Decimals

Updated 4 October 2026

A decimal is not a new kind of number. It is a fraction in disguise — one whose bottom number is ten, or a hundred, or a thousand. 0.7 is $\frac{7}{10}$7/10. 0.35 is $\frac{35}{100}$35/100. The point is just the marker where whole numbers end and tenths begin.

What do decimal places mean?

Each decimal place is worth one tenth of the place to its left, so the places after the point are tenths, hundredths and thousandths. In 3.47, the 3 means 3 ones, the 4 means 4 tenths and the 7 means 7 hundredths. It continues the whole-number pattern: a ten is ten ones, and a one is ten tenths, so each step right divides by 10. The point only marks where the ones end; it is not a column of its own. A common slip is reading the places like whole-number names. The first place after the point is tenths, not tens, and there is no place called oneths, because the ones column sits to the left of the point. Zeros hold places open here too: a 0 in the tenths place keeps a 7 in the hundredths, so it is worth $\frac{7}{100}$7/100, not $\frac{7}{10}$7/10. To check, read the digits without the point and name the last place: 347 hundredths.

$3+\frac{4}{10}+\frac{7}{100}=\frac{347}{100}$3 + 4/10 + 7/100 = 347/100.

Why doesn't more digits mean a bigger decimal?

More digits do not make a decimal bigger, because each digit's value comes from its place, not from how many digits follow it. To compare 0.9 and 0.35, give them the same number of places by writing a zero on the end, since 0.9 = 0.90. Now both count hundredths, and 90 hundredths is more than 35 hundredths, so nine tenths is the bigger number even though 0.35 has more digits. It works because the first place after the point is tenths, and 9 tenths beats 3 tenths before the hundredths are even read. So compare whole-number parts first, then tenths, then hundredths. When the tenths tie, the hundredths decide, and a shorter number is padded with zeros until the lengths match. A zero on the end adds nothing, so it never changes the value. To check, place both on a number line from 0 to 1: nine tenths lies nearer to 1.

1. Compare whole parts first.
2. Whole parts equal? Compare tenths, then hundredths, place by place.
3. Ragged lengths? Pad with zeros until they match.

6 tenths vs 58 hundredths $\to$→ pad $\to 60$→ 60 hundredths vs 58 hundredths $\to 6$→ 6 tenths wins.

Adding and subtracting: line up the point

1. Write the numbers with their decimal points in one vertical line (pad short ones with zeros).
2. Add or subtract column by column, exactly as with whole numbers — carrying and borrowing work unchanged, straight across the point.
3. Drop the point into the answer, in the same vertical line.

$3.7+1.25\to 3.70+1.25=4.95$3.7 + 1.25 → 3.70 + 1.25 = 4.95. $5.2-1.84\to 5.20-1.84=3.36$5.2 − 1.84 → 5.20 − 1.84 = 3.36 (borrowing crosses the point without ceremony).

How do you multiply decimals?

To multiply decimals, multiply the numbers as if the points were not there, then count the decimal places in both numbers and give the answer that many places. For 0.4 × 0.3, work out 4 × 3 = 12. Each number has one decimal place, two in total, so the answer has two places: 0.12. It works because the numbers are 4 tenths and 3 tenths, and tenths times tenths makes hundredths, so the answer is 12 hundredths. That is also why the answer is smaller than both numbers: taking part of a part leaves less than either. A product with fewer digits than places needs one more step. For 5 hundredths times 3 tenths, 5 × 3 = 15 needs three places, so a zero goes in front of the 15 to make 15 thousandths. To check, estimate: 4 tenths of 3 tenths must be less than 3 tenths.

1. Ignore the points; multiply the whole numbers.
2. Count the decimal places in both factors, added together.
3. Place the point that many places in from the right.

6 × 5 tenths $\to 6\times 5=30\to$→ 6 × 5 = 30 → one place $\to 30$→ 30 tenths, which is 3.

$2.5\times 0.3\to 25\times 3=75\to$2.5 × 0.3 → 25 × 3 = 75 → two places $\to 0.75$→ 0.75.

How do you divide decimals?

To divide a decimal by a whole number, use short division and put the point in the answer directly above the point in the number you are dividing: 7.5 ÷ 3 = 2.5. To divide by a decimal, multiply both numbers by 10 or 100 until the number you divide by is whole. For 12 ÷ 0.4, multiplying both numbers by 10 gives 120 ÷ 4 = 30. Scaling both by the same amount keeps the answer, because the question is how many times one fits into the other, and 4 fits into 120 as often as 4 tenths fits into 12. When a division does not come out even, write a zero after the last digit and keep dividing, since 3 ones are also 30 tenths. Dividing by a number below 1 makes the answer bigger, so 30 is more than 12. To check, multiply back: 30 × 4 = 120 tenths, which is 12.

A decimal divided by a whole number: the point rises straight up

Do short division as usual.

7 ones $÷3\to 2$÷ 3 → 2 remainder $1\to$1 → the 1 carries as 10 tenths. (The carry makes 15 tenths; 15 ÷ 3 = 5 tenths.)

Dividing BY a decimal: scale both up until the divisor is whole

1. Multiply BOTH numbers by 10 (again and again) until the divisor is whole.
2. Divide as normal.

both $\times 10\to 120÷4\to$× 10 → 120 ÷ 4 → 30.

How do you round decimals?

To round a decimal, look at the one digit just after the place you are keeping: if it is 5 or more, round the kept digit up; if it is 4 or less, leave it as it is, and drop every digit after the kept place. To round 4.56 to one decimal place, keep the tenths, look at the hundredths digit, 6, and round up to 4.6. It works because 6 hundredths is past the halfway point of a tenth, which is 5 hundredths, so the number is nearer the higher tenth. Take care when a 9 rounds up, because it carries into the ones: 2 and 97 hundredths rounded to one decimal place becomes 3, written with a 0 in the tenths so that it still shows one place. To check, make sure the answer has exactly the number of places you were asked for and sits within half a step of the start.

next digit $6\to$6 → past halfway $\to$→ round up. $4.32\to$4.32 → next digit $2\to 4.3$2 → 4.3.

There is a full guide to this — and to using rounding to estimate answers before you compute them — in Estimation and Rounding.

The famous ones, by heart

These few unlock most everyday arithmetic — see the percentages guide for how far 10% alone can carry you. (And 0.333… recurring is not an accident: whether a fraction's decimal stops or repeats forever is decided by its denominator — a story the app's Why Every Fraction Terminates or Recurs lesson tells properly.)

The three mistakes worth naming

• Longer means bigger. 0.35 < 0.9. Compare place by place from the left.
• Lining up the ends to add. The points line up; the ends can be as ragged as they like.
• Expecting × to grow and ÷ to shrink. Below 1, they swap jobs — 0.4 × 0.3 = 0.12, but 12 ÷ 0.4 = 30.

Decimals also carry on into money and growth. Percentage change, reverse percentages, simple and compound interest, inflation and loan repayments are all the same place value applied over time, and they are covered in decimals, percentages and interest.