How to Find the Reciprocal

Co-authored by David Jia

Last Updated: September 20, 2022 References

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Reciprocals are helpful in all sorts of algebraic equations. For example when you are dividing one fraction by another you multiply the first by the Reciprocal of the 2nd. You might also need reciprocals when finding equations of lines.

Method 1

Method 1 of 3:

Finding the Reciprocal of a Fraction or Whole Number

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$Step 1 Find the reciprocal of a fraction by flipping it.$

1
Find the reciprocal of a fraction by flipping it. The definition of "reciprocal" is simple. To find the reciprocal of any number, just calculate "1 ÷ (that number)."^{[1]
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Expert Source

David Jia
Academic Tutor

Expert Interview} For a fraction, the reciprocal is just a different fraction, with the numbers "flipped" upside down (inverted).^{[2]
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Research source}
- For instance, the reciprocal of ³/₄ is ⁴/₃.
- Any number times its reciprocal will give you 1.
$Step 2 Write the reciprocal of a whole number as a fraction.$

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Write the reciprocal of a whole number as a fraction. Again, the reciprocal of a number is always 1 ÷ (that number).^{[3]
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Research source} For a whole number, write that as a fraction; there's no point in calculating it out to a decimal.
- For instance, the reciprocal of 2 is 1 ÷ 2 = ¹/₂.

Method 2

Method 2 of 3:

Finding the Reciprocal of a Mixed Number

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Mixed numbers are part whole number and part fraction, such as 2⁴/₅.^{[4]
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Research source} There are two steps to finding the reciprocal of a mixed number, explained below.
2
Change it to an improper fraction.^{[5]
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Expert Source

David Jia
Academic Tutor

Expert Interview} Remember, the number 1 can always be written as (number)/(the same number), and fractions with the same denominator (lower number) can be added together.^{[6]
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Research source} Here's an example with 2⁴/₅:
- 2⁴/₅
- = 1 + 1 + ⁴/₅
- = ⁵/₅ + ⁵/₅ + ⁴/₅
- = ^(5+5+4)/₅
- = ¹⁴/₅.
$Step 3 Flip the fraction.$

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Flip the fraction. Once the number is written entirely as a fraction, you can find the reciprocal just like you would with any fraction: by flipping it.^{[7]
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Research source}^{[8]
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Expert Source

David Jia
Academic Tutor

Expert Interview}
- In the example above, the reciprocal of ¹⁴/₅ is ⁵/₁₄.

Method 3

Method 3 of 3:

Finding the Reciprocal of a Decimal

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$Step 1 Change it to a fraction if possible.$

1
Change it to a fraction if possible. You might recognize some common decimal numbers that can easily be turned into fractions.^{[9]
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Research source} For instance, 0.5 = ¹/₂, and 0.25 = ¹/₄. Once in fraction form, just flip the fraction to find the reciprocal.
- For instance, the reciprocal of 0.5 is ²/₁ = 2.
2
Write out a division problem. If you can't change it to a fraction, calculate the reciprocal of that number as a division problem: 1 ÷ (the decimal). You can use a calculator to solve this, or continue on to the next step to solve it by hand.^{[10]
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Research source}
- For example, you can find the reciprocal of 0.4 by calculating 1 ÷ 0.4.
3
Change the division problem to use whole numbers. The first step to dividing decimals is to move the decimal point until all the numbers involved are whole numbers. As long as you move the decimal point the same number of spaces for both numbers, you'll get the correct answer.
- For example, you can take 1 ÷ 0.4 and rewrite it as 10 ÷ 4. In this case, you've moved each decimal place one space to the right, which is the same as multiplying each number by ten.
Use long division techniques to calculate the reciprocal. If you calculate it for 10 ÷ 4, you'll get the answer 2.5, the reciprocal of 0.4.

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What is the reciprocal of a squared number?

Donagan

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It is 1 divided by the squared number.

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What is the reciprocal of 8?

Community Answer

1/8. Since every whole number is invisibly divided by 1 (in this case, 8/1), you just interchange the numerator and denominator, putting the denominator over the numerator and there is your reciprocal, 8/1. In other words, a reciprocal takes a number a/b and turns it into b/a.

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What is the reciprocal of a+b?

Donagan

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1 / (a+b).

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A number's negative reciprocal is the same as the regular reciprocal, multiplied by negative one.^{[11]
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Research source} For instance, the negative reciprocal of ³/₄ is -⁴/₃.

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The number 0 does not have a reciprocal, since 1 ÷ 0 is undefined.^{[12]
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Research source}

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The number 1 is its own reciprocal, since 1 ÷ 1 = 1.

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