Mathématiques
What Is a Reciprocal?
The reciprocal of a number is 1 divided by it, so 5 becomes 1/5 and 2/3 becomes 3/2. The check is always the same: the two must multiply to 1.
Reciprocal
The reciprocal of a number is 1 divided by it — the number you multiply by to reach 1, found by turning a fraction upside down.
- Aussi appelé
- multiplicative inverse
- Où les élèves le rencontrent
- Grade 6 to 8, when dividing by a fraction is first taught, and again at GCSE, where negative indices and the graph of y = 1/x both rest on it.
La réponse en bref
The reciprocal of a number is 1 divided by that number, so the reciprocal of 5 is 1/5 and the reciprocal of 2/3 is 3/2. A number multiplied by its reciprocal always gives 1. Zero is the single exception: nothing multiplied by 0 gives 1, so 0 has no reciprocal.
Un exemple
2/3 × 3/2 = 6/6 = 1
This line is not a demonstration of the answer, it is the definition doing its own checking. Whenever you produce a reciprocal, multiply it by the original: if the product is not exactly 1, the flip was wrong. It takes three seconds and catches the sign errors that flipping negatives tends to produce.
Four reciprocals, each with its own check
Whole numbers and decimals feel like they need a different rule from fractions, but they do not. 5 is 5/1, so flipping it gives 1/5, and 0.25 is 1/4, so flipping it gives 4. Rewrite as a fraction, turn it over, verify the product is 1.
The last of the four is where mistakes cluster. Flipping a negative number keeps the minus sign, because a negative multiplied by a positive can never come to 1.
- 5 → 1/5, because 5 × 1/5 = 1
- 2/3 → 3/2, because 2/3 × 3/2 = 1
- 0.25 → 4, because 0.25 is 1/4 and 1/4 × 4 = 1
- −4 → −1/4, because −4 × −1/4 = 1 — the sign stays where it was
Why zero is left out
The reciprocal of 0 would have to be a number that gives 1 when multiplied by 0. There is no such number, because 0 multiplied by anything at all is 0. So zero has no reciprocal — not a very large one, not an undefined one that might be filled in later, none.
This is the same fact as you cannot divide by zero, said from the other side, which is why a calculator answers 1 ÷ 0 with an error rather than a number. It is also why every definition of a rational number carries the condition that the denominator is not zero.
The two places a question really needs it
Dividing by a fraction is the first. Dividing by 3/4 and multiplying by 4/3 are the same operation, which is what the keep-change-flip slogan is pointing at without explaining. The slogan is easier to remember once you know that flip means take the reciprocal, and that the reason it works is that dividing by a number is multiplying by its reciprocal.
Negative indices are the second. x⁻¹ means the reciprocal of x, so 2⁻¹ is 1/2 and 2⁻³ is 1/8. A third use arrives with straight-line graphs: perpendicular gradients are negative reciprocals of each other, so a line of gradient 2/5 meets one of gradient −5/2 at a right angle.
Questions fréquentes
What is the reciprocal of a whole number?
It is 1 over that number: the reciprocal of 7 is 1/7, and of 12 is 1/12. Whole numbers behave like every other case once you remember that 7 means 7/1. Turning 7/1 upside down gives 1/7, and the check confirms it: 7 × 1/7 = 1.
Why does dividing by a fraction mean multiplying by its reciprocal?
Because dividing by a number and multiplying by its reciprocal are two names for the same operation. Halving something and multiplying it by 1/2 give identical results, and the same holds for any non-zero number. The rule is not a trick invented for fractions; fractions are just where it becomes visible.
Does zero have a reciprocal?
No. There is no number that multiplies with 0 to give 1, because 0 times anything is 0. Every other number, positive, negative, fractional or decimal, has exactly one reciprocal. Zero is the only exception in the whole system, which is why it is worth remembering as a special case.
Is the reciprocal of a negative number negative?
Yes. The reciprocal of −4 is −1/4, not 1/4. The product of the two must come to positive 1, and only two negatives multiply to a positive, so the sign has to be kept. This is exactly the error the multiply-and-check habit is designed to catch.
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