Key takeaway: The reciprocal of a nonzero number is the number that multiplies with it to give 1, found by swapping the numerator and denominator of its fraction form, so 7/9 pairs with 9/7.
Grade 66.NS.A.1Numbers and Operations
Definition of Reciprocal
The reciprocal of a number is the partner that brings the product to exactly 1. Two numbers that multiply to 1 are called reciprocals of each other, or multiplicative inverses.
To find it, write the number as a fraction and turn that fraction upside down. A whole number such as 12 is \(\frac{12}{1}\), so its partner is \(\frac{1}{12}\). A mixed number must become an improper fraction first. Zero has no partner, because no number times 0 equals 1.
Formula and notation
For a fraction with \(a \neq 0\) and \(b \neq 0\):
\[\frac{a}{b} \times \frac{b}{a} = 1\]
For any \(n\), the partner is written \(\frac{1}{n}\) or \(n^{-1}\). The sign stays the same: a positive number has a positive partner, and a negative number has a negative one.
Key facts
- A number and its reciprocal always have a product of exactly 1.
- Zero has no partner of this kind, since division by zero is undefined.
- Only 1 and negative 1 are their own partners under this flip.
- Dividing by a fraction is the same as multiplying by its flipped version.
Reciprocal example
Find the reciprocal of \(\frac{7}{9}\), of 12, and of \(2\frac{3}{8}\). Then check the first one.
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\(\frac{7}{9}\) flips to \(\frac{9}{7}\).
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12 is \(\frac{12}{1}\), which flips to \(\frac{1}{12}\).
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\(2\frac{3}{8} = \frac{19}{8}\), which flips to \(\frac{8}{19}\).
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Check: \(\frac{7}{9} \times \frac{9}{7} = \frac{63}{63} = 1\).
The partners are \(\frac{9}{7}\), \(\frac{1}{12}\) and \(\frac{8}{19}\).
Common mistake
Watch out
Hana says the reciprocal of \(-\frac{4}{11}\) is \(\frac{11}{4}\), because “it flips the sign.”
Why it is wrong
That mixes up two ideas. The opposite changes the sign; the reciprocal changes the position of the numbers. Here \(-\frac{4}{11} \times (-\frac{11}{4}) = 1\), so the partner is \(-\frac{11}{4}\). The sign stays negative.
Related terms and lessons
Browse the glossary: A–Z list · Grade 6 vocabulary · Numbers and Operations