A decimal is just a base-ten fraction — tenths, hundredths, and place value to the right of the point.
You already know how to cut a whole into equal parts and name the pieces with a fraction. A decimal is the same idea, but it always cuts by ten. Split one whole into ten equal steps and each step is one tenth; split one of those steps into ten again and each is one hundredth. Because every cut is a cut by ten — the same ten that runs our place-value columns for whole numbers — we can keep writing the pieces in those columns, just to the right of a small point. So a decimal is nothing new: it is a fraction whose denominator is 10, or 100, or 1000, written in place-value shorthand. This lesson shows where the point comes from, what each place is worth, how to read a decimal aloud, why 0.3 and 0.30 are the same amount, and how to compare two decimals without being fooled by their length.
Take one whole and slice it into ten equal parts. Each part is one tenth — and you already have a name for one of ten equal parts: it is the fraction 110. The decimal way to write it is 0.1. The little dot — the decimal point — marks where the whole numbers end and the broken-up pieces begin.
So 0.1 means exactly 110; three of those steps is 0.3, which is 310; seven of them is 0.7 = 710. The point does not change the counting — it only tells you the pieces are tenths, not ones.
A decimal is a fraction whose denominator is a power of ten: 10, 100, 1000, … The digits after the point count those base-ten pieces. 0.1 = 110, and that is all the point is saying.
What if one tenth is still too big? Cut it into ten again. Each new sliver is one tenth of one tenth — one hundredth, the fraction 1100, written 0.01. Cut once more and you reach thousandths, 11000 = 0.001. Each step to the right cuts ten times finer.
That gives the places to the right of the point their names, marching outward from the point: the first place is tenths, the next is hundredths, then thousandths. They line up with the whole-number columns you already use — ones, tens, hundreds — but going the other way, dividing by ten at every step instead of multiplying.
Each place to the right of the point is worth one tenth of the place to its left: tenths, then hundredths, then thousandths. A hundredths grid shows it — the whole is one square, each small cell is 0.01 = 1100.
To read a decimal, name the whole-number part, say "point," then read the digits after the point one at a time. So 0.25 is read "zero point two five." What does it mean? The 2 sits in the tenths place and the 5 in the hundredths place, so
0.25 = 2 tenths + 5 hundredths = 210 + 5100 = 25100 = 14.
That last reduction is honest: 25 and 100 share a factor of 25, so 25100 reduces to 14. A decimal is read off its place value, then — if you like — reduced to lowest terms like any other fraction.
0.05 reads "zero point zero five." There are zero tenths and five hundredths, so it is 5100 = 120. The leading zero in the tenths place is doing real work: it keeps the 5 in the hundredths place. Drop it and you would have 0.5 — ten times bigger.
Is 0.3 the same as 0.30? Yes. Writing 0.3 means three tenths. Writing 0.30 means three tenths and zero hundredths — and zero hundredths add nothing. Another way to see it: three tenths is thirty hundredths, because cutting each tenth into ten gives ten times as many pieces of one-tenth the size. The pictures below cover exactly the same area.
A zero on the right end of the decimal part changes nothing: 0.3 = 0.30 = 0.300. But a zero between the point and a digit — like the one in 0.05 — is not trailing; it holds a place, and removing it changes the value.
Here is the trap. Which is bigger, 0.3 or 0.25? It is tempting to say 0.25, because it "looks longer" — more digits. But length is not size. Compare place by place, from the left, exactly as you compare whole numbers from the highest column down.
Tenths first: 0.3 has 3 tenths, 0.25 has only 2 tenths. Three tenths beats two tenths, so we already have a winner — 0.3 > 0.25 — and we never even reach the hundredths. A handy move is to line the decimals up by writing the same number of places: 0.30 vs 0.25. Now 30 hundredths against 25 hundredths makes the answer obvious.
More digits does not mean bigger. 0.25 > 0.3 is wrong: 3 tenths beats 2 tenths no matter how many digits trail behind. Compare the tenths place first; only if it ties do you move on to the hundredths.
| Idea | What it says | Example |
|---|---|---|
| A decimal is a base-ten fraction | digits after the point count tenths, hundredths, thousandths | 0.1 = 110, 0.01 = 1100 |
| Each step right ÷ 10 | every place is one tenth of the place to its left | tenths → hundredths → thousandths |
| Read off the place value | a decimal means a sum of tenths, hundredths, … | 0.25 = 25100 = 14 |
| Trailing zeros don't matter | a zero on the right end leaves the value unchanged | 0.3 = 0.30 |
| Compare left to right | first differing place from the left decides; not length | 0.3 > 0.25 |
Write each as a fraction with denominator 10, 100, or 1000, then reduce: (a) 0.7, (b) 0.04, (c) 0.6.
(a) 0.7 = 710 (already lowest terms). (b) 0.04 = 4100 = 125. (c) 0.6 = 610 = 35.
Read 0.205 aloud and say what each digit is worth.
"Zero point two zero five." The 2 is 2 tenths, the 0 is 0 hundredths (a place-holder), and the 5 is 5 thousandths. Altogether 0.205 = 2051000 = 41200.
True or false: 0.5 = 0.50 = 0.500. Explain.
True. Each extra zero is a trailing zero — it adds zero hundredths, then zero thousandths, which change nothing. All three name five tenths, the same amount: 510 = 12.
Put in order from smallest to largest: 0.8, 0.08, 0.18, 0.2.
Line them up as hundredths: 0.80, 0.08, 0.18, 0.20. Comparing left to right: 0.08 < 0.18 < 0.2 < 0.8. (0.08 has 0 tenths; 0.18 has 1 tenth; 0.2 has 2 tenths; 0.8 has 8 tenths.)
Which is larger, 0.7 or 0.65? Decide place by place, and don't be fooled by length.
0.7 is larger. Tenths place first: 0.7 has 7 tenths, 0.65 has only 6 tenths. Seven beats six, so 0.7 > 0.65 — even though 0.65 "looks longer." (As hundredths: 70 > 65.)
A grid of 100 cells has 40 shaded. Write the shaded amount as a decimal and as a reduced fraction.
40 of 100 cells is 0.40, which is the same value as 0.4 (the trailing zero changes nothing). As a fraction, 0.40 = 40100 = 25.
Six questions to lock it in. Tap the answer you think is right.
This lesson introduces decimal notation as a way of writing base-ten fractions, the heart of CCSS 4.NF.C.6 (express fractions with denominators 10 and 100 in decimal notation) and 5.NBT.A.1 (each place is ten times the place to its right, and one tenth of the place to its left). The trailing-zero idea and the place-by-place comparison set up 4.NF.C.7 and 5.NBT.A.3.b (compare two decimals to thousandths using place value). The most useful habit to reinforce at home: when comparing decimals, line them up to the same number of places and read from the left — never judge by how long the number looks.