Reciprocals
Reciprocals are pairs of numbers whose product equals 1. To find the reciprocal of a fraction, swap the numerator and denominator; to find the reciprocal of a whole number, write it as a fraction over 1 and then flip it. In Grade 6 Saxon Math Course 1 (Chapter 5: Number and Operations), students use reciprocals in fraction division: dividing by a number is the same as multiplying by its reciprocal (Keep-Change-Flip). The reciprocal of 3/4 is 4/3; the reciprocal of 5 is 1/5; the reciprocal of 2/3 is 3/2. The number 1 is its own reciprocal; zero has no reciprocal.
Key Concepts
Property Two numbers whose product is 1 are reciprocals. For example, $\frac{1}{2} \times 2 = 1$, so $\frac{1}{2}$ and $2$ are reciprocals.
Examples The reciprocal of $\frac{5}{8}$ is $\frac{8}{5}$, because $\frac{5}{8} \times \frac{8}{5} = \frac{40}{40} = 1$. The reciprocal of the whole number 7 (or $\frac{7}{1}$) is $\frac{1}{7}$, because $7 \times \frac{1}{7} = 1$. The reciprocal of $\frac{1}{10}$ is 10 (or $\frac{10}{1}$), because $\frac{1}{10} \times 10 = 1$.
Explanation Reciprocals are math’s dynamic duo! They are two numbers that, when multiplied, give you a perfect 1. To find a fraction's reciprocal, you just flip it upside down. For example, $\frac{2}{5}$ becomes $\frac{5}{2}$. This 'flipping' trick is the secret weapon that lets you transform tricky fraction division problems into much friendlier multiplication tasks. It is very useful.
Common Questions
What is a reciprocal?
The reciprocal of a number is what you multiply it by to get 1. For a fraction a/b, the reciprocal is b/a. For a whole number n, it is 1/n.
What is the reciprocal of 3/4?
Flip numerator and denominator: reciprocal = 4/3. Check: 3/4 x 4/3 = 12/12 = 1.
What is the reciprocal of 7?
Write 7 as 7/1, then flip: reciprocal = 1/7. Check: 7 x 1/7 = 7/7 = 1.
How are reciprocals used in fraction division?
To divide by a fraction, multiply by its reciprocal. For example, 5 / (2/3) = 5 x (3/2) = 15/2 = 7 and 1/2.
Does zero have a reciprocal?
No. Zero has no reciprocal because any number times 0 equals 0, never 1. Division by zero is undefined.