Scientific Notation: Powers of 10

The speed of light in a vacuum is approximately 300 000 000 meters per second. The mass of an electron is 0.000 000 000 000 000 000 000 000 000 000 91 kg. The difficulty of printing- and understanding- such large and small numbers is evident. To make our lives simpler, we use scientific notation, a shorthand notation that makes use of powers of 10.

Recall that if you raise 10 to a power, say N, it is equivalent to writing the numeral 1 followed by a number of zeros equal to N. That is,

102 = 100

105 = 100 000

and 1012 = 1 000 000 000 000

Note that 100 follows this rule as well, since it equals 1 ( a one and no zeros following).

Negative powers of 10 are evaluated by remembering that a negative exponent implies a fraction:

a-N = 1/aN

Following this rule,

10-2 = 1/100 = 0.01

10-5 = 1/10 000 = 0.000 01

and 10-12 = 1/1 000 000 000 000 = 0.000 000 000 001

Note that:
- we use spaces instead of commas to group zeros in very large and very small numbers
- a zero is placed to the left before the decimal point (in the "ones" position) if the number is less than one
-105 is the same as 1 x 105. The inclusion of the 1x doesn't change the value

 

EXERCISES TO TRY:

Write in decimal notation:

108

10-6

10-3

Write using powers of 10:

1 000 000

0.000 000 1

0.000 1

ANSWERS:

100 000 000; 0.000 001; 0.001

106; 10-7; 10-4