Basic arithmetic operations have a number of properties.
Associativity is a property of operations that allows the order of the calculations to be changed by grouping terms together using brackets, without changing the answer of the operation.
This property applies to both addition and multiplication.
The order of operations applies in the examples below. The calculation in brackets must always be performed first.
Associativity of addition||\begin{align}(10+20)+30&=10+(20+30)\\ 30+30&=10+50\\60&=60\end{align}||
Associativity of multiplication||\begin{align}(10\times20)\times30&=10\times(20\times30)\\ 200\times30&=10\times600\\6000&=6000\end{align}||
Subtraction and division are not associative operations.
||\begin{align}(30-20)-10&\overset{?}{=}30-(20-10)\\ 10-10&\overset{?}{=}30-10\\0&\color{#ec0000}{\large\neq}20\end{align}||
||\begin{align}(100\div20)\div5&\overset{?}{=}100\div(20\div5)\\ 5\div5&\overset{?}{=}100\div4\\1&\color{#ec0000}{\large\neq}25\end{align}||
Commutativity is a property of operations that allows the order of terms to be changed without changing the answer.
This property applies to addition and multiplication.
Commutativity of addition||\begin{align}2+3&=3+2\\5&=5\end{align}||
Commutativity of multiplication||\begin{align}2\times3&=3\times2\\6&=6\end{align}||
Subtraction and division are not commutative operations.
||\begin{align}10-2&\overset{?}{=}2-10\\8&\color{#ec0000}{\large\neq}-8\end{align}||
||\begin{align}10\div2&\overset{?}{=}2\div10\\5&\color{#ec0000}{\large\neq}0.2\end{align}||
Distributivity of multiplication is a property of operations that allows multiplication to be distributed over addition or subtraction.
The distributivity of multiplication transforms a multiplication of sums (or differences) into an addition (or subtraction) of products.
The distributivity of multiplication over addition||\begin{align}\color{#3b87cd}{\boldsymbol{2\ \times}}\ (10\color{#fa7921}+5)&=\color{#3b87cd}{\boldsymbol{2\ \times}}\ 10\color{#fa7921}+\color{#3b87cd}{\boldsymbol{2\ \times}}\ 5\\2\times15&=20+10\\30&=30\end{align}||
The distributivity of multiplication over subtraction||\begin{align}\color{#3b87cd}{\boldsymbol{2\ \times}}\ (10\color{#fa7921}{\large-}5)&=\color{#3b87cd}{\boldsymbol{2\ \times}}\ 10\color{#fa7921}{\large-}\color{#3b87cd}{\boldsymbol{2\ \times}}\ 5\\2\times5&=20-10\\10&=10\end{align}||
The distributivity of multiplication can be helpful when performing mental calculations. For example, if we want to find the product of |22 \times 13,| we can rewrite the number |13| as the sum of |10+3| and distribute the multiplication by |22.| The multiplication then becomes a sum of |2| multiplications which are easier to do mentally.||\begin{align}22 \times 13 &= 22 \times (10+3)\\&=22 \times 10+ 22 \times 3 \\&= 220+66\\&=286\end{align}||
The distributivity of multiplication only applies to addition and subtraction. You cannot distribute a multiplication over another multiplication. Here's an example to prove this:||\begin{align}\color{#3b87cd}{\boldsymbol{2\ \times}}\ (10\color{#fa7921}\times5)&\overset{?}{=}\color{#3b87cd}{\boldsymbol{2\ \times}}\ 10\ \color{#fa7921}\times\ \color{#3b87cd}{\boldsymbol{2\ \times}}\ 5\\2\times50&\overset{?}{=}20\times 10\\100&\color{#ec0000}{\large\neq}200\end{align}||
The distributivity of division is the property of operations that allows you to distribute the same divisor over each term in an addition or subtraction.
The distributivity of division transforms a division of sums (or differences) into an addition (or subtraction) of quotients.
The distributivity of division over addition||\begin{align}(10\color{#fa7921}+15)\color{#3b87cd}{\boldsymbol{\div5}}&=10\color{#3b87cd}{\boldsymbol{\div5}}\color{#fa7921}+15\color{#3b87cd}{\boldsymbol{\div5}}\\25\div5&=2+3\\5&=5\end{align}||
The distributivity of division over subtraction||\begin{align}(60\color{#fa7921}-15)\color{#3b87cd}{\boldsymbol{\div3}}&=60\color{#3b87cd}{\boldsymbol{\div3}}\color{#fa7921}-15\color{#3b87cd}{\boldsymbol{\div3}}\\45\div3&=20-5\\15&=15\end{align}||
Division is only distributive when the addition or subtraction is the dividend. It is not distributive if the addition or subtraction is the divisor. Here's an example to prove this:||\begin{align}30\div(3+2)&\overset{?}{=}(30\div3)+(30\div2)\\30\div5&\overset{?}{=}10+15\\6&\color{#ec0000}{\large\neq}25\end{align}||
The same property applies when the division is written in the form of a fraction.
Addition is in the numerator (dividend)||\begin{align}\dfrac{20+30}{5}&=\dfrac{20}{5}+\dfrac{30}{5}\\\dfrac{50}{5}&=4+6\\10&=10\end{align}||
Addition is in the denominator (divisor)||\begin{align}\dfrac{60}{2+3}&\overset{?}{=}\dfrac{60}{2}+\dfrac{60}{3}\\\dfrac{60}{5}&\overset{?}{=}30+20\\12&\color{#ec0000}{\large\neq}50\end{align}||
Like the distributivity of numbers, the distributivity of algebraic expressions applies to each term inside the brackets when multiplying or dividing algebraic expressions.
||\begin{alignat}{20}2(2y+3)&=2&&\times2y&&+2&&\times3\\&=&&\ 4y&&\ +&&\ 6\end{alignat}||
||\begin{align}\boldsymbol{\color{#3b87cd}{-}}(2x+5y-10)&=\boldsymbol{\color{#3b87cd}{-1\,\times}}\,(2x+5y-10)\\&=(\boldsymbol{\color{#3b87cd}{-1\,\times}}\, 2x)+(\boldsymbol{\color{#3b87cd}{-1\,\times}}\, 5y)-(\boldsymbol{\color{#3b87cd}{-1\,\times}}\, 10)\\&=-2x+-5y--10\\&=-2x\ -\ \ 5y\ \, +\ \,10\end{align}||
||\begin{alignat}{20}\dfrac{21a+35}{7} &= \dfrac{21a}{7} &&+ \dfrac{35}{7} \\&=\ \ 3a&&+\ \ 5\end{alignat}||
||\begin{alignat}{20}\dfrac{6(10+b)}{2}&=\dfrac{6\times 10+6\times b}{2}\\&=\dfrac{60+6b}{2}\\&=\dfrac{60}{2}+\dfrac{6b}{2}\\&=\ 30\,+\ 3b\end{alignat}||
The identity element, or neutral element, is a number that, when used in an operation on another number, results in the other number itself.
For addition and subtraction, the neutral element is |0,| whereas for multiplication and division, the neutral element is |1.|
Neutral element of addition||\begin{align}3+0&=3\\ -62+0& =-62\end{align}||
Neutral element of subtraction||\begin{align}5-0&=5\\ -14-0&=-14\end{align}||
Neutral element of multiplication||\begin{align}15\times1&=15\\ \dfrac{2}{3}\times1&=\dfrac{2}{3}\end{align}||
Neutral element of division||\begin{align}8\div1&=8\\ \dfrac{3}{4}\div1&=\dfrac{3}{4}\end{align}||
Addition and multiplication are commutative operations. This means that the neutral (identity) element can be found in any position in an addition or multiplication operation.
On the other hand, since subtraction and division are not commutative operations, the neutral element only works when it is located to the right of the subtraction or division.
||\begin{align}\boldsymbol{\color{#3b87cd}{10}}-0&=\boldsymbol{\color{#3b87cd}{10}}\\ 0-\boldsymbol{\color{#3b87cd}{10}}&= \boldsymbol{\color{#ec0000}{-10}}\ \text{and not}\ \boldsymbol{\color{#3b87cd}{10}} \end{align}||
||\begin{align}\boldsymbol{\color{#3b87cd}{2}}\div 1&=\boldsymbol{\color{#3b87cd}{2}}\\ 1\div\boldsymbol{\color{#3b87cd}{2}}&= \boldsymbol{\color{#ec0000}{0.5}}\ \text{and not}\ \boldsymbol{\color{#3b87cd}{2}} \end{align}||
When reducing algebraic expressions, we generally don't write the neutral elements.
If we obtain |0| as the coefficient for a term, we don't write out the term, since |0| is the neutral element of addition and subtraction.||\begin{align}5x+2y-2x-3x&=\boldsymbol{\color{#3b87cd}{0x}}+2y\\&=2y\end{align}||
If we obtain |1| as the coefficient for a term, we don't write this coefficient, since |1| is the neutral element of multiplication.||\begin{align}5x+2y-4x&=\boldsymbol{\color{#3b87cd}{1}}x+2y\\&=x+2y\end{align}||
If we obtain |1| as the denominator, we don't write it since |1| is the neutral element of division.||\begin{align}\dfrac{6x}{2}&=\dfrac{3x}{\boldsymbol{\color{#3b87cd}{1}}}\\&=3x\end{align}||
In addition, the neutral (identity) element is obtained by adding a number with its opposite. For a multiplication, the neutral element is obtained by multiplying a number by its reciprocal.
||\begin{align}1 + (-1) &= 0\\ -\dfrac{4}{3} + \dfrac{4}{3}&=0\end{align}||
\begin{align}6\times \dfrac{1}{6}&=1\\ \dfrac{2}{5} \times \dfrac{5}{2} &= 1\end{align}
The absorbing element, or annihilating element, is the number which, when combined by an operation with all the other numbers, gives the absorbing element.
The absorbing element for multiplication and division is |0.|
If we take |0| and multiply or divide it by any other number, the answer is always |0.|
Absorbing element of multiplication||\begin{align}10 \times 0&=0\\ 3 \times 5 \times 0 \times 2 &= 0\end{align}||
Absorbing element of division||\begin{align}0 \div 14 &= 0\\ 0 \div \dfrac{1}{5}&=0\end{align}||
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Multiplication is a commutative operation. This means that the absorbing element can be found in any position of the multiplication.
On the other hand, since division is not a commutative operation, the absorbing element must always be placed to the left of the division, because it is impossible to divide by |0.|||\begin{align}&0\div9= 0\\ &\boldsymbol{\color{#ec0000}{9\div0\rightarrow\ \textbf{Impossible}}}\end{align}|| -
Addition and subtraction have no absorbing element.