Properties of arithmetic operations

Commutativity

An operation is called commutative if and only if we can switch the position of its operands (whatever these operands are) without affecting the result.

• Addition is commutative. Consider \begin{align} 3+5 &= 8 \\ 5+3 &= 8 \end{align} They both give the same result. This is true for any two numbers.
• Subtraction is not commutative. Consider \begin{align} 4-7 &= -3 \\ 7-4 &= 3 \end{align} They do not give the same result.
• Multiplication is commutative. Consider \begin{align} 2\times 4 &= 8 \\ 4\times 2 &= 8 \end{align} They both give the same result. This is true for any two numbers.
• Division is not commutative. Consider \begin{align} 8\div 2 &= 4 \\ 2\div 8 &= \frac{1}{4} \end{align} They do not give the same result.

Associativity

An operation is called associative if and only if ‘chaining’ two (or more) of the same operation can be evaluated from left to right or from right to left.

• Addition is associative. Consider \begin{align} \color{blue}{2 + 5} + 3 &= \color{blue}{7} + 3 &&= 10 \\ 2 + \color{green}{5 + 3} &= 2 + \color{green}{8} &&= 10 \end{align} The two evaluations give the same result. This is true regardless of what numbers are used.
• Subtraction is not associative. Consider \begin{align} \color{blue}{8 - 5} - 2 &= \color{blue}{3} - 2 &&= 1 \tag{correct}\\ 8 - \color{green}{5 - 2} &= 8 - \color{green}{3} &&= 5 \tag{incorrect} \end{align} The two evaluations do not give the same result. We evaluate subtraction from left to right in such cases; therefore the first evaluation is the correct one, and the second is wrong.
• Multiplication is associative. Consider \begin{align} \color{blue}{2 \times 3} \times 4 &= \color{blue}{6} \times 4 &&= 24 \\ 2 \times \color{green}{3 \times 4} &= 2 \times \color{green}{12} &&= 24 \end{align} The two evaluations give the same result. This is true regardless of what numbers are used.
• Division is not associative. Consider \begin{align} \color{blue}{12 \div 6} \div 3 &= \color{blue}{2} \div 3 &&= \frac{2}{3} \tag{correct} \\ 12 \div \color{green}{6 \div 3} &= 12 \div \color{green}{2} &&= 6 \tag{incorrect} \end{align} The two evaluations do not give the same result. We evaluate division from left to right in such cases; therefore the first evaluation is the correct one, and the second is wrong.

Distributivity

We say that multiplication distributes over addition and subtraction, which means if we are multiplying a number by the sum or the difference of two others, we can equivalently multiply it by each of them first, then take the sum or the difference. This is best illustrated by an example.

\begin{align} 3\times (\color{red}{4 + 2}) &= 3 \times \color{red}{6} &&= 18 \\ \color{blue}{3\times 4} + \color{green}{3 \times 2} &= \color{blue}{12} + \color{green}{6} &&= 18 \\ \text{Therefore } 3\times(4+2) &= 3\times 4 + 3\times 2 \end{align}

This is true for any three numbers. This is also true whether the multiplication is done on the right or on the left, and whether we are dealing with subtraction or addition. Consider the following

\begin{align} (\color{red}{4 - 2}) \times 3 &= \color{red}{2} \times 3 &&= 6 \\ \color{blue}{4\times 3} - \color{green}{2 \times 3} &= \color{blue}{12} + \color{green}{6} &&= 6 \\ \text{Therefore } (4-2)\times 3 &= 4\times 3 - 2\times 3 \end{align}

We cannot say the same about division, particularly when it is done from the left. Consider the following.

\begin{align} 24 \div (\color{red}{2 + 4}) &= 24 \div \color{red}{6} &&= 4 \\ \color{blue}{24\div 2} + \color{green}{24\div 4} &= \color{blue}{12} + \color{green}{6} &&= 18 \\ \text{Therefore } 24\div(2 + 4) &\neq 24\div 2 + 24\div 4 \end{align}

Therefore, division is not distributive from the left on addition or subtraction.

Inverse element

Every number has an additive inverse. When a number is added to its additive inverse, the result is always zero.

• The additive inverse of zero is zero, because \(0+0=0\).
• The additive inverse of a positive number is the negative number with the same magnitude. For example, the additive inverse of \(5\) is \(-5\) because \(5+(-5)=-5+5=0\).
• The additive inverse of a negative number is the positive number with the same magnitude. For example, the additive inverse of \(-4\) is \(4\) becauase \(4+(-4)=-4+4=0\).

Subtraction is the same as adding the additive inverse.

Every number except zero has a multiplicative inverse. When a number is multiplied by its multiplicative inverse, the result is always one.

• Zero has no multiplicative inverse.
• The multiplicative inverse of any non-zero number is obtained by dividing \(1\) by that number. This is also called the ‘reciprocal’ of that number.
• For integers, the reciprocal is always a rational number. For example, the reciprocal of \(3\) is \(\frac{1}{3}\).
• For rational numbers that are not integers, the reciprocal is obtained by switching the numerator and the denominator. For example, the reciprocal of \(\frac{4}{7}\) is \(\frac{7}{4}\).
• For irrational numbers like \(\pi\), the reciprocal is also an irrational number. For example, the reciprocal of \(\pi\) is \(\frac{1}{\pi}\), and the that of \(\sqrt{2}\) is \(\frac{1}{\sqrt{2}}\).

Division is the same as multiplying by the multiplicative inverse (the reciprocal).