Reciprocals
Reciprocals is a Grade 6 math skill in Saxon Math, Course 1 that defines the reciprocal of a number as 1 divided by that number, or equivalently, flipping the numerator and denominator of a fraction. The reciprocal of 3/4 is 4/3; the reciprocal of 5 is 1/5. A number multiplied by its reciprocal always equals 1: 3/4 × 4/3 = 12/12 = 1. Reciprocals are the foundation of fraction division—dividing by a number is identical to multiplying by its reciprocal. This 'keep-change-flip' (KCF) rule for division requires students to first identify the reciprocal of the divisor.
Key Concepts
Property Reciprocals are two numbers whose product equals 1. To find the reciprocal of a fraction, you reverse its terms. The reciprocal of $\frac{a}{b}$ is $\frac{b}{a}$.
Examples The reciprocal of 5 is $\frac{1}{5}$, because you can write 5 as $\frac{5}{1}$ and then flip it. The reciprocal of $\frac{3}{4}$ is $\frac{4}{3}$, because $\frac{3}{4} \times \frac{4}{3} = \frac{12}{12} = 1$. To solve the equation $\frac{5}{6} \times \square = 1$, you use the reciprocal of $\frac{5}{6}$, which is $\frac{6}{5}$.
Explanation Think of a reciprocal as a number's 'upside down' twin! When you multiply a number by its reciprocal, they team up to become 1. It's a handy trick for division problems. For any fraction, just flip the numerator and the denominator to find its twin.
Common Questions
What is the reciprocal of a fraction?
The reciprocal is formed by flipping the fraction—swapping numerator and denominator. The reciprocal of 3/7 is 7/3. The reciprocal of 2/5 is 5/2.
What is the reciprocal of a whole number?
Write the whole number as a fraction over 1, then flip it. The reciprocal of 6 is 1/6. The reciprocal of 12 is 1/12.
What happens when you multiply a number by its reciprocal?
The product is always 1. For example, 4/5 × 5/4 = 20/20 = 1. This is the defining property of reciprocals.
How are reciprocals used in fraction division?
To divide by a fraction, multiply by its reciprocal. The rule is: keep the first fraction, change ÷ to ×, flip the second fraction. For example, 2/3 ÷ 4/5 = 2/3 × 5/4 = 10/12 = 5/6.
Does 0 have a reciprocal?
No. The reciprocal of 0 would be 1/0, which is undefined. Division by zero is not allowed in mathematics.