Reciprocal
In short: Reciprocal describes the partner you multiply a nonzero number by to land on exactly 1, and for a fraction it is the same fraction turned upside down, such as 8/5 for 5/8.
6th GradeNumbers and Operations6.NS.A.1
What is Reciprocal?
A number and its reciprocal cancel each other under multiplication. Take \(\frac{5}{8}\) of some amount, then take \(\frac{8}{5}\) of what you got, and the original amount returns.
Formally, for \(a \neq 0\) and \(b \neq 0\), the partner of \(\frac{a}{b}\) is \(\frac{b}{a}\), since \(\frac{a}{b} \times \frac{b}{a} = 1\). Any whole number \(n\) pairs with \(\frac{1}{n}\); the number zero has no partner at all.
How to spot it
- A division by a fraction must be rewritten as a multiplication.
- A problem says two numbers have a product of 1.
- You are asked for the “multiplicative inverse” or told to flip a fraction.
- A whole number is turned into a fraction over 1 before being flipped.
The method
- Write the number as a fraction; a whole number \(n\) becomes \(\frac{n}{1}\), a mixed number becomes an improper fraction.
- Swap the numerator and the denominator.
- Do not change the sign: a negative starting number gives a negative result, so \(-\frac{2}{7}\) becomes \(-\frac{7}{2}\).
- Check by multiplying the two numbers: the product must be exactly 1.
Reciprocal: solved example
Problem
Find the reciprocal of \(3\frac{1}{5}\) and verify it.
Solution
Convert to an improper fraction: \(3\frac{1}{5} = \frac{16}{5}\).
Flip it: \(\frac{5}{16}\). Check: \(\frac{16}{5} \times \frac{5}{16} = \frac{80}{80} = 1\). The answer is \(\frac{5}{16}\).
Try it
Question 1
What number multiplies with 0.25 to give 1?
Show the answer
\(0.25 = \frac{1}{4}\), so its partner is 4, since \(\frac{1}{4} \times 4 = 1\).
Question 2
Does 0 have a partner that multiplies with it to give 1?
Show the answer
No. Multiplying 0 by anything gives 0, never 1, so no partner exists.
Reciprocal vs. Opposite Numbers
Opposites add to 0 and sit on either side of zero on the number line, like 5 and \(-5\). Reciprocals multiply to 1 and come from flipping a fraction, like 5 and \(\frac{1}{5}\). The two ideas use different operations and different targets.
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Related words
Whole dictionary: A–Z · 6th Grade · Numbers and Operations