Reciprocal
$$\text{Reciprocal of } \frac{a}{b} = \frac{b}{a}, \quad \frac{a}{b} \times \frac{b}{a} = 1$$
What is Reciprocal?
The reciprocal of a fraction a/b is b/a - formed by swapping numerator and denominator. For whole numbers, the reciprocal of n is 1/n. Every non-zero number has exactly one reciprocal, and a number multiplied by its reciprocal always equals 1: $$\frac{a}{b} \times \frac{b}{a} = \frac{ab}{ab} = 1$$ Zero has no reciprocal because 1/0 is undefined.
Reciprocals are the foundation of fraction division. Dividing by any fraction is identical to multiplying by its reciprocal: $$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$$ The keep-change-flip method taught in school is simply a name for this reciprocal substitution - keep the first fraction, change ÷ to ×, flip the second fraction.
For negative fractions, the reciprocal carries the same sign: the reciprocal of -3/4 is -4/3. For mixed numbers, you must convert to an improper fraction first: the reciprocal of 2½ is not ½/2. Convert to 5/2, then flip to get 2/5.
Reciprocal explained to a beginner
Think of a reciprocal as the "flip" of a fraction. If you have 3/4, flipping it gives 4/3. Multiply them together: 3/4 × 4/3 = 12/12 = 1. That "comes back to 1" property is what makes reciprocals so useful - it's how division gets turned into multiplication.
A good way to picture it: if you walk 3 steps forward for every 4 steps sideways (ratio 3/4), the reciprocal 4/3 describes the exact opposite journey. One undoes the other, which is why multiplying gives 1.
When to use Reciprocal
Use reciprocals when dividing fractions (replace ÷ with × and flip the second fraction), when solving equations of the form (a/b)x = c (multiply both sides by the reciprocal b/a), and when working with rates where you need to invert a ratio - for example, converting km per hour to hours per km.
Worked examples for Reciprocal
This table quickly gives you the overview you need to understand Reciprocal and its most important comparisons.
| Number | Reciprocal | Product | Notes |
|---|---|---|---|
| 3/4 | 4/3 | 12/12 = 1 | Standard fraction |
| 5 | 1/5 | 5/5 = 1 | Whole number: write as 5/1, then flip |
| -2/7 | -7/2 | 14/14 = 1 | Negative: sign is preserved |
| 1 | 1 | 1/1 = 1 | Reciprocal of 1 is 1 |
| 2½ (= 5/2) | 2/5 | 10/10 = 1 | Mixed number: convert to improper first |
Common pitfalls
The most common error with mixed numbers: students take the reciprocal of 2½ as ½/2 = 1/4. You must convert to an improper fraction first (2½ = 5/2), then flip to get 2/5. Also, zero has no reciprocal - 1/0 is undefined. And the reciprocal of a negative fraction is also negative: the reciprocal of -3/4 is -4/3, not 4/3.
Frequently asked questions about Reciprocal
What is the reciprocal of a fraction?
The reciprocal of a fraction a/b is b/a - numerator and denominator are swapped. Multiplying a fraction by its reciprocal always gives 1: 3/4 × 4/3 = 12/12 = 1. This is called the multiplicative inverse.
How do reciprocals relate to fraction division?
Dividing by a fraction is identical to multiplying by its reciprocal. Instead of computing a/b ÷ c/d directly, you compute a/b × d/c. The keep-change-flip method is just a name for this substitution: flip the divisor and change the operation to multiplication.
What is the reciprocal of a whole number?
Write the whole number as a fraction over 1, then flip. The reciprocal of 5 is 1/5; the reciprocal of 8 is 1/8. Multiplying any whole number n by its reciprocal 1/n gives 1: n × 1/n = n/n = 1.
Does every number have a reciprocal?
Every non-zero real number has exactly one reciprocal. Zero is the only exception: 1/0 is undefined because no real number multiplied by 0 can equal 1.
Quiz: how well do you know reciprocals?
1. What is the reciprocal of 3/4, and what is their product?
2. From the examples table, what is the reciprocal of the whole number 5?
3. What is the correct reciprocal of the mixed number 2½? The pitfalls section identifies the common error.
4. Does every number have a reciprocal?
5. How do reciprocals relate to fraction division, according to the FAQ?