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Scientific Notation

(also called Exponential Notation)

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Depending on one's choice of units, numbers in physics can be very  big or very  small! For example, the radius of the Earth (useful in gravity problems) is approximately 6,380,000 m; its mass is 5,970,000,000,000,000,000,000,000 kg. On the other hand, the wavelength of blue light is about 0.000 000 4 m and the mass of a proton is 0.000 000 000 000 000 000 000 000 001 675 kg. To make these numbers easier to write -- and to make fewer mistakes in writing them -- we make use of the fact that most of the digits are zeros!

First, observe that
  • 10 = 101
  • 100 = 10 X 10 = 102
  • 1000 = 10 X 10 X 10 = 103
  • 10000 = 10 X 10 X 10 X 10 = 104
  • and so on...
  • 0.1 = 1/10 = 1/101=10-1
  • 0.01 = 1/100 = 1/102=10-2
  • 0.001 = 1/1000 = 1/103=10-3
  • 0.0001 = 1/1000 = 1/104=10-4
  • and so on...

Rule of thumb #1: If you have a 1 followed by n  0's, then you have 10n. Eg. 100,000,000 = 108

Rule of thumb #2: If you have a decimal point followed by n  0's, followed by a 1, then you have 10n+1. Eg. 0.000 000 01 = 10-8

With this observation we can now write the radius of the Earth as 638 X 104 m or the mass of the proton as 0.1675 X 10-26 kg.

We can improve on this method of writing numbers by requiring that all numbers multiplying the powers of 10 be between 1 and 10. (Strictly speaking, we want this factor (the number multiplying the powers of 10) to be greater than or equal to 1 and less than 10.) So the radius of the Earth is now 6.38 X 106 m and the mass of the proton is 1.675 X 10-27 kg. The exponent on the power of 10 tells us how many places to move the decimal point. The sign of the exponent tells us how many places to move it - for a positive exponent we move the decimal to the right (we want a number bigger than one); for a negative exponent we move the decimal to the left (we want a number less than one).

Test yourself.
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