Maths
What is the reciprocal of a number, and how do you find it?
Quick answer
The reciprocal of a number is 1 divided by that number, so the two multiply to make 1. To find the reciprocal of a fraction, flip it upside down: the reciprocal of 3/4 is 4/3.
In one sentence
The reciprocal of a number is what you multiply it by to get 1: for a fraction, you just turn it upside down.
Explained
Cut a pizza into 8 equal slices. One slice is 1/8 of the pizza, and 8 slices make 1 whole pizza. So 8 × 1/8 = 1. The numbers 8 and 1/8 are reciprocals of each other.
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The rule is short:
reciprocal of x = 1 ÷ x, and x × (1/x) = 1 (for any number x except 0)
The reciprocal is also called the multiplicative inverse, because it undoes multiplying. With a calculator you can find it with the 1/x or x⁻¹ button: the power −1 means "one over". (See powers and indices.)
How to find it depends on how the number is written:
- A fraction: swap the numerator and denominator. The reciprocal of 2/7 is 7/2.
- A whole number: write it over 1, then flip. 5 = 5/1, so its reciprocal is 1/5.
- A unit fraction: the reciprocal of 1/9 is 9/1 = 9.
- A mixed number: change it to an improper fraction first, then flip. 2½ = 5/2, so the reciprocal is 2/5.
- A decimal: change it to a fraction, or work out 1 ÷ the decimal. 0.25 = 1/4, so its reciprocal is 4.
- A negative number: keep the minus sign. The reciprocal of −3 is −1/3, because −3 × −1/3 = +1.
Two special cases: 0 has no reciprocal (no number times 0 makes 1, and you cannot divide by zero), and 1 and −1 are their own reciprocals. Flipping twice brings you back: the reciprocal of the reciprocal of 3/5 is 3/5 again.
Big numbers have small reciprocals, and small numbers have big ones. The reciprocal of 100 is 0.01; the reciprocal of 0.01 is 100. A number between 0 and 1 always has a reciprocal bigger than 1.
The most common use is dividing fractions: dividing by a number is the same as multiplying by its reciprocal. That is the "keep, change, flip" rule: 3/4 ÷ 2/5 = 3/4 × 5/2 = 15/8. See dividing fractions for more.
Examples
| Number | Written as a fraction | Reciprocal | Check |
|---|---|---|---|
| 6 | 6/1 | 1/6 | 6 × 1/6 = 1 |
| 3/8 | 3/8 | 8/3 (= 2⅔) | 3/8 × 8/3 = 24/24 = 1 |
| 1/10 | 1/10 | 10 | 1/10 × 10 = 1 |
| 1¾ | 7/4 | 4/7 | 7/4 × 4/7 = 1 |
| 0.5 | 1/2 | 2 | 0.5 × 2 = 1 |
| 0.2 | 1/5 | 5 | 0.2 × 5 = 1 |
| −2/9 | −2/9 | −9/2 | −2/9 × −9/2 = 1 |
| 1 | 1/1 | 1 | 1 × 1 = 1 |
| 0 | – | none | – |
Worked example: find the reciprocal of 3⅓. As an improper fraction, 3⅓ = (3 × 3 + 1)/3 = 10/3. Flip it: 3/10, which is 0.3. Check: 10/3 × 3/10 = 30/30 = 1.
Common mistakes
- Flipping only the fraction part of a mixed number. The reciprocal of 2½ is not 2 and 2/1. Convert first: 2½ = 5/2, so the reciprocal is 2/5.
- Changing the sign. The reciprocal of −4 is −1/4, not 1/4. A number and its reciprocal always have the same sign. (Changing the sign gives the opposite, or additive inverse: the opposite of 4 is −4.)
- Mixing up reciprocal and opposite. Reciprocals multiply to 1 (5 and 1/5). Opposites add to 0 (5 and −5).
- Flipping the wrong fraction when dividing. In 2/3 ÷ 4/5, keep the first fraction and flip only the one you divide by: 2/3 × 5/4.
Step by step
Step by step
- Write it as a fractionWhole number: put it over 1. Mixed number: make it an improper fraction. Decimal: write it as a fraction.
- Flip itSwap the numerator and the denominator.
- Keep the signIf the number was negative, its reciprocal is negative too.
- SimplifyWrite the answer in its simplest form, or as a whole number or mixed number if you prefer.
- CheckMultiply the number by your answer. You should get exactly 1.
For a decimal such as 1.25, you can also divide: 1 ÷ 1.25 = 0.8. Or use the fraction route: 1.25 = 5/4, and the reciprocal is 4/5 = 0.8.
Practice questions
Find the reciprocal
Try each one, then check your answer.
- What is the reciprocal of 7?
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7 = 7/1, flipped: 1/7 - What is the reciprocal of 1/9?
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1/9 flipped is 9/1: 9 - What is the reciprocal of 3/8?
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8/3 (= 2⅔) - What is the reciprocal of 2½?
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2½ = 5/2, flipped: 2/5 - What is the reciprocal of 0.5?
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0.5 = 1/2, flipped: 2 - What is the reciprocal of 0.25?
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0.25 = 1/4, flipped: 4 - What is the reciprocal of −6?
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Keep the sign: −1/6 - What is the reciprocal of −2/3?
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Flip and keep the sign: −3/2 (= −1½) - What is the reciprocal of 1?
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1 × 1 = 1, so it is 1 - What is the reciprocal of 1.25?
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1.25 = 5/4, flipped: 4/5 = 0.8 - What is the reciprocal of 3¼?
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3¼ = 13/4, flipped: 4/13 - Work out 3/4 × 4/3.
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12/12 = 1 (they are reciprocals) - Work out 2/3 ÷ 4/5 using the reciprocal.
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2/3 × 5/4 = 10/12 = 5/6 - Work out 6 ÷ 3/4.
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6 × 4/3 = 24/3 = 8 - What is the negative reciprocal of 5/2?
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Flip and change the sign: −2/5 - Which number has no reciprocal, and why?
Show answer
0, because no number times 0 equals 1 (1 ÷ 0 is undefined).
Word problems
Reciprocals in real life
Try each one, then check your answer.
- How many 3/4-litre bottles can you fill from 6 litres of juice?
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6 ÷ 3/4 = 6 × 4/3 = 8 bottles - A ribbon is 5 m long. How many pieces 2/3 m long can you cut from it?
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5 ÷ 2/3 = 5 × 3/2 = 7.5, so 7 whole pieces (with 1/3 m left over) - One batch of cookies uses 1/4 kg of flour. How many batches can you make with 3 kg?
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3 ÷ 1/4 = 3 × 4 = 12 batches - A line has a slope of 2/3. What is the slope of a line perpendicular to it?
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The negative reciprocal: −3/2 - Two 6-ohm resistors are connected in parallel, so 1/R = 1/6 + 1/6. What is the total resistance R?
Show answer
1/R = 2/6 = 1/3, so R is the reciprocal of 1/3: 3 ohms
Questions people ask
What is the reciprocal of a number?
It is 1 divided by the number. A number times its reciprocal is always 1: the reciprocal of 4 is 1/4, and the reciprocal of 2/3 is 3/2.
What is the reciprocal also called?
The multiplicative inverse. Some people also say "inverse" for short, but in maths "inverse" can mean other things too (like inverse operations or inverse functions), so "reciprocal" is clearer.
How do you find the reciprocal of a whole number?
Write it as a fraction over 1 and flip it. The reciprocal of 12 is 1/12. With a calculator, press 1 ÷ 12 or the x⁻¹ button: 0.0833…
How do you find the reciprocal of a mixed number?
Turn it into an improper fraction first, then flip. 1⅔ = 5/3, so its reciprocal is 3/5.
What is the reciprocal of a negative number?
It is negative too. The reciprocal of −5 is −1/5, because a negative times a negative is positive: −5 × −1/5 = 1.
What is a negative reciprocal?
Flip the number and change its sign. The negative reciprocal of 3 is −1/3. It matters for graphs: two lines are perpendicular when their slopes are negative reciprocals, like 2 and −1/2. See slope.
Is the reciprocal the same as the opposite?
No. The opposite (additive inverse) of 5 is −5, because 5 + (−5) = 0. The reciprocal (multiplicative inverse) of 5 is 1/5, because 5 × 1/5 = 1.
Do complex numbers have reciprocals?
Yes, every complex number except 0 does. For a + bi, the reciprocal is (a − bi) ÷ (a² + b²). For example, 1/(3 + 4i) = (3 − 4i)/25. This is upper-secondary maths.
Quick quiz
Reciprocals
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Question 1 of 5
What is the reciprocal of 5/6?
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Related topics
Dividing fractions
To divide by a fraction, multiply by its reciprocal. Keep the first fraction, change ÷ to ×, flip the second fraction, then simplify. 3/4 ÷ 1/2 = 3/4 × 2/1 = 3/2.
Multiplying fractions
To multiply fractions, multiply the numerators (tops) together and the denominators (bottoms) together, then simplify. You do not need a common denominator: 2/3 × 4/5 = 8/15.
Fractions
A fraction is part of a whole that has been split into equal parts. In ¾, the bottom number says how many equal parts and the top says how many you have.
Improper fractions and mixed numbers
An improper fraction has a numerator bigger than its denominator, like 7/4. A mixed number shows the same amount as a whole number plus a fraction, like 1 3/4.
Adding and subtracting fractions
To add or subtract fractions, first give them the same denominator. Then add or subtract only the numerators and keep the denominator. For example, 1/2 + 1/3 = 3/6 + 2/6 = 5/6.
Comparing fractions
To compare fractions, check whether they share a denominator or a numerator, or rewrite them with a common denominator. The bigger fraction is the one with more of the whole.