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 both a quantity (nothing) and a crucial placeholder in our number system.

Common stuck point: Zero isn't 'nothing'—it's a number with properties (additive identity).

Sense of Study hint: Try testing zero in different operations: what happens when you add it, multiply by it, or put it in a place value? Notice the different behaviors.

Worked Examples

Example 1

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

Solution

  1. 1
    (a) 17 + 0 = 17 — adding zero changes nothing (additive identity).
  2. 2
    (b) 17 \times 0 = 0 — any number times zero is zero (zero product property).
  3. 3
    (c) 0 \div 17 = 0 — zero divided by any nonzero number is zero.

Answer

(a)\; 17 \qquad (b)\; 0 \qquad (c)\; 0
Zero has three distinct roles in arithmetic: additive identity (a + 0 = a), annihilator for multiplication (a \times 0 = 0), and gives zero when divided by a nonzero number. These are separate properties, not one rule.

Example 2

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

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

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

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