Why Indices Matter: Earthquakes, pH and More

You’ve probably seen those little numbers sitting above a big number—like 2³ or 10⁶—and wondered why they matter. Those tiny superscripts are called indices (or exponents), and they’re a shorthand for multiplying a number by itself. What seems like a simple notation actually helps us handle massive ranges of values in science, engineering and everyday life without writing out endless digits.

A brief history: from Chuquet to Descartes

• Late 1400s: French mathematician Nicolas Chuquet scribbled the first known use of exponents on paper, using tiny raised numbers in his private notes. • 1637: René Descartes popularised the notation we use today in his work "La Géométrie." He used a small number to the right of the variable to show repeated multiplication, making algebra much more compact and readable.

Where you'll see this in real life

• Earthquake magnitudes (Richter scale): Each whole-number jump means about ten times bigger ground motion—and about 31.6 times more energy—so a 6.0 really is a lot stronger than a 5.0. • Acidity (pH scale): pH is the negative logarithm (–log₁₀) of hydrogen ion concentration. A pH of 3 is ten times more acidic than a pH of 4. • Sound levels (decibels): We measure loudness on a log scale (dB = 10 × log₁₀ of power ratio) so our ears can handle a billion-to-one range of intensities. • File sizes: Kilo-, mega- and giga-bytes are based on powers of two (or sometimes ten), so 1 KB is roughly 2¹⁰ bytes, 1 MB is 2²⁰ bytes, and so on.

A common misconception

People often trip up on negative and fractional indices. • Negative indices (a⁻ⁿ) mean “reciprocal,” so a⁻³ = 1⁄a³. • Fractional indices (a^{1/2}, a^{3/4}) link to roots: a^{1/2} is the square root of a, a^{3/4} is the fourth root of a cubed. Once you see them as "just another way to rewrite multiplication and roots," they start making sense.