قدر مطلق (ریاضی)

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

${\displaystyle (1)}$

${\displaystyle |a|\ge 0}$	${\displaystyle (2)}$	نامنفی بودن
${\displaystyle |a|=0⟺a=0}$	${\displaystyle (3)}$	صفر بودن
${\displaystyle |ab|=|a||b|}$	${\displaystyle (4)}$	ضرب‌پذیری
${\displaystyle |a+b|\le |a|+|b|}$	${\displaystyle (5)}$	جمع‌پذیری

${\displaystyle |-a|=|a|}$	${\displaystyle (6)}$	تقارن
${\displaystyle |a-b|=0⟺a=b}$	${\displaystyle (7)}$	گرفته شده از صفر بودن
${\displaystyle |a-b|\le |a-c|+|c-b|}$	${\displaystyle (8)}$	نامساوی مثلث گرفته شده از جمع‌پذیری
${\displaystyle |a/b|=|a|/|b|\text{ (if}b\ne 0)}$	${\displaystyle (9)}$	تقسیم پذیری گرفته شده از ضرب‌پذیری
${\displaystyle |a-b|\ge ||a|-|b||}$	${\displaystyle (10)}$

${\displaystyle |a|\le b⟺-b\le a\le b}$

${\displaystyle |a|\ge b⟺a\le -b\text{ or}b\le a}$

${\displaystyle |x-3|\le 9}$	${\displaystyle ⟺-9\le x-3\le 9}$
	${\displaystyle ⟺-6\le x\le 12}$

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

${\displaystyle z=x+iy}$

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

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

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

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

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

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

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

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

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

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

${\displaystyle a=(a_1,a_2,\dots ,a_n)}$

${\displaystyle b=(b_1,b_2,\dots ,b_n)}$

${\displaystyle \sqrt{(a_1-b_1{)}^2+(a_2-b_2{)}^2+\cdots +(a_n-b_n{)}^2}.}$

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

${\displaystyle a=a_1+i{a}_2}$

${\displaystyle b=b_1+i{b}_2}$

${\displaystyle |a-b|}$	${\displaystyle =|(a_1+i{a}_2)-(b_1+i{b}_2)|}$
	${\displaystyle =|(a_1-b_1)+i(a_2-b_2)|}$
	${\displaystyle =\sqrt{(a_1-b_1{)}^2+(a_2-b_2{)}^2}.}$

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

${\displaystyle v(a)\ge 0}$	نامنفی بودن
${\displaystyle v(a)=0⟺a=\mathbf{0}}$	صفر بودن
${\displaystyle v(ab)=v(a)v(b)}$	تفکیک‌پذیری در ضرب
${\displaystyle v(a+b)\le v(a)+v(b)}$	نامساوی مثلثی

d(a, b) = v(a − b)

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