قدر مطلق (ریاضی)

${\displaystyle |a|={\begin{cases}a,&{\mbox{if}}a>0\\-a,&{\mbox{if}}a<0.\end{cases}}}$

${\displaystyle |a|={\sqrt {a^{2}}}}$

${\displaystyle (1)}$

${\displaystyle |a|\geq 0}$	${\displaystyle (2)}$	نامنفی بودن
${\displaystyle |a|=0\iff a=0}$	${\displaystyle (3)}$	صفر بودن
${\displaystyle |ab|=|a||b|\,}$	${\displaystyle (4)}$	ضرب‌پذیری
${\displaystyle |a+b|\leq |a|+|b|}$	${\displaystyle (5)}$	جمع‌پذیری

${\displaystyle |-a|=|a|\,}$	${\displaystyle (6)}$	تقارن
${\displaystyle |a-b|=0\iff a=b}$	${\displaystyle (7)}$	گرفته شده از صفر بودن
${\displaystyle |a-b|\leq |a-c|+|c-b|}$	${\displaystyle (8)}$	نامساوی مثلث گرفته شده از جمع‌پذیری
${\displaystyle |a/b|=|a|/|b|{\mbox{ (if}}b\neq 0)\,}$	${\displaystyle (9)}$	تقسیم پذیری گرفته شده از ضرب‌پذیری
${\displaystyle |a-b|\geq ||a|-|b||}$	${\displaystyle (10)}$

${\displaystyle |a|\leq b\iff -b\leq a\leq b}$

${\displaystyle |a|\geq b\iff a\leq -b{\mbox{ or}}b\leq a}$

${\displaystyle |x-3|\leq 9}$	${\displaystyle \iff -9\leq x-3\leq 9}$
	${\displaystyle \iff -6\leq x\leq 12}$

${\displaystyle |a|={\sqrt {a^{2}}}}$

${\displaystyle z=x+iy}$

${\displaystyle |z|={\sqrt {x^{2}+y^{2}}}}$

${\displaystyle |x+i0|={\sqrt {x^{2}+0^{2}}}={\sqrt {x^{2}}}=|x|}$

${\displaystyle z=x+iy=r(\cos \phi +i\sin \phi )}$

${\displaystyle {\overline {z}}=x-iy}$

${\displaystyle {\begin{aligned}|z|&=r,\\|z|&=|{\overline {z}}|\end{aligned}}}$

${\displaystyle |z|={\sqrt {\overline {z}}}}$

${\displaystyle |z|^{2}=z{\overline {z}}=x^{2}+y^{2}}$

${\displaystyle \operatorname {sgn}(x)={\frac {x}{|x|}},}$

${\displaystyle u(x)={\begin{cases}0,&x<0\\{\frac {1}{2}},&x=0\\1,&x>0,\end{cases}}}$

${\displaystyle u(x)={\frac {\operatorname {sgn}(x)+1}{2}}.\,}$

${\displaystyle \int |x|dx={\frac {x|x|}{2}}+C,}$

${\displaystyle {\int |x|dx=x|x|-\int {\frac {x^{2}}{|x|}}dx=x|x|-\int |x|dx\iff 2\int |x|dx=x|x|\iff \int |x|dx={\frac {x|x|}{2}}+C.}}$

${\displaystyle a=(a_{1},a_{2},\dots ,a_{n})}$

${\displaystyle b=(b_{1},b_{2},\dots ,b_{n})}$

${\displaystyle {\sqrt {(a_{1}-b_{1})^{2}+(a_{2}-b_{2})^{2}+\cdots +(a_{n}-b_{n})^{2}}}.}$

${\displaystyle |a-b|={\sqrt {(a-b)^{2}}}.}$

${\displaystyle a=a_{1}+ia_{2}\,}$

${\displaystyle b=b_{1}+ib_{2}\,}$

${\displaystyle |a-b|\,}$	${\displaystyle =|(a_{1}+ia_{2})-(b_{1}+ib_{2})|\,}$
	${\displaystyle =|(a_{1}-b_{1})+i(a_{2}-b_{2})|\,}$
	${\displaystyle ={\sqrt {(a_{1}-b_{1})^{2}+(a_{2}-b_{2})^{2}}}.}$

${\displaystyle d(a,b)\geq 0}$	نامنفی بودن
${\displaystyle d(a,b)=0\iff a=b}$	نشان دهندهٔ تساوی با عدد دیگر
${\displaystyle d(a,b)=d(b,a)\,}$	جابجایی
${\displaystyle d(a,b)\leq d(a,c)+d(c,b)}$	نامساوی مثلثی

${\displaystyle |a|={\begin{cases}a,&{\mbox{if}}a\geq 0\\-a,&{\mbox{if}}a<0,\end{cases}}}$

${\displaystyle v(a)\geq 0}$	نامنفی بودن
${\displaystyle v(a)=0\iff a=\mathbf {0} }$	صفر بودن
${\displaystyle v(ab)=v(a)v(b)\,}$	تفکیک‌پذیری در ضرب
${\displaystyle v(a+b)\leq v(a)+v(b)}$	نامساوی مثلثی

d(a, b) = v(a − b)

• d(x, y) < max{d(x, z), d(y, z)}
• ${\displaystyle {\big \{}v{\Big (}{\textstyle \sum _{k=1}^{n}}\mathbf {1} {\Big )}:n\in \mathbb {N} {\big \}}}$ is bounded in R
• ${\displaystyle v{\Big (}{\textstyle \sum _{k=1}^{n}}\mathbf {1} {\Big )}\leq 1{\text{ for every}}n\in \mathbb {N} .}$
• ${\displaystyle v(a)\leq 1\Rightarrow v(1+a)\leq 1{\text{ for all}}a\in F.}$
• ${\displaystyle v(a+b)\leq \mathrm {max} \{v(a),v(b)\}{\text{ for all}}a,b\in F.}$