Decimal place value

Before the decimal point, each step to the left multiplies by ten. After the point, each step to the right divides by ten. That single fact unlocks every decimal.

Idea

In a whole number like $234$, the columns are hundreds, tens, ones. In a decimal like $234.567$, we continue past the point: tenths, hundredths, thousandths.

So $234.567 = 2(100) + 3(10) + 4(1) + 5(\tfrac{1}{10}) + 6(\tfrac{1}{100}) + 7(\tfrac{1}{1000})$.

Important: the digit just after the point is the tenths digit, not the "first decimal". The second is hundredths, the third is thousandths, and so on. A common mistake is to read $0.07$ as "seven tenths" , it is actually seven hundredths.

Trailing zeros after a decimal don't change the value: $0.5 = 0.50 = 0.500$. They just say "and zero more hundredths, and zero more thousandths". Sometimes we add them to align two numbers for adding or comparing.

Worked examples

Example 1. Write $7.408$ in expanded form.

$7.408 = 7 + \frac{4}{10} + \frac{0}{100} + \frac{8}{1000}$. Notice the $0$ in the hundredths column , it must be kept as a placeholder.

Example 2. What is the value of the digit $5$ in $32.85$?

$5$ sits in the hundredths column, so its value is $\tfrac{5}{100} = 0.05$.

Example 3. Convert $\tfrac{3}{10} + \tfrac{7}{100}$ into a decimal.

$\tfrac{3}{10} = 0.3$ and $\tfrac{7}{100} = 0.07$. Sum: $0.3 + 0.07 = 0.37$.

Example 4. Are $0.4$ and $0.40$ the same number? Justify.

$0.4 = \tfrac{4}{10} = \tfrac{40}{100} = 0.40$. Yes, they are the same , just written with different trailing zeros.

Try it yourself

1. Write $0.59$ in expanded form.
2. What is the place value of the digit $7$ in $5.073$?
3. Convert $\tfrac{6}{10}$ and $\tfrac{6}{100}$ into decimals.
4. Write $25.04$ as a sum of place-value pieces.
5. Are $0.7$ and $0.07$ the same? Why or why not?
6. Write the decimal in which $4$ is in the hundredths and $9$ in the tenths and $5$ in the ones , and read it aloud.
7. Express $3 + \tfrac{2}{10} + \tfrac{5}{1000}$ as a decimal.
8. Write $0.250$ without trailing zeros.

Activity

Take a measuring tape or ruler. Pick five real-world lengths (your pencil, table-edge, book height). Record each to the nearest tenth or hundredth of a centimetre. Write each measurement in expanded form.