lvert z \right\rvert =\sqrt{13}$ $\mathbf{v}=\langle -5,12 \rangle, \; \left\lvert\mathbf{v} \right\rvert =13$ . `\|-7\| = 7` `z=2-3i, \|z\|=sqrt(13)` `v=<-5,12>, \|v\|=13`
`arg(z)`	For a complex number of the form `z = r[cos(theta)+i*sin(theta)]`, denotes the argument of `z` where `arg(z)=theta`.	$z=2(\cos{\frac{\pi}{3}}+i\sin{\frac{\pi}{3}})$ $\arg{z}=\frac{\pi}{3}$ `z=2[cos(pi/3)+i*sin(pi/3)]` `arg(z) = pi/3`
`[[x]] or floor(x)`	Denotes the greatest integer of x (also called `floor x`), where `[[x]]` = the largest integer $\le x.$	$\pmb{[[}-3.2\pmb{]]}=-4$ $\pmb{[[}3.2\pmb{]]}=3$ $\lfloor -3.2 \rfloor=-4$ $\lfloor 3.2 \rfloor=3$ `[[-3.2]] = -4` `[[3.2]] = 3` `floor(-3.2) = -4` `floor(3.2) = 3`
`\|`	Divides	$2\vert 4$ `2\|4 (2 divides 4)`
`==>`	Implies	$x \ge 4 \Rightarrow x > 2$ `x >= 4 ==> x > 2`
`<==`	Is implied by	$x > 2 \Leftarrow x \ge 4$ `x > 2 <== x >= 4`
`<==> or iff`	Implies and is implied by. If and only if.	$x=2 \iff 2x=4$ `x = 2 <==> 2x = 4 x = 2 iff 2x = 4`