The most common number system we use is based on powers of 10

Power of 10

10^{3}

10^{2}

10^{1}

10^{0}

10^{-1}

10^{-2}

10^{-3}

^{ } = 1 x 10 x 10 x 10

^{ } = 1 x 10 x 10

^{ } = 1 x 10

^{ } = 1

^{ } = 1 ÷ 10

^{ } = 1 ÷ 10 ÷ 10

^{ } = 1 ÷ 10 ÷ 10 ÷ 10

^{ }

^{ }

^{ }

^{ }

^{ } (= 1 ÷ 10)

^{ } (= 1 ÷ 100)

^{ } (= 1 ÷ 1000)

Number

^{ } = 1000

^{ } = 100

^{ } = 10

^{ } = 1

^{ } = 0.1

^{ } = 0.01

^{ } = 0.001

In science we have to deal with both very large and very small numbers which are cumbersome to express using conventional numbers. Scientific notation is a way of expressing numbers in a much more convenient format. Calculations can also be performed with simpler processes by using scientific notation (also referred to as standard form)

consider the following number 325,000

- This is the same value as 3.25 x 100,000
- 100,000 can be expressed as 10
^{5} - Therefore we can express the 325,000 as
**3.25 x 10**^{5} - This is known as scientific notation

- Consider another example 0.045
- This is the same as 4.5 x 0.01
- 0.01 = 10
^{-2} - Therefore we can express the 0.045 as
**4.5 x 10**^{-2}

To convert a number from general format to scientific notation:

- Move the decimal point until there is a single digit before it.
- Count the number of places you moved the decimal point, this will determine the power of 10
- If you moved the decimal point to the left the power is positive if you have moved it to the right the power will be negative

**Examples**

- 670 = 6.70 x 10
^{2}(*decimal point has moved 2 places to the left*) - 0.4 = 4 x 10
^{-1}(*decimal point has moved 1 place to the right*) - 0.007 = 7 x 10
^{-3}(*decimal point has moved 3 places to the right*) - 9750 = 9.75 x 10
^{3}(*decimal point has moved 3 places to the left*)

**Important**

When converting a number to scientific notation we need to decide whether to include the trailing zeros. See the examples above.(
670 = 6.70 x 10^{2} and 9750 = 9.75 x 10^{3})

Whether to include the trailing zeros or not depends on the precision of the number. This is explained in the section on measurement theory
(chapter 7). (This is another advantage of using scientific notation, it gives us a convenient way of expressing the precision of a number.)

Unless you are justified in showing greater precision you should normally discard any trailing zeros.