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Decimals

A decimal is a fraction with the denominator left out — 0.25 is twenty-five hundredths, and the point is what tells you so. Children who read the digits after the point as an ordinary number get 0.5 < 0.25, because 5 is less than 25, and that mistake survives a long time if it is not named early.

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Adding decimals with different places

7.3 + 2.82

  1. Line up the decimal points, not the last digits.
  2. 7.3 is 7.30 — writing the extra zero makes the columns match, and changes nothing.
  3. Add the hundredths, then the tenths, then the ones: 0 + 2, 3 + 8 carrying one, 7 + 2 plus the carry.
  4. The answer is 10.12.

Worth knowing. The zero on the end of 7.30 is the whole trick. It is not a new digit, it is the one that was always there and did not need writing until now.

Comparing two decimals

0.5 and 0.25

  1. Compare place by place from the left, exactly as with whole numbers.
  2. Tenths first: 5 tenths against 2 tenths.
  3. 5 is more than 2, so 0.5 is bigger, and the digits after that do not matter.
  4. 0.5 > 0.25.

Worth knowing. Reading them as "five" and "twenty-five" gives the wrong answer. Written as 0.50 and 0.25 the comparison is obvious, which is why writing the extra zero is worth the moment it takes.

Where it usually goes wrong

  • Comparing the digits after the point as whole numbers, so 0.25 is put above 0.5.
  • Lining up the right-hand edges instead of the decimal points when adding.
  • Losing the point in the answer, so 7.3 + 2.82 comes out as 1012.

The Professor lines up the points before he adds, not the right-hand edges. The point is the fixed thing; the digits move around it.

Practise it

All level paths

Questions teachers and parents ask

Is 0.5 the same as 0.50?

Yes. Both are five tenths, and the extra zero adds no hundredths. It is worth writing when it makes columns line up, and it is worth knowing that a longer decimal is not automatically a bigger one.

Why do the decimal points have to line up?

Because addition works column by column, and the columns are the places. Lining up the right-hand edges instead puts tenths above hundredths, which adds two different things together.

When are decimals taught?

Reading, writing and comparing decimals to thousandths is grade 5 (5.NBT.A.3), adding and subtracting them is grade 5 (5.NBT.B.7), and fluency with all four operations arrives in grade 6 (6.NS.B.3).

How do decimals and fractions relate?

Every decimal is a fraction over a power of ten: 0.25 is 25/100, which simplifies to 1/4. It goes the other way too, but only sometimes — 1/3 has no exact decimal, and the 3s go on forever.

Where this shows up

Standards

  • 4.NF · Number and operations — fractions
  • 5.NBT · Number and operations in base ten
  • 6.NS · The number system

Formats