Orders of Magnitude & Estimation
From protons to quasars, and how to estimate anything to the nearest power of ten.
Lesson at a glanceShow the key points
- Write any number as a × 10ᵇ with 1 ≤ a < 10.
- Order of magnitude: round a to 1 (a ≤ 5) or 10 (a > 5); the resulting power of ten's exponent is the order of magnitude.
- The Sun's diameter (1.39 × 10⁹ m) is of order 10⁹ m; a hydrogen atom (1.06 × 10⁻¹⁰ m) of order 10⁻¹⁰ m: 19 orders apart.
- Order-of-magnitude estimates replace every number by its power of ten to get a quick, roughly right answer.
A mountain and a pinhead, a galaxy and a hydrogen atom, the age of the universe and an electron’s orbit. The numbers in physics are so wildly different that the most useful first question is often: roughly how big, to the nearest power of ten?
Powers of ten
Write any number as with and b a whole number. The Sun’s diameter is m; a hydrogen atom’s is m.
So the Sun’s diameter is of order 10⁹ m and the atom’s of order 10⁻¹⁰ m. The Sun is 9 − (−10) = 19 orders of magnitude larger: ten million million million times.
Order-of-magnitude estimates
To get a quick, roughly right answer, replace every number by its power of ten and do the arithmetic with exponents. You won’t get the exact value, but you will rarely be off by more than one order of magnitude, which is often all you need to decide whether something is plausible.
Roughly how many times does a human heart beat in a 75-year life?
- 1Write numbers as with 1 ≤ a < 10.
- 2Order of magnitude: round a to 1 (≤ 5) or 10 (> 5); the exponent of the result.
- 3Estimate by turning every number into its power of ten and adding exponents.
Check yourself
Keep exploring
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