## Multiplication of decimals

### Multiplying by powers of 10

When any number is multiplied by a power of 10, each digit is multiplied by the same power of 10. For example

24.06 × 10 = 20 × 10 + 4 × 10 + 0.06 × 10 = 200 + 40 + 0.6 = 240.6

Multiplying by 10 corresponds to moving the decimal point one place to the right and inserting a zero where necessary.

Similarly, multiplying by 100 (= 10^{2}) corresponds to moving the decimal point two places to the right and inserting zeros where necessary. For example

8.7 × 100 = 870

Multiplying by 1000 (= 10^{3}) corresponds to moving the decimal point three places to the right and inserting zeros where necessary. For example

8.7 × 1000 = 8700

### Dividing by powers of 10

When any number is divided by a power of 10, each digit is divided by the same power of 10. For example:

\begin{align}78.4 ÷ 10&=70÷10+8÷10+0.4÷10\\\\ &=70×\dfrac{1}{10}+8×\dfrac{1}{10}+0.4×\dfrac{1}{10}\\\\ &=7.0+0.8+0.04\\\\ &=7.84\end{align}Dividing by 10 corresponds to moving the decimal point one place to the left and inserting a zero where necessary.

Dividing by 100 (= 10^{2}) corresponds to moving the decimal point two places to the left and inserting zeros where necessary. For example

78.4 ÷ 100 = 0.784

Dividing by 1000 (= 10^{3}) corresponds to moving the decimal point three places to the left and inserting zeros where necessary. For example

78.4 ÷ 1000 = 0.0784