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Arithmetic operations v t e

Addition (+) ${\displaystyle \scriptstyle \left.{\begin{matrix}\scriptstyle {\text{term}}\,+\,{\text{term}}\\\scriptstyle {\text{summand}}\,+\,{\text{summand}}\\\scriptstyle {\text{addend}}\,+\,{\text{addend}}\\\scriptstyle {\text{augend}}\,+\,{\text{addend}}\end{matrix}}\right\}\,=\,}$ ${\displaystyle \scriptstyle {\text{sum}}}$ Subtraction (−) ${\displaystyle \scriptstyle \left.{\begin{matrix}\scriptstyle {\text{term}}\,-\,{\text{term}}\\\scriptstyle {\text{minuend}}\,-\,{\text{subtrahend}}\end{matrix}}\right\}\,=\,}$ ${\displaystyle \scriptstyle {\text{difference}}}$ Multiplication (×) ${\displaystyle \scriptstyle \left.{\begin{matrix}\scriptstyle {\text{factor}}\,\times \,{\text{factor}}\\\scriptstyle {\text{multiplier}}\,\times \,{\text{multiplicand}}\end{matrix}}\right\}\,=\,}$ ${\displaystyle \scriptstyle {\text{product}}}$ Division (÷) ${\displaystyle \scriptstyle \left.{\begin{matrix}\scriptstyle {\frac {\scriptstyle {\text{dividend}}}{\scriptstyle {\text{divisor}}}}\\\scriptstyle {\frac {\scriptstyle {\text{numerator}}}{\scriptstyle {\text{denominator}}}}\end{matrix}}\right\}\,=\,}$ ${\displaystyle \scriptstyle \left\{{\begin{matrix}\scriptstyle {\text{fraction}}\\\scriptstyle {\text{quotient}}\\\scriptstyle {\text{ratio}}\end{matrix}}\right.}$ Exponentiation (^) ${\displaystyle \scriptstyle {\text{base}}^{\text{exponent}}\,=\,}$ ${\displaystyle \scriptstyle {\text{power}}}$ nth root (√) ${\displaystyle \scriptstyle {\sqrt{\scriptstyle {\text{radicand}}}}\,=\,}$ ${\displaystyle \scriptstyle {\text{root}}}$ Logarithm (log) ${\displaystyle \scriptstyle \log _{\text{base}}({\text{anti-logarithm}})\,=\,}$ ${\displaystyle \scriptstyle {\text{logarithm}}}$

v t e

Division is one of the four basic operations of arithmetic. The other operations are addition, subtraction, and multiplication. What is being divided is called the dividend, which is divided by the divisor, and the result is called the quotient.

At an elementary level the division of two natural numbers is, among other possible interpretations, the process of calculating the number of times one number is contained within another.^[1]: 7 For example, if 20 apples are divided evenly between 4 people, everyone receives 5 apples (see picture). However, this number of times or the number contained (divisor) need not be integers.

The division with remainder or Euclidean division of two natural numbers provides an integer quotient, which is the number of times the second number is completely contained in the first number, and a remainder, which is the part of the first number that remains, when in the course of computing the quotient, no further full chunk of the size of the second number can be allocated. For example, if 21 apples are divided between 4 people, everyone receives 5 apples again, and 1 apple remains.

For division to always yield one number rather than an integer quotient plus a remainder, the natural numbers must be extended to rational numbers or real numbers. In these enlarged number systems, division is the inverse operation to multiplication, that is a = c / b means a × b = c, as long as b is not zero. If b = 0, then this is a division by zero, which is not defined.^{[a][4]: 246} In the 21-apples example, everyone would receive 5 apple and a quarter of an apple, thus avoiding any leftover.

Both forms of division appear in various algebraic structures, different ways of defining mathematical structure. Those in which a Euclidean division (with remainder) is defined are called Euclidean domains and include polynomial rings in one indeterminate (which define multiplication and addition over single-variabled formulas). Those in which a division (with a single result) by all nonzero elements is defined are called fields and division rings. In a ring the elements by which division is always possible are called the units (for example, 1 and −1 in the ring of integers). Another generalization of division to algebraic structures is the quotient group, in which the result of "division" is a group rather than a number.

Introduction

The simplest way of viewing division is in terms of quotition and partition: from the quotition perspective, 20 / 5 means the number of 5s that must be added to get 20. In terms of partition, 20 / 5 means the size of each of 5 parts into which a set of size 20 is divided. For example, 20 apples divide into five groups of four apples, meaning that "twenty divided by five is equal to four". This is denoted as 20 / 5 = 4, or 20/5 = 4.^[2] In the example, 20 is the dividend, 5 is the divisor, and 4 is the quotient.

Unlike the other basic operations, when dividing natural numbers there is sometimes a remainder that will not go evenly into the dividend; for example, 10 / 3 leaves a remainder of 1, as 10 is not a multiple of 3. Sometimes this remainder is added to the quotient as a fractional part, so 10 / 3 is equal to 3+1/3 or 3.33..., but in the context of integer division, where numbers have no fractional part, the remainder is kept separately (or exceptionally, discarded or rounded).^[5] When the remainder is kept as a fraction, it leads to a rational number. The set of all rational numbers is created by extending the integers with all possible results of divisions of integers.

Unlike multiplication and addition, division is not commutative, meaning that a / b is not always equal to b / a.^[6] Division is also not, in general, associative, meaning that when dividing multiple times, the order of division can change the result.^[7] For example, (24 / 6) / 2 = 2, but 24 / (6 / 2) = 8 (where the use of parentheses indicates that the operations inside parentheses are performed before the operations outside parentheses).

Division is traditionally considered as left-associative. That is, if there are multiple divisions in a row, the order of calculation goes from left to right:^[8][9]

${\displaystyle a/b/c=(a/b)/c=a/(b\times c)\;\neq \;a/(b/c)=(a\times c)/b.}$

Division is right-distributive over addition and subtraction, in the sense that

${\displaystyle {\frac {a\pm b}{c}}=(a\pm b)/c=(a/c)\pm (b/c)={\frac {a}{c}}\pm {\frac {b}{c}}.}$

This is the same for multiplication, as ${\displaystyle (a+b)\times <!--\; \times \; -->c}$Zdroj:https://en.wikipedia.org?pojem=Division_(mathematics)
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