Formelsamling/Matematik/Algebra

Räknelagar

$a+b=b+a\,\!$	(kommutativa lagen under addition)
$a\cdot b=b\cdot a\,\!$	(kommutativa lagen under multiplikation)
$(a+b)+c=a+(b+c)\,\!$	(associativa lagen under addition)
$(a\cdot b)\cdot c=a\cdot (b\cdot c)\,\!$	(associativa lagen under multiplikation)
$a\cdot (b+c)=a\cdot b+a\cdot c\,\!$	(distributiva lagen)
$a+c=b+c\ \Leftrightarrow \ a=b\,\!$	(annulleringslagen under addition)
$a\cdot c=b\cdot c\ \Leftrightarrow \ a=b\quad om\ c\neq 0\,\!$	(annulleringslagen under multiplikation)

Bråkregler

$a\cdot {\frac {b}{c}}={\frac {a}{1}}\cdot {\frac {b}{c}}={\frac {ab}{c}}$	$c\neq 0$
${\frac {a}{b}}\cdot {\frac {c}{d}}={\frac {ac}{bd}}$	$b\neq 0,d\neq 0$
${\frac {a}{b}}{\Big /}{\frac {c}{d}}={\frac {\frac {a}{b}}{\frac {c}{d}}}={\frac {a}{b}}\cdot {\frac {d}{c}}={\frac {a}{b}}{\frac {d}{c}}={\frac {ad}{bc}}$	$b\neq 0,c\neq 0,d\neq 0$
${\frac {a}{b}}+{\frac {c}{d}}={\frac {ad}{bd}}+{\frac {bc}{bd}}={\frac {ad+bc}{bd}}$	$b\neq 0,d\neq 0$

Parentesregler

$a+(-b)=a-b\,\!$

$a\cdot b=ab\,\!$

$a-(-b)=a+b\,\!$

$a\cdot (-b)=a(-b)=-ab\,\!$

$(-a)\cdot (-b)=(-a)(-b)=ab\,\!$

Algebra

Låt $a,b,c\in \mathbb {R}$ och $m,n\in \mathbb {Z}$.

$(a+b)^{2}=a^{2}+2ab+b^{2}\,\!$	(första kvadreringsregeln)
$(a-b)^{2}=a^{2}-2ab+b^{2}\,\!$	(andra kvadreringsregeln)
$(a+b)(a-b)=a^{2}-b^{2}\,\!$	(konjugatregeln)
$(a+b)^{3}=a^{3}+3a^{2}b+3ab^{2}+b^{3}\,\!$
$(a-b)^{3}=a^{3}-3a^{2}b+3ab^{2}-b^{3}\,\!$
$a^{3}+b^{3}=(a+b)(a^{2}-ab+b^{2})\,\!$
$a^{3}-b^{3}=(a-b)(a^{2}+ab+b^{2})\,\!$

$n!=(1\cdot 2\cdot 3\cdot ...\cdot n)=\prod _{k=1}^{n}k$	(fakultet)
$(a+b)^{n}=\sum _{k=0}^{n}{n \choose k}a^{n-k}b^{k}=\sum _{k=0}^{n}{\frac {n!}{k!(n-k)!}}a^{n-k}b^{k}$	(binomialteoremet)
$(a_{1}+a_{2}+...+a_{m})^{n}=\sum _{k_{1}+k_{2}+...+k_{m}=n}^{}{\frac {n!}{{k_{1}}!{k_{2}}!...{k_{m}}!}}a_{1}^{k_{1}}a_{2}^{k_{2}}...a_{m}^{k_{m}}$	(multinomialteoremet)

Kvadratkomplettering

$x^{2}+px=x^{2}+px+({\frac {p}{2}})^{2}-({\frac {p}{2}})^{2}=(x+{\frac {p}{2}})^{2}-({\frac {p}{2}})^{2}$

Förstagradsekvationen

$ax+b=0\,\!$	$a\neq 0\,\!$
$x=-{\frac {b}{a}}\,\!$

Andragradsekvationen

Rötterna till andragradsekvationen på formen $x^{2}+px+q=0$ ges av:

$x_{1}=-{\frac {p}{2}}+{\sqrt {\left({\frac {p}{2}}\right)^{2}-q}}\quad och\quad x_{2}=-{\frac {p}{2}}-{\sqrt {\left({\frac {p}{2}}\right)^{2}-q}}$

då gäller

$x_{1}+x_{2}=-p\,\!$

$x_{1}\cdot x_{2}=q\,\!$

Kvadratrötter

För $a\geq 0,\ b\geq 0,\ c>0$:

${\sqrt {a}}\cdot {\sqrt {a}}=a\,\!$

${\sqrt {a}}\cdot {\sqrt {b}}={\sqrt {ab}}\,\!$

$b{\sqrt {a}}={\sqrt {b^{2}a}}\,\!$

${\frac {\sqrt {a}}{\sqrt {c}}}={\sqrt {\frac {a}{c}}}\,\!$

${\frac {a}{\sqrt {c}}}={\frac {a{\sqrt {c}}}{c}}\,\!$

${\sqrt[{n}]{ab}}={\sqrt[{n}]{a}}{\sqrt[{n}]{b}}\,\!$

${\sqrt[{n}]{\frac {a}{c}}}={\frac {\sqrt[{n}]{a}}{\sqrt[{n}]{c}}}\,\!$

${\sqrt[{m}]{\sqrt[{n}]{a}}}={\sqrt[{mn}]{a}}\,\!$

$a{\sqrt[{n}]{b}}={\sqrt[{n}]{a^{n}b}}\,\!$

${\sqrt[{m}]{\sqrt[{n}]{a}}}={\sqrt[{n}]{\sqrt[{m}]{a}}}\,\!$

${\sqrt[{nq}]{a^{mq}}}={\sqrt[{n}]{a^{m}}}\,\!$

Potensregler

$1^{n}=1\,\!$
$a^{n}=\underbrace {a\cdot a\cdot \ldots \cdot a} _{n}$
$a^{1}=a\,\!$
$a^{0}=1\,\!$	$a\neq 0\,\!$
$a^{-n}={\frac {1}{a^{n}}}\,\!$	$a\neq 0\,\!$
$a^{m/n}=(a^{m})^{1/n}={\sqrt[{n}]{a^{m}}}\,\!$	$m,\ n\ >0\,\!$
$a^{m}\cdot a^{n}=a^{m+n}\,\!$
${\frac {a^{m}}{a^{n}}}=a^{m-n}\,\!$	$a\neq 0\,\!$
$(ab)^{m}=a^{m}\cdot b^{m}\,\!$
$\left({\frac {a}{b}}\right)^{m}={\frac {a^{m}}{b^{m}}}\,\!$	$b\neq 0\,\!$
$(a^{m})^{n}=a^{m\cdot n}=(a^{n})^{m}\,\!$

Logaritmer

För $y>0,\ a>0,\ a\neq 1$:

$y=10^{x}\Leftrightarrow x=\log _{10}\ y=\lg \ y\,\!$

$y=a^{x}\Leftrightarrow x=\log _{a}\ y\,\!$

$y=e^{x}\Leftrightarrow x=\ln \ y\,\!$

$\ln \ y=\ln \ 10\cdot \lg \ y\approx 2,3026\ \lg \ y\,\!$

$\lg \ y=\lg \ e\cdot \ln \ y\approx 0,4343\ \ln \ y\,\!$

Logaritmlagar

$a^{\log _{a}x}=x\,\!$
$\log(ab)=\log a+\log b\,\!$	$a>0\ och\ b>0\,\!$
$\log {\frac {a}{b}}=\log a-\log b\,\!$
$\log _{a}a^{n}=n\log a\,\!$
$\log _{a}{\sqrt[{n}]{a}}={\frac {1}{n}}\log a\,\!$
$a^{\frac {\log b}{\log a}}=b\,\!$