Identity Elements Examples: 42 Problems with Answers

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Identity Elements.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

Special numbers that leave any other number unchanged under a given operation: 0 for addition, 1 for multiplication.

Adding 0 leaves any number unchanged; multiplying by 1 also leaves it unchanged. Both are 'do-nothing' values.

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: An identity element leaves any number unchanged: add 0, or multiply by 1.

Common stuck point: The procedure for identity elements is the easy part; the trap is using 0 as the multiplicative identity. Asking "Does this number leave every other number unchanged under the given operation?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Does this number leave every other number unchanged under the given operation?

Worked Examples

Example 1

easy

Show that adding 0 to any number doesn't change it. Verify with

14 + 0

and

0 + 14

Answer

14 in both cases

First step

14 + 0 = 14

. Adding nothing changes nothing.

Full solution

2
$0 + 14 = 14$ . Same result.
3
The identity element for addition is 0: $a + 0 = 0 + a = a$ .

Zero is the additive identity. Adding 0 to any number leaves it unchanged. It is like adding an empty set.

Example 2

medium

Show that multiplying any number by 1 doesn't change it. Use

23 \times 1

and

1 \times 23

. Also show what happens with

23 \times 0

Example 3

medium

To convert

\frac{2}{3}

to a fraction with denominator 15, multiply by what form of 1?

Example 4

medium

Solve

7 \cdot 1 \cdot x = 35

Example 5

hard

Show that

a \star b = a + b - ab

has identity

e=0

, and find the inverse of

a

Example 6

challenge

Does the operation

a\star b=a^b

on positive reals have an identity? Explain.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

What is

387 + 0

? What property does this demonstrate?

Example 2

medium

Fill in the blank and name the property:

456 \times \square = 456

and

456 \times \square = 0

Example 3

easy

What is

14 + 0

Example 4

easy

What is

23 \times 1

Example 5

easy

Which number is the additive identity?

Example 6

easy

Which number is the multiplicative identity?

Example 7

easy

What is

7 \times 0

Example 8

easy

Does

5 + 1

equal 5?

Example 9

easy

What is

\frac{9}{1}

Example 10

easy

Fill the blank:

46 \times \underline{\ \ } = 46

Example 11

medium

Why does multiplying a fraction by

\frac{3}{3}

not change its value?

Example 12

medium

Simplify

0 + 8x + 0

Example 13

medium

What is

1 \times 1 \times 1 \times 7

Example 14

medium

In the operation

a\star b = a+b-3

, which value of

b

acts as an identity (leaves

a

unchanged)?

Example 15

medium

Explain why

a + 0 = a

but

a \times 0 = 0

in one sentence each.

Example 16

medium

Solve

x \times 1 = 12 - 5

Example 17

medium

Which is the identity for the operation 'maximum',

a\star b=\max(a,b)

, over non-negative integers?

Example 18

medium

Rewrite

\frac{5}{8}

with denominator 24 by multiplying by a form of 1.

Example 19

challenge

For the operation

a\star b = ab + a + b

, find the identity element

e

Example 20

challenge

Does the operation

a\star b = a + b + 1

have an identity? Find it.

Example 21

challenge

Why does

7^0 = 1

relate to the multiplicative identity?

Example 22

medium

Simplify

1\times(x+0)

Example 23

easy

Fill the blank:

42 + \square = 42

Example 24

easy

Fill the blank:

\square \times 73 = 73

Example 25

easy

True or false:

0 \times 999 = 0

Example 26

easy

Is the additive identity for whole numbers 1 or 0?

Example 27

medium

Simplify

5x \cdot 1 + 0 \cdot y

Example 28

medium

For the operation

a \star b = a \cdot b

, what is the identity element?

Example 29

medium

For

a \star b = a + b - 5

, find the identity element.

Example 30

medium

Simplify

(x+0)\cdot 1 - 0

Example 31

medium

What is the identity for the operation

a \star b = \min(a,b)

on positive integers up to 100?

Example 32

medium

Simplify

\frac{8}{8}\cdot 23 + 0

Example 33

hard

For the operation

a \star b = a + b + ab

, find the identity element.

Example 34

hard

For the operation

a \star b = 2ab

, does an identity element exist?

Example 35

hard

Why is the identity element in a group always unique?

Example 36

challenge

For the operation

a\star b=\sqrt{a^2+b^2}

on non-negative reals, find the identity element.

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Related Concepts

Addition Multiplication Inverse Operations Algebra as Structure

Background Knowledge

These ideas may be useful before you work through the harder examples.

additionmultiplication