Zero and Negative Indices

Scientists write a virus as 10^{-7} metres wide and a data rate in 10^{-3} seconds — negative powers are how we tame the tiniest numbers without a page of leading zeros, and the power 0 turns up the moment you divide a quantity by itself. So both had better mean something sensible.

Here's a puzzle that stops people in their tracks. A power like 2^3 means "multiply three 2s together". So what on earth could 2^0 mean — "multiply zero 2s together"? And 2^{-3}, "multiply minus three 2s"? You can't write down a negative number of factors. It sounds like nonsense.

But mathematics has a beautiful trick up its sleeve. Instead of throwing these cases out, it asks a better question: what values would keep the laws of indices working? When you follow that thread, it forces two crisp answers: anything (nonzero) to the power 0 is 1, and a negative index means "one over" the positive power. No guessing, no decree — the rules simply leave no other choice.

Anything to the power zero is 1

Watch the pattern as the index steps down by one. Each step, we're dividing by the base once more:

2^3 = 8, \quad 2^2 = 4, \quad 2^1 = 2, \quad 2^0 = \,?

Going 8 \to 4 \to 2, every step halves the number. Keep the pattern going and the next step must be 2 \div 2 = 1. So 2^0 = 1.

a^0 = 1 \quad (\text{for any non-zero } a)

The laws prove it directly, too. Any non-zero number divided by itself is 1. But by the division law, a^n \div a^n = a^{n-n} = a^0. Two names for the same thing, so a^0 = 1. This holds for every non-zero base: 5^0 = 1, 100^0 = 1, even (0.5)^0 = 1. The index 0 says "no copies of the base" — and an empty product is 1, the number that changes nothing when you multiply by it.

A negative index means "one over"

Don't stop at zero — let the index keep falling. Each step still divides by the base:

2^1 = 2, \quad 2^0 = 1, \quad 2^{-1} = \tfrac{1}{2}, \quad 2^{-2} = \tfrac{1}{4}, \quad 2^{-3} = \tfrac{1}{8}

Below zero the numbers don't go negative — they turn into fractions, getting smaller and smaller. A negative index is the signal to take the reciprocal ("one over") of the matching positive power:

a^{-n} = \frac{1}{a^{n}}

The division law confirms it. What is 2^2 \div 2^5? By the law it's 2^{2-5} = 2^{-3}. By cancelling factors it's \frac{2 \times 2}{2 \times 2 \times 2 \times 2 \times 2} = \frac{1}{2^3} = \frac{1}{8}. The two must agree, so 2^{-3} = \frac{1}{8}.

Worked examples

Evaluate a few. Read off the base, then apply the rule:

Simplify a mixed expression. The ordinary laws still apply — negative indices just come along for the ride. Simplify a^5 \times a^{-3}. Add the indices as usual: a^{5 + (-3)} = a^{2}. And a^2 \div a^{6} = a^{2-6} = a^{-4} = \dfrac{1}{a^4} — a negative index popping out simply tells you the answer is a fraction.

Write a small decimal as a power of ten. Negative powers of ten are how we capture tiny numbers neatly: 0.001 = \dfrac{1}{1000} = \dfrac{1}{10^3} = 10^{-3}. This is the very machinery behind standard form, which writes a number like 0.0006 as 6 \times 10^{-4}.

Negative powers describe the very small

Positive powers of ten count the very big — a thousand is 10^3, a million 10^6. Negative powers do the opposite job: they count the very small, and scientists lean on them constantly. Look at the ladder either side of zero:

\ldots,\; 10^{2}=100,\; 10^{1}=10,\; 10^{0}=1,\; 10^{-1}=0.1,\; 10^{-2}=0.01,\; 10^{-3}=0.001,\;\ldots

Each step right divides by ten and shifts the decimal point one place. That's exactly how a millimetre is 10^{-3} metres, a width of human hair is around 10^{-4} metres, and the atoms it's made of are about 10^{-10} metres across. Writing "10^{-10}" beats writing "0.0000000001" and counting the zeros by candlelight — and it never goes negative, just smaller and smaller and smaller.

Two surprises trip up nearly everyone the first time:

Nobody sat down and decided that a^0 = 1 to be annoying. It's the single value that keeps every other rule consistent. Suppose you wanted 2^3 \times 2^0 to obey Law 1: it should equal 2^{3+0} = 2^3. That can only happen if 2^0 is a number that leaves 2^3 unchanged when you multiply — and the only such number is 1. Any other choice would break the laws you already trust.

This is one of the most powerful moves in all of mathematics: when a definition runs out (what is zero copies of a thing?), you extend the pattern by demanding the existing rules keep working, and let that pin down the answer. The exact same logic later forces the meaning of fractional indices — where a^{1/2} turns out to be a square root — and, further on, even the startling idea of raising a number to an imaginary power. One insistence — "the laws must hold" — carries you a very long way.

See it explained