The reciprocal of a number is 1 divided by that number, so the two multiply to 1. For a fraction, swap the numerator and denominator: the reciprocal of 34 is 43, and the reciprocal of 5 is 15. Zero has no reciprocal. Dividing by a fraction is the same as multiplying by its reciprocal.
Flip a fraction to get its reciprocal; the pair always multiplies to 1. That is why dividing by a fraction means multiplying by its reciprocal.
Formal definition and notation
For any nonzero numberx, its reciprocal is 1x, the unique number with x × 1x = 1; it is also called the multiplicative inverse and written x−1. For a fraction ab with a ≠ 0 and b ≠ 0, the reciprocal is ba. Zero has no reciprocal, because no number times 0 equals 1.
Symbol
Read it as
Meaning
1x
one over x
the reciprocal of x, for x ≠ 0
x−1
x to the negative one
another way to write the reciprocal: 4−1 = 14
ba
b over a
the reciprocal of the fraction ab
Key facts
A number times its reciprocal is always 1: 29 × 92 = 1.
To find the reciprocal of a whole number, write it over 1 and flip: 7 = 71, so its reciprocal is 17.
The additive inverse is another name for the opposite (sum 0). The reciprocal is the multiplicative inverse (product 1).
Reciprocalvs.Inverse operations
Inverse operations undo each other, like multiplying and dividing. The reciprocal is a number, not an operation.
Quick check: 3 questions
Question 1 of 3
What is the reciprocal of 135?
Show answer and explanation
Answer: B.135 = 85, and flipping gives 58. 153 flips only the fraction part, −85 is the opposite, and 53 ignores the whole number.
Question 2 of 3
What is 49 ÷ 23?
Show answer and explanation
Answer: C.49 × 32 = 1218 = 23. 827 multiplies without flipping, 278 flips the wrong fraction, and 32 is the reciprocal of the divisor itself.
Question 3 of 3
Which number has no reciprocal?
Show answer and explanation
Answer: D. Zero has no reciprocal because nothing times 0 equals 1. 1 and −1 are their own reciprocals, and the reciprocal of 0.5 is 2.
Reciprocal: frequently asked questions
What is the reciprocal of a whole number?
It is 1 over that number. Write the whole number as a fraction over 1, then flip it: 8 is 81, so its reciprocal is 18. Check by multiplying: 8 × 18 = 1. Every whole number except 0 has a reciprocal, and that reciprocal is a unit fraction.
Why do we multiply by the reciprocal to divide fractions?
Dividing by a number undoes multiplying by it, and so does multiplying by its reciprocal. For example, 6 ÷ 12 asks how many halves fit in 6. Each whole holds 2 halves, so the answer is 6 × 2 = 12, and 2 is the reciprocal of 12. The same reasoning works for every nonzero divisor.
Is the reciprocal the same as the multiplicative inverse?
Yes. Both names mean the number you multiply by to get 1, the identity for multiplication. “Reciprocal” is the everyday word, and “multiplicative inverse” is the formal one used in algebra. Do not confuse it with the additive inverse, or opposite, which is the number you add to get 0.
NumberA number is an amount or value used to count, measure, order and compare. We write numbers with digits (7), words (seven) or other symbols, but the number itself is the amount: 7 dots, the word seven and the numeral 7 all show the same number. Numbers include whole numbers, fractions, decimals and negatives.Number System · Kindergarten, Grade 1, Grade 2Whole NumberA whole number is 0 or a positive counting number: 0, 1, 2, 3, and so on. Whole numbers do not include fractions, decimals, or negative numbers, so 12 is a whole number but 8.5, 34 and −3 are not. Every whole number is also an integer.Number System · Grade 3, Grade 4, Grade 5Mixed NumberA mixed number has a whole number and a fraction together, such as 234, which means 2 + 34. It represents a value greater than one whole when the whole-number part is positive. Every mixed number can be rewritten as an improper fraction, and back: 234 = 114.Number System · Grade 4, Grade 5, Grade 6Improper FractionAn improper fraction has a numerator that is greater than or equal to its denominator, so its value is 1 or more. Examples are 74, 92 and 55. Every improper fraction can be rewritten as a whole number or a mixed number: 74 = 134. Improper does not mean wrong.Number System · Grade 4, Grade 5, Grade 6SimplifySimplify means to rewrite a number or expression in a shorter, easier form without changing its value. To simplify a fraction, divide the numerator and denominator by a common factor until none is left: 1824 = 34. To simplify an expression, combine like terms and do the operations you can: 3x + 2x = 5x.Number System · Grade 4, Grade 5, Grade 6DivisorA divisor is the number you divide by. In 56 ÷ 8 = 7, the divisor is 8: it tells how many equal groups to make, or how big each group is. The divisor can never be 0. A divisor of a whole number is also one of its factors, because it divides that number with no remainder.Operations · Grade 3, Grade 4, Grade 5UndefinedUndefined describes an expression that has no value, most often because it divides by zero. 70 is undefined, since no number times 0 gives 7. A vertical line has an undefined slope, and 3x − 2 is undefined at x = 2. Undefined does not mean zero: 07 = 0, but 70 has no value.Problem Solving · Grade 6, Grade 8, Algebra 1OppositeThe opposite of a number is the number the same distance from 0 on the other side of the number line. The opposite of 7 is −7, and the opposite of −7 is 7. Opposites add to 0, and 0 is its own opposite. The opposite of a negative number is positive.Number System · Grade 6, Grade 7, Grade 8Additive InverseThe additive inverse of a number is the value that adds to it to give zero. The additive inverse of 9 is −9, because 9 + (−9) = 0, so it is the same thing as its opposite. To find it, change the sign: the additive inverse of −14 is 14, and 0 is its own.Number System · Grade 6, Grade 7, Grade 8Inverse OperationsInverse operations are operations that undo each other. Addition and subtraction are inverses: 9 + 6 − 6 = 9. Multiplication and division are inverses: 7 × 4 ÷ 4 = 7. You use them to check answers and to solve equations, undoing each step in reverse order until the variable is alone.Operations · Grade 1, Grade 3, Grade 6Unit FractionA unit fraction is a fraction with a numerator of 1, such as 12, 15 or 112. It names one equal part of a whole split into that many parts. The bigger the denominator, the smaller the unit fraction, and every fraction ab is a copies of 1b.Number System · Grade 3, Grade 4, Grade 5MultiplicationMultiplication is the operation that combines equal groups to find a total. The fact 4 × 6 = 24 means 4 groups of 6, or 6 + 6 + 6 + 6. The numbers you multiply are called factors, and the result is the product. Order does not matter: 6 × 4 is also 24.Operations · Grade 2, Grade 3, Grade 4FractionA fraction is a number that names equal parts of a whole, written ab. The bottom number, the denominator, tells how many equal parts make one whole; the top number, the numerator, tells how many of those parts you have. So 34 means 3 of 4 equal parts. Every fraction is also a point on the number line.Number System · Grade 3, Grade 4, Grade 5NumeratorThe numerator is the top number in a fraction. It tells how many parts are being counted, while the denominator below the fraction bar tells how many equal parts make one whole. In 35, the numerator is 3: three of five equal parts are taken.Number System · Grade 3, Grade 4, Grade 5DenominatorThe denominator is the bottom number in a fraction. It tells how many equal parts make one whole: in 35, the 5 means the whole is cut into 5 equal parts, and the numerator 3 counts the parts you have. The denominator can never be 0.Number System · Grade 3, Grade 4, Grade 5ExponentAn exponent tells how many times to use a base as a factor. It is the small raised number to the upper right of the base: in 53, the base is 5 and the exponent is 3, so 53 = 5 × 5 × 5 = 125. Any nonzero number with an exponent of 0 equals 1.Algebra · Grade 6, Grade 7, Grade 8Negative ExponentA negative exponent tells you to take the reciprocal of the base raised to the matching positive exponent: a−n = 1an for a ≠ 0. So 2−3 = 18 and 10−2 = 0.01. A negative exponent makes a number smaller in size, but it never makes the value negative.Algebra · Grade 8, Algebra 1DivisionDivision is the operation of splitting a number into equal groups. In 12 ÷ 3 = 4, the dividend 12 is shared into 3 equal groups (the divisor), and each group gets 4 (the quotient). Division undoes multiplication, because 3 × 4 = 12. If the groups do not come out even, the amount left over is the remainder.Operations · Grade 3, Grade 4, Grade 5