Powers of 10

Why powers of 10?

In science, we sometimes deal with enormous numbers (Earth-Sun distance: 150,000,000,000 m) or tiny ones (size of an atom: 0.0000000001 m). Writing them out with all their zeros is long and error-prone. Powers of 10 make it possible to simplify them.

Reminder: what is a power of 10?

10^n means multiplying 10 by itself n times.

• 10^1 = 10
• 10^2 = 100
• 10^3 = 1000
• 10^0 = 1

For numbers smaller than 1, we use negative powers:

• 10^-1 = 1/10 = 0.1
• 10^-2 = 1/100 = 0.01
• 10^-3 = 1/1000 = 0.001

Multiplying by a power of 10

Multiplying a number by 10^n amounts to moving the decimal point n places to the right (if n>0) or to the left (if n<0).

Example: 3.2 x 10^4 = 32000 (the decimal point is moved 4 places to the right) Example: 3.2 x 10^-3 = 0.0032 (the decimal point is moved 3 places to the left)

Summary table

Power	Value	Common name
10^3	1000	thousand
10^6	1000000	million
10^9	1000000000	billion
10^-3	0.001	thousandth
10^-6	0.000001	millionth

Classic pitfall

Watch the sign of the exponent: 10^-3 is not -1000, it is 0.001 (a very small positive number). The minus sign applies to the exponent, not to the number itself.