MISSION · Number sense
Decimals as Parts of One
Place value carrying on past the point
LEVEL 1 · EXPLORE
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LEVEL 2 · LEARN
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The idea in one sentence
The columns don’t stop at the ones — they keep going, each one worth a tenth of the column to its left, and the point just marks where the whole numbers end.
The columns keep dividing by ten
Going left, each place is worth ten times the last: ones, tens, hundreds. Going right, each place is worth a tenth of the last: tenths, hundredths, thousandths. It’s one pattern running in both directions, and the decimal point is the marker that tells you where you are in it.
\[0.47 = \frac{4}{10} + \frac{7}{100} = \frac{47}{100}\]Four tenths and seven hundredths. Or, if you prefer, forty-seven hundredths — because four tenths is forty hundredths, which is the same regrouping trade as anywhere else in place value.
Comparing decimals
Line up the points and compare from the left, one column at a time. The first column where they differ decides it, and nothing further right can overturn that.
Which is bigger, \(0.7\) or \(0.65\)? Longer is not bigger. \(0.7\) is seven tenths; \(0.65\) is six tenths and a bit. The tenths column settles it before the hundredths get a say, so \(0.7\) wins.
If the mismatched lengths bother you, pad the short one with a zero: \(0.70\) against \(0.65\). Adding that zero doesn’t change the value — it’s the equivalent-fraction trick, \(\frac{7}{10} = \frac{70}{100}\) — but it makes both numbers countable in the same unit.
Decimals and fractions are the same thing
Every decimal is a fraction whose bottom is 10, 100, 1000, and so on. That’s the whole translation:
| Decimal | Fraction | In words |
|---|---|---|
| 0.5 | \(\frac{5}{10} = \frac{1}{2}\) | five tenths |
| 0.25 | \(\frac{25}{100} = \frac{1}{4}\) | twenty-five hundredths |
| 0.08 | \(\frac{8}{100} = \frac{2}{25}\) | eight hundredths |
Going the other way, a fraction becomes a decimal by dividing the top by the bottom. Some, like \(\frac{1}{3} = 0.333\ldots\), never finish. That’s not a failure of the decimal — it’s a genuine fact about thirds and a base-ten system that has no whole number of tenths in a third.
Common mix-ups
- Reading 0.7 as smaller than 0.65 because it has fewer digits. Compare by column, not by length.
- Reading the decimal part as a whole number. \(0.9\) is nine tenths, not nine.
- Losing the point when adding. Line the points up vertically, not the last digits, or you’ll add tenths to hundredths.
- Thinking a zero on the end matters. \(0.5\) and \(0.50\) are the same number. A zero between the point and the digits is a different story: \(0.05\) is a tenth the size of \(0.5\).
Check yourself
Put these in order, smallest first: \(0.3\), \(0.29\), \(0.301\). Then say what \(0.29\) is as a fraction.
\(0.29\), then \(0.3\), then \(0.301\). Padding them to \(0.300\), \(0.290\), and \(0.301\) makes the order plain. \(0.29 = \frac{29}{100}\).
LEVEL 3 · REFLECT
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