Formula collection arithmetic

${\displaystyle {\sqrt[{n}]{x}}}$

This article is a collection of formulas for arithmetic . Math symbols are used, which are explained in the article List of Math Symbols .

notation

• Letters at the beginning of the alphabet stand for any number . ${\displaystyle (a,b,c,\ldots )}$
• Letters in the middle of the alphabet represent natural numbers . ${\displaystyle (i,j,m,n,\ldots )}$
• Letters at the end of the alphabet represent variables . ${\displaystyle (x,y,\ldots )}$
• The operator precedence applies ( point calculation before line calculation ): arithmetic operations of the second level (multiplication and division) bind more strongly than those of the first level (addition and subtraction) and arithmetic operations of the third level (extraction of the roots and exponentiation) are stronger than those of the second level.
• It is the staple rule : standing operations in brackets, they will be executed first. If operations of the same level are consecutive without parentheses, the operations are carried out from left to right.

Basic arithmetic

Arithmetic operations

addition

${\displaystyle a+b=c}$   (Summand + summand = sum)

subtraction

${\displaystyle a-b=c}$   (Minuend - Subtrahend = difference)

multiplication

${\displaystyle a\cdot b=c}$   (Factor factor = product)

division

${\displaystyle a:b=c}$   (Dividend: divisor = quotient)

The division by zero is undefined.

Rules of brackets

${\displaystyle a+(b+c)=a+b+c}$

${\displaystyle a+(b-c)=a+b-c}$

${\displaystyle a-(b+c)=a-b-c}$

${\displaystyle a-(b-c)=a-b+c}$

Laws of Calculation

Associative Laws

${\displaystyle a+\left(b+c\right)=\left(a+b\right)+c}$

${\displaystyle a\cdot \left(b\cdot c\right)=\left(a\cdot b\right)\cdot c}$

Commutative laws

${\displaystyle a+b=b+a\,}$

${\displaystyle a\cdot b=b\cdot a}$

Distributive laws

${\displaystyle a\cdot \left(b+c\right)=a\cdot b+a\cdot c}$

${\displaystyle \left(a+b\right)\cdot c=a\cdot c+b\cdot c}$

Neutrality of and ${\displaystyle 0}$ ${\displaystyle 1}$

${\displaystyle a+0=0+a=a}$

${\displaystyle a\cdot 1=1\cdot a=a}$

Binomial formulas

${\displaystyle (a+b)^{2}=a^{2}+2\cdot a\cdot b+b^{2}}$

${\displaystyle (a-b)^{2}=a^{2}-2\cdot a\cdot b+b^{2}}$

${\displaystyle (a+b)\cdot (a-b)=a^{2}-b^{2}}$

Fractions

Designations

definition

${\displaystyle {\frac {a}{b}}=a:b}$   (Numerator: denominator)

The numerator and denominator are whole numbers, whereby the denominator cannot be zero.

Special cases

• Trunk break: ${\displaystyle a=1}$
• Real break: ${\displaystyle a<b}$
• Improper fraction: ${\displaystyle a>b}$
• Pseudo fraction: with a whole number ${\displaystyle a=b\cdot c}$ ${\displaystyle c}$
• Reversal break: and are swapped ${\displaystyle a}$ ${\displaystyle b}$

Calculation rules

sign

${\displaystyle {\frac {-a}{b}}={\frac {a}{-b}}=-{\frac {a}{b}}}$

${\displaystyle {\frac {-a}{-b}}={\frac {a}{b}}}$

Expand and shorten

${\displaystyle {\frac {a}{b}}={\frac {a\cdot c}{b\cdot c}}}$   For ${\displaystyle c\neq 0}$

addition

${\displaystyle {\frac {a}{b}}+{\frac {c}{d}}={\frac {a\cdot d+c\cdot b}{b\cdot d}}}$

subtraction

${\displaystyle {\frac {a}{b}}-{\frac {c}{d}}={\frac {a\cdot d-c\cdot b}{b\cdot d}}}$

multiplication

${\displaystyle {\frac {a}{b}}\cdot {\frac {c}{d}}={\frac {a\cdot c}{b\cdot d}}}$

division

${\displaystyle {\frac {a}{b}}:{\frac {c}{d}}={\frac {a}{b}}\cdot {\frac {d}{c}}={\frac {a\cdot d}{b\cdot c}}}$

Percentage calculation

Definitions

${\displaystyle p\,\%={\frac {p}{100}}}$   (Percentage = percentage value: base value)

${\displaystyle p\,{}^{0\!}\!/\!_{00}={\frac {p}{1{\,}000}}}$   (Per mille rate = per mille value: basic value)

Percentages of frequently used parts

Share in the basic value	${\displaystyle {\frac {1}{100}}}$	${\displaystyle {\frac {1}{50}}}$	${\displaystyle {\frac {1}{40}}}$	${\displaystyle {\frac {1}{25}}}$	${\displaystyle {\frac {1}{20}}}$	${\displaystyle {\frac {1}{16}}}$	${\displaystyle {\frac {1}{15}}}$	${\displaystyle {\frac {1}{12}}}$	${\displaystyle {\frac {1}{11}}}$	${\displaystyle {\frac {1}{10}}}$
percentage	1 %	2%	2.5%	4%	5%	6.25%	≈6.67%	≈8.33%	≈9.09%	10%
Share in the basic value	${\displaystyle {\frac {1}{9}}}$	${\displaystyle {\frac {1}{8}}}$	${\displaystyle {\frac {1}{7}}}$	${\displaystyle {\frac {1}{6}}}$	${\displaystyle {\frac {1}{5}}}$	${\displaystyle {\frac {1}{4}}}$	${\displaystyle {\frac {1}{3}}}$	${\displaystyle {\frac {1}{2}}}$	${\displaystyle {\frac {2}{3}}}$	${\displaystyle {\frac {3}{4}}}$
percentage	≈11.11%	12.5%	≈14.29%	≈16.67%	20%	25%	≈33.33%	50%	≈66.67%	75%

Elementary arithmetic operations

power

Definitions

Natural exponent:

${\displaystyle a^{n}=\underbrace {a\cdot a\dotsm a} _{n\ \mathrm {Faktoren} }}$   ( Power = base to the power of exponent)

Negative exponent:

${\displaystyle a^{-n}={\frac {1}{a^{n}}}}$

Rational exponent:

${\displaystyle x=a^{m/n}\;\Leftrightarrow \;x^{n}=a^{m}}$

Here is a nonnegative rational number and are natural numbers. ${\displaystyle a}$ ${\displaystyle m,n}$

Special cases

${\displaystyle a^{0}=1}$   for , see zero to the power of zero ${\displaystyle a\neq 0}$

${\displaystyle 0^{n}=0}$   For ${\displaystyle n\neq 0}$

Power laws

${\displaystyle a^{m}\cdot a^{n}=a^{m+n}}$

${\displaystyle {\frac {a^{m}}{a^{n}}}=a^{m-n}}$

${\displaystyle ({a^{m}})^{n}=a^{m\cdot n}}$

${\displaystyle a^{n}\cdot b^{n}=(a\cdot b)^{n}}$

${\displaystyle {\frac {a^{n}}{b^{n}}}=\left({\frac {a}{b}}\right)^{n}}$

The definition and calculation rules can be extended to real numbers.

root

definition

${\displaystyle x={\sqrt[{n}]{a}}\;\Leftrightarrow \;x^{n}=a}$   ( nth root , a is called radicand, n root exponent)

Here a nonnegative real number and a natural number is greater than one ${\displaystyle a}$ ${\displaystyle n}$

Special cases

${\displaystyle {\sqrt {a}}={\sqrt[{2}]{a}}}$   ( Square root )

${\displaystyle {\sqrt[{3}]{a}}}$   ( Cube root )

Root Laws

${\displaystyle {\sqrt[{n}]{a}}=a^{\frac {1}{n}}}$

${\displaystyle {\sqrt[{n}]{a^{m}}}=({\sqrt[{n}]{a}})^{m}=a^{\frac {m}{n}}}$

${\displaystyle {\sqrt[{n}]{a}}\cdot {\sqrt[{n}]{b}}={\sqrt[{n}]{a\cdot b}}}$

${\displaystyle {{\sqrt[{n}]{a}} \over {\sqrt[{n}]{b}}}={\sqrt[{n}]{a \over b}}}$

${\displaystyle {\sqrt[{n}]{\sqrt[{m}]{a}}}={\sqrt[{n\cdot m}]{a}}}$

${\displaystyle {\sqrt[{n}]{a}}\cdot {\sqrt[{m}]{a}}={\sqrt[{n\cdot m}]{a^{n+m}}}}$

${\displaystyle {\frac {\sqrt[{n}]{a}}{\sqrt[{m}]{a}}}={\sqrt[{n\cdot m}]{a^{m-n}}}}$

logarithm

definition

${\displaystyle x=\log _{b}a\;\Leftrightarrow \;a=b^{x}}$   ( Logarithm of number a to base b)

Where are positive real numbers. ${\displaystyle a,b}$

Special cases

${\displaystyle \log _{2}a=\operatorname {lb} a}$   ( binary logarithm )

${\displaystyle \log _{e}a=\ln a}$   ( natural logarithm )

${\displaystyle \log _{10}a=\lg a}$   ( decadic logarithm )

${\displaystyle \log _{b}1=0}$

${\displaystyle \log _{b}b=1}$

Logarithmic Laws

${\displaystyle \log _{b}(a\cdot c)=\log _{b}a+\log _{b}c}$

${\displaystyle \log _{b}\left({\frac {a}{c}}\right)=\log _{b}a-\log _{b}c}$

${\displaystyle \log _{b}\left(a^{c}\right)=c\cdot \log _{b}a}$

${\displaystyle \log _{b}a={\frac {\log _{c}a}{\log _{c}b}}}$

Elementary functions

amount

definition

${\displaystyle |a|={\begin{cases}\;\;\,a&\mathrm {f{\ddot {u}}r} \quad a>0\\\;\;\,0&\mathrm {f{\ddot {u}}r} \quad a=0\\-a&\mathrm {f{\ddot {u}}r} \quad a<0\\\end{cases}}}$

properties

${\displaystyle |a|=0\;\Leftrightarrow \;a=0}$

${\displaystyle |a\cdot b|=|a|\cdot |b|}$

${\displaystyle |a+b|\leq |a|+|b|}$   ( Triangle inequality )

sign

definition

${\displaystyle \operatorname {sgn}(a)={\begin{cases}\;\;\,1&\mathrm {f{\ddot {u}}r} \quad a>0\\\;\;\,0&\mathrm {f{\ddot {u}}r} \quad a=0\\-1&\mathrm {f{\ddot {u}}r} \quad a<0\\\end{cases}}}$

properties

${\displaystyle \operatorname {sgn}(a)={\frac {a}{|a|}}}$   For ${\displaystyle a\neq 0}$

${\displaystyle \operatorname {sgn} |a|=|\operatorname {sgn} a|}$

${\displaystyle \operatorname {sgn}(a\cdot b)=\operatorname {sgn}(a)\cdot \operatorname {sgn}(b)}$

Rounding up and down

Definitions

${\displaystyle \lfloor a\rfloor =\max\{k\in \mathbb {Z} \mid k\leq a\}}$   (Rounding)

${\displaystyle \lceil a\rceil =\min\{k\in \mathbb {Z} \mid k\geq a\}}$   (Rounding up)

properties

${\displaystyle {\bigl \lfloor }\lfloor a\rfloor {\bigr \rfloor }={\bigl \lceil }\lfloor a\rfloor {\bigr \rceil }=\lfloor a\rfloor }$

${\displaystyle {\bigl \lceil }\lceil a\rceil {\bigr \rceil }={\bigl \lfloor }\lceil a\rceil {\bigr \rfloor }=\lceil a\rceil }$

${\displaystyle \lfloor a\rfloor +\lfloor b\rfloor \leq \lfloor a+b\rfloor \leq \lfloor a\rfloor +\lfloor b\rfloor +1}$

${\displaystyle \lceil a\rceil +\lceil b\rceil \geq \lceil a+b\rceil \geq \lceil a\rceil +\lceil b\rceil -1}$

Equations

Equivalent transformations

Solving equations

${\displaystyle a=b\;\Leftrightarrow \;b=a}$

${\displaystyle a=b\;\Leftrightarrow \;a+c=b+c}$

${\displaystyle a=b\;\Leftrightarrow \;a-c=b-c}$

${\displaystyle a=b\;\Leftrightarrow \;a\cdot c=b\cdot c}$   For ${\displaystyle c\neq 0}$

${\displaystyle a=b\;\Leftrightarrow \;a:c=b:c}$   For ${\displaystyle c\neq 0}$

${\displaystyle a=b\;\Leftrightarrow \;f(a)=f(b)}$   for every bijective function ${\displaystyle f}$

Linear equations

General form

${\displaystyle a\cdot x=b}$

solutions

${\displaystyle x={\frac {b}{a}}}$   if ${\displaystyle a\neq 0}$

no solution if ${\displaystyle a=0,b\neq 0}$

infinite solutions if ${\displaystyle a=0,b=0}$

Quadratic equations

General form

${\displaystyle ax^{2}+bx+c=0}$   With ${\displaystyle a\neq 0}$

Discriminant

${\displaystyle D=b^{2}-4ac}$

solutions

${\displaystyle x_{1,2}={\frac {-b\pm {\sqrt {b^{2}-4ac}}}{2a}}}$   if ${\displaystyle D>0}$

${\displaystyle x=-{\frac {b}{2a}}}$   if ${\displaystyle D=0}$

no real solution if ${\displaystyle D<0}$

Square addition

${\displaystyle ax^{2}+bx+c=a\left(x+{\frac {b}{2a}}\right)^{2}+\left(c-{\frac {b^{2}}{4a}}\right)}$

pq form

${\displaystyle x^{2}+px+q=0}$

Discriminant

${\displaystyle D={\frac {p^{2}}{4}}-q}$

solutions

${\displaystyle x_{1,2}=-{\frac {p}{2}}\pm {\sqrt {{\frac {p^{2}}{4}}-q}}}$   if ${\displaystyle D>0}$

${\displaystyle x=-{\frac {p}{2}}}$   if ${\displaystyle D=0}$

no real solution if ${\displaystyle D<0}$

Theorem of Vieta

${\displaystyle p=-(x_{1}+x_{2})}$

${\displaystyle q=x_{1}\cdot x_{2}}$

Algebraic equations

General form

${\displaystyle a_{n}x^{n}+a_{n-1}x^{n-1}+a_{n-2}x^{n-2}+\dotsb +a_{2}x^{2}+a_{1}x^{1}+a_{0}=0}$

solutions

${\displaystyle x_{1},\ldots ,x_{n}}$as complex solutions, not necessarily different ( fundamental theorem of algebra )

Decomposition into linear factors

${\displaystyle a_{n}(x-x_{1})(x-x_{2})\dotsm (x-x_{n})=0}$

Polynomial division

${\displaystyle p(x)=s(x)q(x)+r(x)}$   in which ${\displaystyle \operatorname {grad} p\geq \operatorname {grad} q}$

${\displaystyle {\frac {p(x)}{q(x)}}=s(x)+{\frac {r(x)}{q(x)}}}$   in which ${\displaystyle \operatorname {grad} q\geq 0}$

Inequalities

Equivalent transformations

Solving inequalities

${\displaystyle a<b\;\Leftrightarrow \;b>a}$

${\displaystyle a<b\;\Leftrightarrow \;a+c<b+c}$

${\displaystyle a<b\;\Leftrightarrow \;a-c<b-c}$

${\displaystyle a<b\;\Leftrightarrow \;{\begin{cases}a\cdot c<b\cdot c&{\text{falls}}~c>0\\a\cdot c>b\cdot c&{\text{falls}}~c<0\end{cases}}}$

${\displaystyle a<b\;\Leftrightarrow \;{\begin{cases}a:c<b:c&{\text{falls}}~c>0\\a:c>b:c&{\text{falls}}~c<0\end{cases}}}$

${\displaystyle a<b\;\Leftrightarrow \;{\begin{cases}f(a)<f(b)&{\text{falls}}~f~{\text{bijektiv streng monoton steigend ist}}\\f(a)>f(b)&{\text{falls}}~f~{\text{bijektiv streng monoton fallend ist}}\end{cases}}}$

The transformation rules also apply to . ${\displaystyle \leq ,\geq }$

Special inequalities

Triangle inequality

${\displaystyle |a+b|\leq |a|+|b|}$   for all ${\displaystyle a,b}$

Bernoulli's inequality

${\displaystyle (1+a)^{n}\geq 1+a\cdot n}$   for and ${\displaystyle a\geq -1}$ ${\displaystyle n=0,1,2,\ldots }$

Young's inequality

${\displaystyle a\cdot b\leq {\frac {a^{p}}{p}}+{\frac {b^{q}}{q}}}$   for and with ${\displaystyle a,b\geq 0}$ ${\displaystyle p,q>1}$ ${\displaystyle {\tfrac {1}{p}}+{\tfrac {1}{q}}=1}$

Inequalities in means

Inequality of the arithmetic and geometric mean

${\displaystyle {\sqrt[{n}]{a_{1}\cdot \ldots \cdot a_{n}}}\leq {\frac {1}{n}}(a_{1}+\ldots +a_{n})}$   for and ${\displaystyle a_{1},\ldots ,a_{n}\geq 0}$ ${\displaystyle n=2,3,\ldots }$

Inequality of harmonic and geometric mean

${\displaystyle {\frac {n}{{\frac {1}{a_{1}}}+\ldots +{\frac {1}{a_{n}}}}}\leq {\sqrt[{n}]{a_{1}\cdot \ldots \cdot a_{n}}}}$   for and ${\displaystyle a_{1},\ldots ,a_{n}>0}$ ${\displaystyle n=2,3,\ldots }$

Complex numbers

Algebraic form

presentation

${\displaystyle z=a+b\cdot \mathrm {i} }$   with real part , imaginary part and the imaginary unit ${\displaystyle a}$ ${\displaystyle b}$ ${\displaystyle \mathrm {i} }$

${\displaystyle {\bar {z}}=a-b\cdot \mathrm {i} }$   ( Complex conjugation )

Powers of imaginary unity

${\displaystyle \mathrm {i} ^{0}=1}$

${\displaystyle \mathrm {i} ^{1}=\mathrm {i} }$

${\displaystyle \mathrm {i} ^{2}=-1}$

${\displaystyle \mathrm {i} ^{3}=-\mathrm {i} }$

General for : ${\displaystyle n\in \mathbb {Z} }$

${\displaystyle \mathrm {i} ^{4n}=1}$

${\displaystyle \mathrm {i} ^{4n+1}=\mathrm {i} }$

${\displaystyle \mathrm {i} ^{4n+2}=-1}$

${\displaystyle \mathrm {i} ^{4n+3}=-\mathrm {i} }$

Arithmetic operations

${\displaystyle (a+\mathrm {i} b)+(c+\mathrm {i} d)=(a+c)+\mathrm {i} (b+d)}$

${\displaystyle (a+\mathrm {i} b)-(c+\mathrm {i} d)=(a-c)+\mathrm {i} (b-d)}$

${\displaystyle (a+\mathrm {i} b)\cdot (c+\mathrm {i} d)=ac-bd+\mathrm {i} (ad+bc)}$

${\displaystyle (a+\mathrm {i} b):(c+\mathrm {i} d)={\frac {ac+bd}{c^{2}+d^{2}}}+\mathrm {i} \,{\frac {bc-ad}{c^{2}+d^{2}}}}$   For ${\displaystyle c^{2}+d^{2}\neq 0}$

Polar shape

presentation

${\displaystyle z=r\cdot (\cos(\varphi )+\mathrm {i} \cdot \sin(\varphi ))}$   with the amount and the argument ${\displaystyle r}$ ${\displaystyle \varphi }$

amount

${\displaystyle r=|z|={\sqrt {z\cdot {\bar {z}}}}={\sqrt {a^{2}+b^{2}}}}$

argument

${\displaystyle \varphi ={\begin{cases}\arctan {\frac {b}{a}}&\mathrm {f{\ddot {u}}r} \ a>0\\\arctan {\frac {b}{a}}+\pi &\mathrm {f{\ddot {u}}r} \ a<0,b\geq 0\\\arctan {\frac {b}{a}}-\pi &\mathrm {f{\ddot {u}}r} \ a<0,b<0\\\pi /2&\mathrm {f{\ddot {u}}r} \ a=0,b>0\\-\pi /2&\mathrm {f{\ddot {u}}r} \ a=0,b<0\end{cases}}}$

or

${\displaystyle \varphi ={\begin{cases}\arccos {\frac {a}{r}}&\mathrm {f{\ddot {u}}r} \ b\geq 0\\\arccos \left(-{\frac {a}{r}}\right)-\pi &\mathrm {f{\ddot {u}}r} \ b<0\end{cases}}}$

Exponential form

presentation

${\displaystyle z=r\cdot e^{\mathrm {i} \varphi }}$   with Euler's number ${\displaystyle e}$

${\displaystyle e^{\mathrm {i} \varphi }=\cos \varphi +\mathrm {i} \,\sin \varphi }$   ( Euler's formula )

Conversion formulas

${\displaystyle \sin \varphi ={\frac {e^{\mathrm {i} \varphi }-e^{-\mathrm {i} \varphi }}{2\mathrm {i} }}}$

${\displaystyle \cos \varphi ={\frac {e^{\mathrm {i} \varphi }+e^{-\mathrm {i} \varphi }}{2}}}$

Arithmetic operations

${\displaystyle (r\cdot e^{\mathrm {i} \varphi })\pm (s\cdot e^{\mathrm {i} \psi })={\sqrt {r^{2}+s^{2}\pm 2rs\cos(\varphi -\psi )}}\cdot e^{\mathrm {i} \operatorname {atan2} \left(r\sin \varphi \pm s\sin \psi ,r\cos \varphi \pm s\cos \psi \right)}}$

${\displaystyle (r\cdot e^{\mathrm {i} \varphi })\cdot (s\cdot e^{\mathrm {i} \psi })=(r\cdot s)\cdot e^{\mathrm {i} (\varphi +\psi )}}$

${\displaystyle (r\cdot e^{\mathrm {i} \varphi }):(s\cdot e^{\mathrm {i} \psi })=(r:s)\cdot e^{\mathrm {i} (\varphi -\psi )}}$

Potencies

${\displaystyle (r\cdot e^{\mathrm {i} \varphi })^{n}=r^{n}\cdot e^{\mathrm {i} n\varphi }}$

root

${\displaystyle x^{n}=1\,\Leftrightarrow \,x=e^{2\pi \mathrm {i} k/n}}$   for   ( roots of unity ) ${\displaystyle k=0,1,\dots ,n-1}$

${\displaystyle x^{n}=z\,\Leftrightarrow \,x={\sqrt[{n}]{|z|}}\cdot e^{(\mathrm {i} \arg(z)+2\pi \mathrm {i} k)/n}}$   For ${\displaystyle k=0,1,\dots ,n-1}$

Molecular formulas

Calculation rules

${\displaystyle \sum _{i=1}^{n}c=n\cdot c}$

${\displaystyle \sum _{i=m}^{n}c=(n-m+1)\cdot c}$

${\displaystyle \sum _{i=m}^{n}c\cdot a_{i}=c\cdot \sum _{i=m}^{n}a_{i}}$

${\displaystyle \sum _{i=m}^{n}(a_{i}+b_{i})=\sum _{i=m}^{n}a_{i}+\sum _{i=m}^{n}b_{i}}$

${\displaystyle \sum _{i=m}^{n}a_{i}=\sum _{i=m-r}^{n-r}a_{i+r}}$

${\displaystyle \sum _{i=1}^{n}(a_{i}-a_{i-1})=a_{n}-a_{0}}$   ( Telescope sum )

Arithmetic series

${\displaystyle \sum _{i=1}^{n}i={\frac {n(n+1)}{2}}}$   ( Gaussian empirical formula )

Geometric series

${\displaystyle \sum _{i=0}^{n}k^{i}={\frac {1-k^{n+1}}{1-k}}}$

Power sums

${\displaystyle \sum _{i=1}^{n}i^{2}={\frac {n(n+1)(2n+1)}{6}}}$

${\displaystyle \sum _{i=1}^{n}i^{3}={\frac {n^{2}(n+1)^{2}}{4}}}$

For further power sums see Faulhaber's formula .

Combinatorial Sums

Binomial theorem

${\displaystyle (a+b)^{n}=\sum _{k=0}^{n}{n \choose k}a^{n-k}b^{k}}$

Multinomial theorem

${\displaystyle \left(\sum _{i=1}^{k}a_{i}\right)^{n}=\sum _{n_{1}+\ldots +n_{k}=n}{n \choose n_{1},\ldots ,n_{k}}\,\cdot \,a_{1}^{n_{1}}\cdot a_{2}^{n_{2}}\cdots a_{k}^{n_{k}}}$

Inequalities in sums

Cauchy-Schwarz inequality

${\displaystyle \left(\sum _{i=1}^{n}a_{i}\cdot b_{i}\right)^{2}\leq \left(\sum _{i=1}^{n}a_{i}^{2}\right)\cdot \left(\sum _{i=1}^{n}b_{i}^{2}\right)}$   for everyone and ${\displaystyle a_{1},\ldots ,a_{n}}$ ${\displaystyle b_{1},\ldots ,b_{n}}$

Chebyshev inequalities

${\displaystyle n\cdot \left(\sum _{i=1}^{n}a_{i}\cdot b_{i}\right)\geq \left(\sum _{i=1}^{n}a_{i}\right)\cdot \left(\sum _{i=1}^{n}b_{i}\right)}$   for everyone and ${\displaystyle a_{1}\geq \ldots \geq a_{n}}$ ${\displaystyle b_{1}\geq \ldots \geq b_{n}}$

${\displaystyle n\cdot \left(\sum _{i=1}^{n}a_{i}\cdot b_{i}\right)\leq \left(\sum _{i=1}^{n}a_{i}\right)\cdot \left(\sum _{i=1}^{n}b_{i}\right)}$   for everyone and ${\displaystyle a_{1}\geq \ldots \geq a_{n}}$ ${\displaystyle b_{1}\leq \ldots \leq b_{n}}$

Minkowski inequality

${\displaystyle \left(\sum _{i=1}^{n}|a_{i}+b_{i}|^{p}\right)^{1/p}\leq \left(\sum _{i=1}^{n}|a_{i}|^{p}\right)^{1/p}+\left(\sum _{i=1}^{n}|b_{i}|^{p}\right)^{1/p}}$   for everyone and as well ${\displaystyle a_{1},\ldots ,a_{n}}$ ${\displaystyle b_{1},\ldots ,b_{n}}$ ${\displaystyle p\geq 1}$

Holder's inequality

${\displaystyle \sum _{i=1}^{n}|a_{i}\cdot b_{i}|\leq \left(\sum _{i=1}^{n}|a_{i}|^{p}\right)^{1/p}\cdot \left(\sum _{i=1}^{n}|b_{i}|^{q}\right)^{1/q}}$   for everyone and as well as with ${\displaystyle a_{1},\ldots ,a_{n}}$ ${\displaystyle b_{1},\ldots ,b_{n}}$ ${\displaystyle p,q\geq 1}$ ${\displaystyle {\tfrac {1}{p}}+{\tfrac {1}{q}}=1}$

Jensen's inequality

${\displaystyle f\left(\sum _{i=1}^{n}a_{i}\cdot b_{i}\right)\leq \sum _{i=1}^{n}a_{i}\cdot f(b_{i})}$   for every convex function , with and all ${\displaystyle f}$ ${\displaystyle a_{1},\ldots ,a_{n}\geq 0}$ ${\displaystyle a_{1}+\ldots +a_{n}=1}$ ${\displaystyle b_{1},\ldots ,b_{n}}$

literature

• Bronstein et al .: Paperback of Mathematics . 7th edition. Harri Deutsch, 2008, ISBN 978-3-8171-2007-9 .