The Sun is about 150,000,000,000 metres from Earth. A water molecule is about 0.000000000275 metres wide. Neither number is easy to read, write, or calculate with — which is exactly the problem scientific notation solves.
Writing out the distance to the Sun in full — 150,000,000,000 metres — takes real effort just to count the zeros correctly, and it gets worse the moment two such numbers need to be multiplied together. Scientific notation solves this by rewriting any number, however large or small, in a short, standard form that is actually easy to read, compare, and calculate with.
Scientific notation writes a number as a value between 1 and 10, multiplied by a power of 10:
a × 10n, where 1 ≤ a < 10, and n is a whole number
A positive exponent means the original number is large (10 or more); a negative exponent means the original number is small (less than 1). The exponent simply tells you how many places to move the decimal point.
150,000,000,000 can be rewritten as 1.5 × 1011. The decimal point moves 11 places to the left to get from 150,000,000,000 down to 1.5, so the exponent is 11.
0.000000000275 can be rewritten as 2.75 × 10-10. The decimal point moves 10 places to the right to get from 0.000000000275 up to 2.75, so the exponent is negative 10.
Scientific notation makes it possible to compare wildly different scales on the same terms.
Multiplying numbers in scientific notation is often simpler than multiplying the full numbers directly: multiply the two leading values together, and add the exponents.
(2 × 103) × (3 × 102) = (2 × 3) × 103+2 = 6 × 105
Chemists regularly work with a number called Avogadro's number, approximately 6.022 × 1023, which describes how many particles are in a standard "mole" of a substance — a foundational unit in chemistry. Written in full, that number would be a 6 followed by 23 digits. Scientific notation is not just a convenience here; without it, calculations at this scale would be almost impossible to write down accurately, let alone work with by hand.
Scientific notation is really a form of scientific measurement in its own right — the SI units covered earlier in this series say what is being measured, and scientific notation makes it possible to actually write and calculate with the result, no matter how enormous or how tiny that measurement turns out to be.
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