Zero Examples in Math

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

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

Concept Recap

The number representing the absence of quantity; the additive identity and placeholder in positional notation.

Zero is the placeholder that makes '10' different from '1'—it marks empty positions.

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: Zero is the amount of nothing, the additive identity, and the mark that keeps empty places from collapsing.

Common stuck point: The procedure for zero is the easy part; the trap is dropping the placeholder zero so 105 becomes 15. Asking "Am I representing the absence of an amount or holding an empty place open?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Am I representing the absence of an amount or holding an empty place open?

Worked Examples

Example 1

easy
Evaluate: (a) 17+017 + 0, (b) 17×017 \times 0, (c) 0÷170 \div 17.

Answer

(a)  17(b)  0(c)  0(a)\; 17 \qquad (b)\; 0 \qquad (c)\; 0

First step

1
(a) 17+0=1717 + 0 = 17 — adding zero changes nothing (additive identity).

Full solution

  1. 2
    (b) 17×0=017 \times 0 = 0 — any number times zero is zero (zero product property).
  2. 3
    (c) 0÷17=00 \div 17 = 0 — zero divided by any nonzero number is zero.
Zero has three distinct roles in arithmetic: additive identity (a+0=aa + 0 = a), annihilator for multiplication (a×0=0a \times 0 = 0), and gives zero when divided by a nonzero number. These are separate properties, not one rule.

Example 2

medium
Why is 50\frac{5}{0} undefined? Explain using the definition of division.

Example 3

easy
How many fish are in an empty fish bowl?

Example 4

medium
Which plate has zero apples: a plate with 2 apples or an empty plate?

Example 5

easy
Sam has 7 grapes. He gets 0 more. How many grapes does Sam have?

Example 6

easy
Lily had 6 shells. She picked up 0 more. How many shells does she have?

Example 7

medium
Ben has 2 apples. He picks up 0 apples. Then he picks up 0 more. How many apples does Ben have?

Example 8

easy
Maya has 12 crayons. She gives 0 crayons away. How many crayons does Maya have?

Example 9

easy
There are 9 cookies on a plate. The dog eats 9 cookies. How many cookies are left?

Example 10

hard
A bus has 18 kids. At the stop, 0 kids get off and then 18 kids get off. How many kids are on the bus now?

Example 11

easy
Write the number with 6 hundreds, 0 tens, and 2 ones.

Example 12

easy
Why is 360360 a different number from 3636?

Example 13

hard
Mr. Lee writes a 3-digit number on the board. It has 0 in the tens place and 0 in the ones place. The hundreds digit is 7. What is the number, and what is the value of each 0?

Practice Problems

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

Example 1

easy
Is zero even or odd? Justify.

Example 2

medium
The number 3,0203{,}020 has a zero in the tens place. How does removing that zero (writing 320 instead) change the number's value?

Example 3

easy
What is 7+07 + 0?

Example 4

easy
What is 0×50 \times 5?

Example 5

easy
Why is the number 307 different from 37?

Example 6

easy
What is 9−09 - 0?

Example 7

easy
Is 5÷05 \div 0 defined?

Example 8

easy
What is 0+00 + 0?

Example 9

easy
How many objects are in an empty box?

Example 10

easy
Place zero correctly: write 'five hundred two' as a numeral.

Example 11

medium
Simplify 0×8+40\times 8 + 4.

Example 12

medium
Solve for xx: x+0=12x + 0 = 12.

Example 13

medium
What is 06\frac{0}{6}?

Example 14

medium
Evaluate (−3)+3(-3) + 3.

Example 15

medium
What is 505^0?

Example 16

medium
If a product a×b=0a\times b = 0, what can you conclude about aa or bb?

Example 17

medium
Compute 100×0×25100 \times 0 \times 25.

Example 18

medium
On a number line, what is the distance from 00 to −6-6?

Example 19

challenge
Solve x(x−4)=0x(x-4)=0 using the zero product property.

Example 20

challenge
Why does 00\frac{0}{0} have no single value?

Example 21

challenge
A number's only digits are 0 and one 3. It is three thousand. Write it and identify how many placeholder zeros it has.

Example 22

medium
Evaluate 7+0×97 + 0 \times 9 using order of operations.

Example 23

easy
Your plate is empty. How many candies are on your plate?

Example 24

easy
You had 1 cookie. You ate it. How many cookies are left?

Example 25

easy
Hold up zero fingers. How many fingers are up?

Example 26

easy
There are 3 birds on a branch. They all fly away. How many birds are on the branch?

Example 27

medium
A cup has no water in it. How much water is in the cup?

Example 28

medium
You have 2 stickers. You give both away. How many stickers do you have?

Example 29

hard
Show zero blocks on the table. What does the table look like?

Example 30

easy
What is 0+50 + 5?

Example 31

easy
What is 4+04 + 0?

Example 32

easy
What is 0+90 + 9?

Example 33

medium
There are 10 ducks in a pond. 0 more swim over. How many ducks are in the pond?

Example 34

easy
Compute 8−08 - 0.

Example 35

easy
Compute 6−66 - 6.

Example 36

easy
Compute 15−015 - 0.

Example 37

medium
Compute 14−1414 - 14.

Example 38

medium
Theo had 11 marbles. He lost all 11 in a hole. How many marbles does Theo have?

Example 39

easy
What does the 0 in 305305 mean?

Example 40

easy
What does the 0 in 480480 mean?

Example 41

easy
Write 'two hundred seven' as a numeral.

Example 42

medium
What does the 0 in 704704 mean, and what would the number become if you removed the 0?

Example 43

medium
In the number 890890, what does each digit stand for?
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Background Knowledge

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

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