# 1.01 Scientific Notation (Standard form).

## Powers of 10.

The decimal numbering system is based on powers of 10.

Power of 10.
103
102
101
100
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 these numbers in a more convenient format. Scientific notation also enable calculations be performed more simply. (Scientific notation is also referred to as standard form.)

## Expressing numbers in scientific notation.

consider the following number, 325,000;

• This is the same value as 3.25 x 100,000 .
• 100,000 can be expressed as 105 .
• Therefore we can express 325,000 as, 3.25 x 105.
• 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 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 102   (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 103   (decimal point has moved 3 places to the left) .

Important.
When converting a number to scientific notation, we need to decide whether to include any trailing zeros. See the examples above. (670 is written as 6.70 x 102, but 9750 is written as 9.75 x 103) (no trailing zero!)
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, we are able to express the precision of a number clearly, by the number of digits after the decimal point including trailing zeros.)