Small numbers

It is useful to look at patterns to try to understand negative indices:

10^0 = 1

10^{-1} = 0.1

10^{-2} = 0.01

10^{-3} = 0.001

10^{-4} = 0.0001

10^{-5} = 0.00001

10^{-6} = 0.000001

curriculum-key-fact
A negative power does not mean that the number is negative. It means that we have gone from multiplying by 10 to dividing by 10.

Example

Write 0.0005 in standard form.

0.0005 can be written as 5 \times 0.0001.

0.0001 = 10^{-4}

So 0.0005 = 5 \times 10^{-4}

Question

What is 0.000009 in standard form?

0.000009 can be written as 9 \times 0.000001.

0.000001 = 10^{-6}

So: 0.000009 = 9 \times 10^{-6}

Example

0.03 = 3 \times 10^{-2} because the 3 is 2 places away from the units column.

0.000039 = 3.9 \times 10^{-5} because the 3 is 5 places away from the units column.

Question

What is 0.000059 in standard form?

5.9 \times 10^{-5} because the 5 is 5 places away from the units column.