Decimals

A price tag reads 3.5; a ruler shows 1.62. What is that little dot between the digits? It is the decimal point — the start of a new chapter for place value.

Whole numbers count complete things; decimals fill the gaps between them.

New places to the right of the point

Place value marches left through tens, hundreds, thousands — and it marches right with rules of its own:

The first place after the point is the tenths place, worth $0.1 = \frac{1}{10}$ ;
The second is the hundredths place, worth $0.01 = \frac{1}{100}$ .

So $0.3$ is $\frac{3}{10}$ — one whole split into 10 equal parts, taking 3 — exactly what fractions describe. Now look at $2.47$ :

$2.47 = 2 + 4 \times 0.1 + 7 \times 0.01$

Every digit still guards its own seat; the stage has simply extended to the right of the point.

InteractiveDecimal Number Line

0 – 103.4

0.01

0.1

3.4 = 3 + 4×0.1 + 0×0.01

Expanded form · 0.01 / 0.1 / 1 / 10

Comparing: line up the places first

Which is bigger, 0.5 or 0.45? Intuition says "0.45 has more digits, so it must be bigger" — wrong.

Pad 0.5 into 0.50 and compare: 0.50 holds 50 hundredths while 0.45 holds only 45, so $0.5 > 0.45$ .

Common trap

When comparing decimals, more digits do not mean a bigger number. Pad the ends with zeros, line up the places, then compare one place at a time from the left.

Adding and subtracting: keep the points in line

To compute $0.7 + 0.45$ , the key is aligning the decimal points:

Pad 0.7 into 0.70;
Then $0.70 + 0.45 = 1.15$ .

It is the same idea as lining up ones under ones in column addition: only matching places may be added. Tenths add to tenths, hundredths to hundredths — mixing them is like stirring a dime straight into the penny jar.

Money is decimals you can touch

12.50 dollars is 12 dollars and 50 cents;
Change of 0.05 dollars is a nickel, 0.10 a dime;
A rate of 0.52 per unit sits between 0.5 and 0.6.

Money runs on tens too — dollars, dimes and cents line up with ones, tenths and hundredths — so receipts are the perfect decimal training ground, every digit doing its own job. Once this feels easy, visit percentages, which are just decimals wearing another outfit.

Check yourself

Quick quiz

1. Which is bigger, 0.5 or 0.45?
2. 0.3 equals which fraction?
3. 0.7 + 0.45 = ?