Reciprocals

Definition of Reciprocals

The reciprocal of a number is defined as one divided by that number. In other words, the reciprocal of a number $x$ is written as:

$$\text{Reciprocal of } x = \frac{1}{x}$$

For example, the reciprocal of 5 is $\frac{1}{5}$. For instance, the reciprocal of 10 is $\frac{1}{10}$.

When the number is a fraction, say $\frac{a}{b}$, its reciprocal is found by swapping the numerator and denominator:

$$\text{Reciprocal of } \frac{a}{b} = \frac{b}{a}$$

This means the reciprocal of $\frac{3}{4}$ is $\frac{4}{3}$.

Finding Reciprocals

Reciprocal of Whole Numbers

To find the reciprocal of a whole number, write it as a fraction with denominator 1, then swap numerator and denominator.

For example, the reciprocal of 7 is $\frac{1}{7}$.

Reciprocal of Fractions

For fractions, simply swap the numerator and denominator to find the reciprocal.

For example, the reciprocal of $\frac{2}{5}$ is $\frac{5}{2}$.

Reciprocal of Decimals

To find the reciprocal of a decimal, write it as a fraction first (if possible), then find the reciprocal by swapping numerator and denominator. Note that some decimals may not convert neatly to fractions, so dividing 1 by the decimal directly is a reliable method.

For example, the reciprocal of 0.25 is:

$$0.25 = \frac{25}{100} = \frac{1}{4} \implies \text{reciprocal} = \frac{4}{1} = 4$$

Alternatively, you can calculate the reciprocal by dividing 1 by the decimal directly:

$$\frac{1}{0.25} = 4$$

For instance, if you want the reciprocal of 0.2, you can write:

$$0.2 = \frac{2}{10} = \frac{1}{5} \implies \text{reciprocal} = 5$$

Properties of Reciprocals

• Multiplying a number by its reciprocal always equals 1: For any number $x \neq 0$,

$$x \times \frac{1}{x} = 1$$

• The reciprocal of the reciprocal returns the original number:

$$\text{If } y = \frac{1}{x}, \text{ then } \frac{1}{y} = x$$

• The reciprocal of zero is undefined: Since division by zero is not allowed, zero has no reciprocal.

For example, the reciprocal of 0 does not exist because $\frac{1}{0}$ is undefined.

Example: Find the reciprocal of 8 and verify the property of multiplication.

The reciprocal of 8 is $\frac{1}{8}$. Multiplying 8 by its reciprocal:

$$8 \times \frac{1}{8} = 1$$

This confirms the property that a number times its reciprocal equals 1.