Why Is a⁰ = 1 and 0! = 1? Simple Proofs for Students

Two facts in maths surprise almost every student:

• Any non-zero number raised to the power 0 equals 1. So 7⁰ = 1, 100⁰ = 1, and even (−5)⁰ = 1.
• 0! (zero factorial) equals 1, not 0.

Most students learn these as rules to memorise. They are not random rules, though. Both facts follow from patterns you already know. This guide gives four short proofs, explained so that a Class 7 student can follow them. It also covers the exceptions examiners like to ask about.

Part 1: Why Is a⁰ = 1?

Proof 1: The Power Pattern

Write down the powers of 2 and watch what happens as the exponent goes down by 1 each time:

Power	Value	What changed?
2⁴	16
2³	8	÷ 2
2²	4	÷ 2
2¹	2	÷ 2
2⁰	1	÷ 2

Each time the exponent goes down by 1, the value is divided by 2. To go from 2¹ to 2⁰, we divide once more: 2 ÷ 2 = 1.

The same works for any base. For powers of 10: 10² = 100, 10¹ = 10, and 10⁰ = 10 ÷ 10 = 1.

Proof 2: The Law of Exponents

When you divide two powers with the same base, you subtract the exponents:

aᵐ ÷ aⁿ = aᵐ⁻ⁿ

Now divide a number by itself, for example a³ ÷ a³:

• By the law of exponents: a³ ÷ a³ = a³⁻³ = a⁰
• But any non-zero number divided by itself is 1

Both answers must be equal, so a⁰ = 1.

The Exception: What About 0⁰?

Both proofs divide by a, and we can never divide by zero. That is why the rule is written as:

a⁰ = 1, for every number a ≠ 0

In school maths, 0⁰ is usually left undefined. In some higher-level topics, such as algebra and series, mathematicians agree to treat 0⁰ as 1 because it makes formulas simpler. In your school exams, though, write a ≠ 0 whenever you use a⁰ = 1.

Part 2: Why Is 0! = 1?

First, What Is a Factorial?

The factorial of a whole number n, written n!, is the product of all whole numbers from n down to 1:

n! = n × (n − 1) × (n − 2) × … × 2 × 1

Factorial	Working	Value
1!	1	1
2!	2 × 1	2
3!	3 × 2 × 1	6
4!	4 × 3 × 2 × 1	24
5!	5 × 4 × 3 × 2 × 1	120

Proof 3: The Factorial Pattern (Working Backwards)

Each factorial is the next one divided by the next number:

n! = (n + 1)! ÷ (n + 1)

Use this rule to walk backwards:

Step	Working	Value
3!	4! ÷ 4 = 24 ÷ 4	6
2!	3! ÷ 3 = 6 ÷ 3	2
1!	2! ÷ 2 = 2 ÷ 2	1
0!	1! ÷ 1 = 1 ÷ 1	1

The pattern leads to 0! = 1.

Proof 4: Counting Arrangements

A factorial counts the number of ways to arrange things in a row:

• 3 different books can be arranged in 3! = 6 ways.
• 1 book can be arranged in 1! = 1 way.
• 0 books can be arranged in exactly one way: leave the shelf empty.

So 0! = 1. This matters a lot in Class 11 permutations and combinations. For example, the formula ⁿCᵣ = n! ÷ (r! × (n − r)!) only gives the right answer for ⁵C₅ = 1 if 0! = 1.

What About Negative Factorials?

Factorials are defined only for whole numbers (0, 1, 2, 3, …). Something like (−1)! is not defined in school maths.

Quick Summary

Fact	Why it is true	Exception
a⁰ = 1	Each lower power divides by a, and aⁿ ÷ aⁿ = a⁰ = 1	Not for a = 0 (0⁰ is undefined in school maths)
0! = 1	1! ÷ 1 = 1, and there is exactly one way to arrange nothing	Negative factorials are not defined

Practice Questions (With Answers)

1. What is 9⁰? Answer: 1
2. What is (−3)⁰ + 5⁰? Answer: 1 + 1 = 2
3. Simplify 2⁵ ÷ 2⁵ using the law of exponents. Answer: 2⁰ = 1
4. What is 0! + 1! + 2!? Answer: 1 + 1 + 2 = 4
5. Find 5! ÷ 4!. Answer: 120 ÷ 24 = 5
6. Is 0! equal to 0 or 1? Answer: 1
7. Find ⁴C₄ = 4! ÷ (4! × 0!). Answer: 24 ÷ (24 × 1) = 1
8. True or false: 0⁰ = 1 in school maths. Answer: False; it is usually left undefined.