A decimal isn't a new kind of number — it's our familiar place-value system, simply continued past the ones into tenths, hundredths and beyond.
Past the decimal point
You already know how place value works to the left of the decimal point: each place is ten times bigger than the one to its right — ones, tens, hundreds, thousands. Decimals do exactly the same thing in the other direction. The places to the right of the point are tenths, then hundredths, then thousandths, and each one is one-tenth of the place to its left.
So the first place after the point counts tenths: \( 0.3 = \frac{3}{10} \). The second place counts hundredths, so a number like \( 0.27 \) is read as 2 tenths and 7 hundredths together — that is, \( 0.27 = \frac{27}{100} \).
It helps to see this. Imagine a single square split into 100 little cells: the whole square is \( 1 \), each full row is \( 0.1 \), and each tiny cell is \( 0.01 \). Drag the slider below to shade in any number of hundredths and watch the decimal grow.
Decimals and fractions are the same thing
Because every decimal place is just a fraction with a denominator of \( 10 \), \( 100 \), \( 1000 \), and so on, every decimal is a fraction wearing different clothes. A few are worth memorising because they appear constantly: \( 0.5 = \frac{1}{2} \), \( 0.25 = \frac{1}{4} \), and \( 0.75 = \frac{3}{4} \).
To turn a decimal into a fraction, read its place value, then simplify. For example \( 0.6 \) is six tenths, so \( 0.6 = \frac{6}{10} = \frac{3}{5} \) once you cancel the common factor of \( 2 \).
To go the other way and turn a fraction into a decimal, just divide the top by the bottom: \( \frac{3}{8} = 3 \div 8 = 0.375 \). A fraction bar has always meant "divide" — the decimal is simply the answer.
In words A decimal is just a fraction whose denominator is \( 10 \), \( 100 \), \( 1000 \), … — and the point tells you which one. Read \( 0.3 \) as "three tenths", \( 0.27 \) as "twenty-seven hundredths", and the hidden fraction appears on its own.
Comparing decimals
To compare two decimals, line up the decimal points and compare place by place, working left to right — the first place where they differ decides the winner.
Take \( 0.4 \) against \( 0.38 \). Write them with matching places as \( 0.40 \) versus \( 0.38 \). Now compare the tenths first: \( 4 \) tenths beats \( 3 \) tenths, so \( 0.4 \) is the larger number, even though \( 0.38 \) shows more digits.
That last point is the trap to watch for: with decimals, more digits does NOT mean a bigger number. The length of the tail is irrelevant — only the value of each place matters.
Tip To compare decimals, write in the invisible trailing zeros so both numbers have the same number of places. Then \( 0.4 \) becomes \( 0.40 \), it lines up neatly beside \( 0.38 \), and the comparison is obvious at a glance.
Adding and subtracting decimals
The whole secret to adding and subtracting decimals is a single habit: line up the decimal points, so that each place sits directly over its match. Fill in any missing zeros so both numbers have the same length, then add or subtract exactly as you always have.
For \( 2.4 + 0.85 \), stack them with the points aligned and pad \( 2.4 \) to \( 2.40 \). Adding column by column gives \( 2.4 + 0.85 = 3.25 \).
Subtraction works the same way. For \( 5.2 - 1.75 \), write \( 5.2 \) as \( 5.20 \) so the places match, then subtract: \( 5.2 - 1.75 = 3.45 \). The aligned point keeps tenths under tenths and hundredths under hundredths, which is all that ever goes wrong.
Multiplying and dividing by 10, 100, 1000
Here is where decimals feel almost magical. Multiplying by \( 10 \) shifts every digit one place to the left — which looks, from the digits' point of view, like the decimal point sliding to the right: \( 3.7 \times 10 = 37 \).
Dividing by \( 10 \) does the reverse, shifting every digit one place to the right: \( 3.7 \div 10 = 0.37 \). Multiply or divide by \( 100 \) and the shift is two places; by \( 1000 \), three places. This clean, ten-by-ten behaviour is exactly why our number system is called base ten.
- Read the place value: \( 0.16 \) is sixteen hundredths, so \( 0.16 = \frac{16}{100} \).
- Look for a common factor. Both \( 16 \) and \( 100 \) divide by \( 4 \).
- Cancel the \( 4 \): \( \frac{16}{100} = \frac{4}{25} \).
So \( 0.16 = \frac{16}{100} = \mathbf{\frac{4}{25}} \) in simplest form.
- Line up the decimal points and pad with a zero so the places match: \( 3.70 + 0.45 \).
- Add the hundredths: \( 0 + 5 = 5 \).
- Add the tenths: \( 7 + 4 = 11 \) — write \( 1 \), carry \( 1 \) into the ones.
- Add the ones: \( 3 + 0 + 1 = 4 \).
So \( 3.7 + 0.45 = \mathbf{4.15} \).
Practice
Try each one yourself, then reveal the full solution.
1. Write \( \frac{3}{4} \) as a decimal.
A fraction bar means "divide", so turn \( \frac{3}{4} \) into a division: \( 3 \div 4 \).
Three divided by four gives seventy-five hundredths.
So \( \frac{3}{4} = 3 \div 4 = \mathbf{0.75} \).
2. Which is larger, \( 0.6 \) or \( 0.59 \)?
Write in the invisible trailing zero so both have the same number of places: \( 0.60 \) versus \( 0.59 \).
Compare the tenths first: \( 6 \) tenths beats \( 5 \) tenths, so the extra digit on \( 0.59 \) does not help it.
So \( 0.60 > 0.59 \), which means the larger number is \( \mathbf{0.6} \).
3. Compute \( 5.2 - 1.75 \).
Line up the decimal points and pad with a zero so the places match: \( 5.20 - 1.75 \).
Subtract place by place, borrowing where needed, keeping tenths under tenths and hundredths under hundredths.
So \( 5.2 - 1.75 = \mathbf{3.45} \).