Identity Elements Examples in Math

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+014 + 0 and 0+140 + 14.

Answer

14 in both cases

First step

1
14+0=1414 + 0 = 14. Adding nothing changes nothing.

Full solution

  1. 2
    0+14=140 + 14 = 14. Same result.
  2. 3
    The identity element for addition is 0: a+0=0+a=aa + 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 23×123 \times 1 and 1×231 \times 23. Also show what happens with 23×023 \times 0.

Example 3

medium
To convert 23\frac{2}{3} to a fraction with denominator 15, multiply by what form of 1?

Example 4

medium
Solve 7⋅1⋅x=357 \cdot 1 \cdot x = 35.

Example 5

hard
Show that a⋆b=a+b−aba \star b = a + b - ab has identity e=0e=0, and find the inverse of aa.

Example 6

challenge
Does the operation a⋆b=aba\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+0387 + 0? What property does this demonstrate?

Example 2

medium
Fill in the blank and name the property: 456×□=456456 \times \square = 456 and 456×□=0456 \times \square = 0.

Example 3

easy
What is 14+014 + 0?

Example 4

easy
What is 23×123 \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 7×07 \times 0?

Example 8

easy
Does 5+15 + 1 equal 5?

Example 9

easy
What is 91\frac{9}{1}?

Example 10

easy
Fill the blank: 46×  ‾=4646 \times \underline{\ \ } = 46.

Example 11

medium
Why does multiplying a fraction by 33\frac{3}{3} not change its value?

Example 12

medium
Simplify 0+8x+00 + 8x + 0.

Example 13

medium
What is 1×1×1×71 \times 1 \times 1 \times 7?

Example 14

medium
In the operation a⋆b=a+b−3a\star b = a+b-3, which value of bb acts as an identity (leaves aa unchanged)?

Example 15

medium
Explain why a+0=aa + 0 = a but a×0=0a \times 0 = 0 in one sentence each.

Example 16

medium
Solve x×1=12−5x \times 1 = 12 - 5.

Example 17

medium
Which is the identity for the operation 'maximum', a⋆b=max⁡(a,b)a\star b=\max(a,b), over non-negative integers?

Example 18

medium
Rewrite 58\frac{5}{8} with denominator 24 by multiplying by a form of 1.

Example 19

challenge
For the operation a⋆b=ab+a+ba\star b = ab + a + b, find the identity element ee.

Example 20

challenge
Does the operation a⋆b=a+b+1a\star b = a + b + 1 have an identity? Find it.

Example 21

challenge
Why does 70=17^0 = 1 relate to the multiplicative identity?

Example 22

medium
Simplify 1×(x+0)1\times(x+0).

Example 23

easy
Fill the blank: 42+□=4242 + \square = 42.

Example 24

easy
Fill the blank: □×73=73\square \times 73 = 73.

Example 25

easy
True or false: 0×999=00 \times 999 = 0.

Example 26

easy
Is the additive identity for whole numbers 1 or 0?

Example 27

medium
Simplify 5x⋅1+0⋅y5x \cdot 1 + 0 \cdot y.

Example 28

medium
For the operation a⋆b=a⋅ba \star b = a \cdot b, what is the identity element?

Example 29

medium
For a⋆b=a+b−5a \star b = a + b - 5, find the identity element.

Example 30

medium
Simplify (x+0)⋅1−0(x+0)\cdot 1 - 0.

Example 31

medium
What is the identity for the operation a⋆b=min⁡(a,b)a \star b = \min(a,b) on positive integers up to 100?

Example 32

medium
Simplify 88⋅23+0\frac{8}{8}\cdot 23 + 0.

Example 33

hard
For the operation a⋆b=a+b+aba \star b = a + b + ab, find the identity element.

Example 34

hard
For the operation a⋆b=2aba \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⋆b=a2+b2a\star b=\sqrt{a^2+b^2} on non-negative reals, find the identity element.
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Background Knowledge

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

additionmultiplication