Indices: The Power Behind Exponential Growth

An index (or exponent or power) tells you how many times to multiply a number by itself. 2³ means 2 × 2 × 2 = 8. The rules for working with indices — multiplying by adding the powers, dividing by subtracting them, raising a power to a power by multiplying — are surprisingly elegant and consistent. Negative indices represent reciprocals. Fractional indices represent roots. One set of rules covers all of them.

Archimedes counted grains of sand

One of the earliest uses of a systematic notation for large numbers appears in Archimedes' The Sand Reckoner (around 250 BC), where he estimated the number of grains of sand that would fill the universe — and invented a number system to handle it. Scottish mathematician John Napier invented logarithms in 1614 precisely to make working with large numbers easier, and logarithms are intimately connected to indices (they're the inverse). René Descartes introduced the superscript notation we use today — writing 2³ rather than 2 × 2 × 2 — in his 1637 Géométrie.

Exponential behaviour in the world

Indices govern any process where something multiplies by a fixed factor at each step. Compound interest grows exponentially — a credit card debt at a high interest rate can feel manageable for a while and then suddenly spiral. Computer storage doubles with each generation roughly following Moore's Law. Earthquake magnitudes on the Richter scale are logarithmic, meaning a magnitude 7 quake releases about 32 times the energy of a magnitude 6. A bacterium doubling every 20 minutes can go from one cell to over a billion in 10 hours. The index laws are the mathematics behind all of this.