Scientific Notation

Scientific Notation is a handy way to write very large and very small numbers. Instead of having to use lots of digits, scientific notation allows shorter versions of the number to be written. It uses the format shown below:

m x 10n
where 1 m < 10
and
n is any integer

Multiplying by a power of ten is a neat, alternate way of writing a number that has many zeros. The table below is a reminder of the value of some powers of ten.

Powers of 10

Power Repeated Multiplication Value
102 10 x 10 100
103 10 x 10 x 10 1,000
104 10 x 10 x 10 x 10 10,000
105 10 x 10 x 10 x 10 x 10 100,000
106 10 x 10 x 10 x 10 x 10 x 10 1,000,000
107 10 x 10 x 10 x 10 x 10 x 10 x 10 10,000,000
108 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 100,000,000

Scientific Notation Examples

2,300,000
can be written as
2.3 x 1,000,000
which can be written as
2.3 x 106

More Examples

Number Scientific Notation
2,000 2 x 103
76,000 7.6 x 104
1,450,000 1.45 x 106
10,412,000 1.0412 x 107
909,100,000 9.091 x 108
4,570,000,000 4.57 x 109

Terminology

illustration of coefficient multiplied by a power of 10 as used in scientific notation

Count the hops!

Finding the value of the exponent is the main challenge when using scientific notation as this tells us the number that we multiply the co-efficient by. One method to help with this is to think of multiplying by powers of 10 as moving the decimal point - one place for each power of ten. If you think of each move as a hop then you can find what power of ten to multiply the coefficient by. The example below shows that 2.3 multiplied by 106 - six hops - makes 2,300,000.

relationship between 2.3 and 2,300,000 shown as 6 hops of the decimal point

So, as we have seen above already, 2,300,000 = 2.3 x 106

Scientific Notation For Small Numbers

All the examples above have been of large numbers. Scientific Notation is also very useful for representing very small numbers. In these cases, instead of multiplying, we are in effect dividing by powers of ten. We use negative exponents to show this.

0.000000657
can be written as
6.57 ÷ 10,000,000
which can be written as
6.57 x 10-7

More Examples

Number Scientific Notation
0.02 2 x 10-2
0.0021 2.1 x 10-3
0.0000051 5.1 x 10-6
0.00000003 3 x 10-8

Again we can use the "count the hops" method to find the power of ten we are multiplying by (multiplying with a negative exponent has the same effect as dividing)

relationship between 3 and 0.00000003 shown as 8 hops of the decimal point

So 0.00000003 = 3 x 10-8

Scientific Notation Worksheets

Try the worksheets below to practice scientific notation..

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