# Place Value

$$\_\,.\_\,\_\,\_\,\_\,\_\,\_\,\_\,\_\,\_\,\_\,\_\,\_\,\_\,\_\,\_\,\_\,\_\,\_\,\_$$

## For integers

We probably mostly understand place value for integers.

$$4321 = 4\times1000 + 3\times100 + 2\times10 + 1$$

### Multiplying by 10

An understanding of place value, together with the distributive property, is the key to understanding why we mulitply by 10 by just slapping a zero on the end.

\begin{align} 10\times321 &= 10\times(3\times100 + 2\times10 + 1) \\ &= 3\times1000 + 2\times100 + 10 = 3210 \end{align}

## For decimals

Place value for decimals is very similar but we use powers of $1/10$ to the right of the decimal point.

$$1.23 = 1 + 2\times\frac{1}{10} + 3\times\frac{1}{100}$$

### Multiplying by 10

The right way to multiply by 10 is now to slide the decimal point to the right.

\begin{align} 10\times1.23 &= 10\times\left(1 + 2\times\frac{1}{10} + 3\times\frac{1}{100}\right) \\ &= 10\times1 + 10\times\frac{2}{10} + 10\times\frac{3}{100} \\ &= 10 + 2 + \frac{3}{10} = 12.3 \end{align}

### Dividing by 10

We can divide by 10 by sliding the decimal point to the left.

\begin{align} \frac{1.23}{10} &= \frac{1}{10}\times\left(1 + 2\times\frac{1}{10} + 3\times\frac{1}{100}\right) \\ &= \frac{1}{10}\times1 + \frac{1}{10}\times\frac{2}{10} + \frac{1}{10}\times\frac{3}{100} \\ &= \frac{1}{10} + \frac{2}{100} + \frac{3}{1000} = 0.123 \end{align}

## Computing tips

• $10\%$: Move the decimal over one spot to the left
• Example: $(0.1)\times\$22.50 = \$2.25$
• $20\%$: Move the decimal over and double
• Example: $(0.2)\times\$22.50 = \$4.50$
• $5\%$: Move the decimal over and cut in half