Reciprocal

The reciprocal of a number is 1 divided by that number. The reciprocal of 5 is 1/5, and the reciprocal of a fraction is the fraction flipped upside down, so the reciprocal of 2/3 is 3/2. A number multiplied by its reciprocal always equals 1. Zero is the only number with no reciprocal.

What is a reciprocal?

For any number x except 0, the reciprocal is:

The reciprocal is also called the multiplicative inverse, because multiplying by it takes you back to 1, just as adding the opposite (additive inverse) takes you back to 0. With powers, the reciprocal of x is written x⁻¹, so 4⁻¹ = 1/4.

How to find the reciprocal

• Whole numbers: put 1 over the number. The reciprocal of 12 is 1/12.
• Fractions: swap the numerator and denominator.
• Mixed numbers: change to an improper fraction first, then flip. 2¼ = 9/4, so its reciprocal is 4/9. Flipping just the ¼ and keeping the 2 is wrong.
• Decimals: divide 1 by the decimal, or change the decimal to a fraction and flip it. 0.4 = 2/5, so its reciprocal is 5/2 = 2.5.
• Negative numbers: the reciprocal keeps the sign, because a negative times a negative is positive.

Why zero has no reciprocal

The reciprocal of 0 would be 1 ÷ 0, and you can't divide by zero. Another way to see it: a reciprocal must multiply with the number to give 1, but anything multiplied by 0 gives 0. So 0 has no reciprocal. The only numbers that are their own reciprocals are 1 and −1.

What reciprocals are used for

Dividing fractions

Dividing by a number is the same as multiplying by its reciprocal. That is why the rule for dividing fractions is "keep, change, flip": keep the first fraction, change ÷ to ×, and flip the second one.

So 3/4 ÷ 2/5 = 15/8, or 1⅞ as a mixed number.

Solving equations

To undo a fraction in front of x, multiply both sides by its reciprocal. For ⅔x = 8, multiply by 3/2:

Perpendicular lines

Two lines are perpendicular when one slope is the negative reciprocal of the other. A line with slope 4 is perpendicular to a line with slope −1/4. See perpendicular lines for the full method.

Reciprocal trig functions

In trigonometry, three functions are defined as reciprocals of the main three: cosecant csc θ = 1/sin θ, secant sec θ = 1/cos θ and cotangent cot θ = 1/tan θ. Since sin 30° = 1/2, csc 30° = 2.

The reciprocal graph y = 1/x

Plot every number against its reciprocal and you get the graph of y = 1/x, called the reciprocal function. In England, AQA's GCSE Maths specification expects students to recognise and sketch it, with x ≠ 0.

Here x is a number and y is its reciprocal. The graph has two separate curves, one in the top right and one in the bottom left, because a number and its reciprocal always have the same sign. As x gets bigger, 1/x gets closer to 0. As x gets close to 0, 1/x shoots off towards infinity. The curve gets ever closer to the x-axis and the y-axis but never touches them. Lines like these are called asymptotes.

Shifting the graph moves the asymptotes. For y = 1/(x − 2) + 3, the vertical asymptote moves to x = 2, where the bottom of the fraction is zero, and the horizontal asymptote moves up to y = 3. Try one, then press Mark to check it in the course.

Common mistakes

• Changing the sign. The reciprocal of −3 is −1/3, not 1/3. That would be the negative reciprocal, a different thing.
• Flipping a mixed number as it stands. Convert to an improper fraction first.
• Giving 0 a reciprocal. 1 ÷ 0 is undefined.
• Mixing up reciprocal and opposite. The opposite of 5 is −5, which adds to 0. The reciprocal of 5 is 1/5, which multiplies to 1.
• Flipping the wrong fraction when dividing. Only the second fraction, the one you divide by, is flipped.

To practise reciprocals among other topics in a short timed paper, try today's Geometry, Algebra 2 & SAT Math paper. In the UK, the same practice opens as GCSE Maths Higher.

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