Inverse

Inverse means the opposite in effect. The reverse of.

Think of it as the mathematical "Undo" command ... something that gets us back to where we started.

It is a general idea in mathematics and has many meanings. Here are a few.

The Inverse of Adding is Subtracting

Adding moves us one way, subtracting moves us the opposite way.

Example: 20 + 9 = 29 can be reversed by 29 − 9 = 20 (back to where we started)

And the other way around:

Example: 15 − 3 = 12 can be reversed by 12 + 3 = 15 (back to where we started)

Additive Inverse

The additive inverse is what we add to a number to get zero.

Example: The additive inverse of −5 is +5, because −5 + 5 = 0.

Another example: the additive inverse of +7 is −7.

And the additive inverse of 0 is also 0, because 0 + 0 = 0.

The Inverse of Multiplying is Dividing

Multiplying can be "undone" by dividing.

Example: 5 × 9 = 45 can be reversed by 45 / 9 = 5

It works the other way around too, dividing can be undone by multiplying.

Example: 10 / 2 = 5 can be reversed by 5 × 2 = 10

Multiplicative Inverse

The multiplicative inverse is what we multiply a number by to get 1.

It is the reciprocal of a number.

Example: The multiplicative inverse of 5 is 15, because 5 × 15 = 1

But Not With 0

We can't divide by 0, so don't try!

Example: 5 × 0 = 0 can't be reversed by 0/0 = ???

Inverse of a Function

Doing a function and then its inverse will give us back the original value:

Diagram showing a function f transforming an apple into a banana, and its inverse function f^-1 transforming the banana back into an apple.
When the function f turns the apple into a banana,
Then the inverse function f^-1 turns the banana back to the apple

Here we have the function f(x) = 2x+3, written as a flow diagram:

Flow diagram for the function f(x) = 2x + 3. Input x, then multiply by 2, then add 3, resulting in 2x+3.

The Inverse Function goes the other way:

Flow diagram for the inverse function. Input y, then subtract 3, then divide by 2, resulting in (y-3)/2.

So the inverse of: 2x+3 is: (y−3)/2

Using function notation:

f(x) = 2x+3
f^-1(y) = (y − 3) / 2

Read Inverse of a Function to discover more.

Inverse Sine, Cosine and Tangent

Example: the sine function

Diagram showing a right triangle with angle theta. The sine function takes theta to the ratio opposite/hypotenuse. The inverse sine function takes the ratio opposite/hypotenuse back to theta.

The sine function sin takes angle θ and gives the ratio oppositehypotenuse

The inverse sine function sin^-1 takes the ratio oppositehypotenuse and gives angle θ

A similar idea applies to cosine, tangent and other trig functions.

sin⁻¹ doesn't mean 1/sin. It is just how we write the inverse function. Also called arcsin, arccos, and arctan.

Read Inverse Sine, Cosine, Tangent to discover more.

The Inverse of an Exponent is a Logarithm

Read logarithms to discover more, but basically:

Equation showing 2 raised to the power of 3 equals 8, transformed into the logarithm base 2 of 8 equals 3.

The logarithm tells us what the exponent is!

Example: Because 2³ = 8, we can say log₂(8) = 3. The logarithm has given us back the exponent.