Your teacher said "the identity of addition is 0" in one breath and "the inverse of 5 is -5" in the next. Both are about addition, both involve zero somehow, and the words sound almost identical — identity, inverse. It is easy to leave class thinking they mean the same thing. They do not. They are opposite ideas playing opposite roles, and mixing them up quietly breaks how you solve equations.

The one-line difference

Identity is a single, universal element of the system. Inverse is paired to a specific input — every element has its own inverse. The identity is "leave alone." An inverse is "undo."

Identity: the do-nothing element

For addition, the identity is 0:

a + 0 = a \qquad \text{for every } a

For multiplication, the identity is 1:

a \times 1 = a \qquad \text{for every } a

There is one identity per operation. Every element of the system sees the same identity. Plug in any a you like — the rule holds.

Inverse: the element-specific undo

For addition, the inverse of a is the number -a that brings a back to the identity:

a + (-a) = 0

For multiplication, the inverse of a (when a \neq 0) is \tfrac{1}{a}:

a \times \tfrac{1}{a} = 1

Notice the pairing. The additive inverse of 5 is -5. The additive inverse of 7 is -7. The additive inverse of -3 is 3. Every element gets its own inverse — the inverse is a function of the element, unlike the identity, which is fixed for the whole system.

Identity versus inverse — do-nothing versus undoTwo contrasting diagrams stacked vertically. The top diagram shows a is added to zero and produces a — the identity leaves a unchanged. The bottom diagram shows a is added to minus a and produces zero — the inverse sends a to the identity element. A side panel lists examples: inverse of five is minus five, inverse of seven is minus seven, inverse of minus three is three — one inverse per input — whereas the identity zero is the same for every input. Identity: changes nothing a + 0 = a same 0 for every a Inverse: undoes to the identity a + (−a) = 0 different −a for every a examples inv(5) = −5 inv(7) = −7 inv(−3) = 3 inv(0) = 0 identity 0 (same for all)
The identity is one element, shared by everyone — add zero and nothing moves. An inverse is *paired*: every input has its own undo partner. Five and minus-five are partners, seven and minus-seven are partners, and so on. The identity is a single fixed fact about the system; inverses form a whole family, one per element.

Try it: add inverse to any input

Dragging a confirms a plus its inverse lands on the identity zeroA horizontal number line from negative ten to ten with a draggable red point labelled a. Readouts above show the current value of a, its additive inverse minus a, and the sum a plus minus a which always equals zero — the additive identity. −10 0 10 ↔ drag a
Drag $a$. Its inverse $-a$ flips to the opposite side. Their sum reads $0$ — the identity — for every single position. The inverse depends on $a$. The identity does not.

Why the mix-up is dangerous

When you solve 5x + 10 = 35, the steps you actually take are both identity-and-inverse moves, done on purpose.

Step 1. You want to remove the +10. You add -10 to both sides — the additive inverse of 10. The left side becomes 5x + 10 + (-10) = 5x + 0 = 5x. The inverse cancels the 10, and the identity 0 quietly disappears under the absorption rule a + 0 = a.

Step 2. You want to remove the coefficient 5. You multiply both sides by \tfrac{1}{5} — the multiplicative inverse of 5. The left side becomes 5x \cdot \tfrac{1}{5} = x \cdot 1 = x. The inverse cancels the 5, and the multiplicative identity 1 disappears under a \cdot 1 = a.

Every algebra step is: apply an inverse to cancel, hit the identity, read off the answer. If you muddle the two concepts, you cannot describe what you are doing — and when an equation gets harder (fractions, matrices, functions), not being able to describe the move is exactly when students get stuck.

Why: "cancel the 10" feels obvious for integers but breaks down when you try it on a matrix or a function. Thinking of it as "apply the additive inverse, which sends that term to the identity" is a rule that keeps working in every system with a well-defined identity and inverse.

A quick test you can run

Fill in the blanks:

Answers: 0, 1; 8, 3. If you answered the first pair with one value each and the second pair with values that changed based on the input, you have the distinction right. The identity is a constant of the system. The inverse is a function of the element.

So — do people use them interchangeably?

Careless writing sometimes blurs the two (you might hear "cancel with the identity" when people mean "cancel with the inverse to get the identity"), but in correct mathematical usage they are always distinct. Every competent text — school, JEE, university — keeps them separate, because they have to. The very reason you can solve equations at all is that the two ideas are separate and each plays a different role.

Short version: identity does nothing; inverse undoes a specific thing. Never swap the words. When you read a proof and the author writes "multiply by the inverse," expect a cancellation that reduces the expression to the identity. When they write "the identity," expect the specific fixed element of the system — 0 for addition, 1 for multiplication, the n \times n identity matrix for matrix multiplication.

Related: Operations and Properties · Identity and Inverse: The Two 'Undo' Buttons of Arithmetic · When Is 1 Not the Multiplicative Identity — Does It Fail in Any System? · Real Numbers and Their Properties