Inverses — Reciprocals and 1/n

The "inverse" of a number, in the way we mean here, is its multiplicative inverse: the number you'd multiply it by to get 1.

Other names: reciprocal, flipped fraction.

Two cases

Case 1 — inverse of a fraction

Flip it.

inverse of  2/3  =  3/2
inverse of  5/8  =  8/5
inverse of  7/4  =  4/7

Check: 2/3 × 3/2 = 6/6 = 1. ✓

Case 2 — inverse of a whole integer

Put it under 1.

inverse of  3  =  1/3
inverse of  7  =  1/7
inverse of  1  =  1/1 = 1

Check: 3 × 1/3 = 3/3 = 1. ✓

The inverse of an integer n is the fraction 1/n. That's it.

Why we say "formal" inverse

There is no rounding, no "kind of", no "almost". The inverse is a fact:

For any non-zero number x, the inverse is 1/x, and x × (1/x) = 1.

This is what mathematicians mean when they say a number is "invertible". Zero has no inverse because dividing by zero is undefined.

Worked examples

inverse of  4/9         →  9/4
inverse of  8           →  1/8
inverse of  1/5         →  5/1   (i.e. 5)
inverse of  -3/7        →  -7/3   (the sign stays)
inverse of  0           →  undefined (no answer exists)

How we will ask this on the exam

You'll see a question like:

What is the inverse of 3/8?

Type 8/3 (numerator 8, denominator 3).

Or:

What is the inverse of 6?

Type 1/6 (numerator 1, denominator 6).

The exam UI gives you a numerator and denominator slot. Use them.

A subtle point: inverse of an inverse

The inverse of the inverse is the original number:

inverse of (inverse of 3/5)
  inverse of 3/5 = 5/3
  inverse of 5/3 = 3/5  ← back where we started

Useful intuition: flipping a fraction twice does nothing.

Try these (answers below)

1. Inverse of 9
2. Inverse of 2/7
3. Inverse of 1
4. Inverse of -5/4

1. 1/9
2. 7/2
3. 1 (1/1 = 1)
4. -4/5