Order of Magnitude

An order of magnitude is a factor of 10. Two numbers differ by one order of magnitude if one is roughly 10 times the other.

Orders of magnitude are the most useful framework for comparing large numbers. Instead of asking "how much bigger is X than Y?" you ask "how many orders of magnitude separate them?"

Examples:

  • 100 vs 1,000: one order of magnitude apart
  • A million vs a billion: three orders of magnitude apart
  • The width of a hair (10^-4 m) vs the Earth's diameter (10^7 m): eleven orders of magnitude apart

In scientific and engineering contexts, estimates are often considered valid if they're within one order of magnitude of the true value. This means being off by a factor of 3 is "close enough," but being off by a factor of 30 is a meaningful error.

Orders of magnitude thinking is a core skill in Fermi estimation (named after physicist Enrico Fermi), where you estimate unknown quantities by combining rough approximations. "How many piano tuners are in Chicago?" is a classic Fermi problem that requires thinking in orders of magnitude.

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More terms

BillionOne billion (1,000,000,000) is 10^9, or one thousand millions. A billion seconds is about 31.7 years.GoogolA googol is 10^100 (1 followed by 100 zeros). It far exceeds the number of atoms in the observable universe (~10^80).GoogolplexA googolplex is 10^googol, or 10^(10^100). It cannot be written out in full because it has more digits than atoms in the universe.InnumeracyInnumeracy is the mathematical equivalent of illiteracy: the inability to understand and work with numbers and basic mathematical concepts.MillionOne million (1,000,000) is 10^6, or one thousand thousands. A million seconds is about 11.5 days.View all glossary terms