Mathematics from zero
Decimals
A price tag reads $0.75. A fraction would write the same amount as 3/4 of a
dollar. Both are correct — they are two ways of writing the same part of a whole. The
one with the dot is a decimal.
After this lesson you can say what a decimal is, name the places after the decimal point, turn a simple fraction into a decimal and back, and explain why 0.5 and 0.50 are the same number.
A decimal writes a fraction using place value. You already know place value for whole numbers: ones, tens, hundreds, each ten times the last. A decimal simply continues that pattern downward, into places smaller than one. It lets you write a part of a whole without a separate top and bottom number.
The decimal point separates the whole part from the part below one. In 3.75, the
dot is the decimal point. Everything left of it is whole units; everything right is
less than one. The first place after the point is tenths (each worth 1/10), the
next is hundredths (each worth 1/100) — each place still ten times smaller than
the one before.
A decimal and a fraction are two names for one amount. 0.7 is 7 tenths, which is
the fraction 7/10. 0.25 is 2 tenths and 5 hundredths, which is 25/100 — and
25/100 simplifies to 1/4. To turn a fraction into a decimal, rewrite it so its
denominator is 10 or 100, then read the digits into the places after the point.
A zero on the right end of a decimal changes nothing. 0.5 is 5 tenths. 0.50 is
5 tenths and 0 hundredths — the extra 0 adds no hundredths, so it is the same amount.
0.5 = 0.50. This is unlike whole numbers, where a 0 on the right does matter (5 and
50 differ). On the right of a decimal, a trailing 0 is just an empty smaller place.
Write the fraction 3/4 as a decimal.
A decimal needs a denominator of 10, 100, and so on. Can 3/4 be rewritten with such
a denominator? Multiply top and bottom by 25: 3/4 = 75/100.
75/100 is 75 hundredths. Read it into the places after the point: 7 in the tenths
place, 5 in the hundredths place. With a 0 for the whole part, that is 0.75.
So 3/4 = 0.75. Check it back: 0.75 is 75/100, and 75/100 simplifies — divide
top and bottom by 25 — to 3/4.
Why this works
Why does the same place-value idea reach below one? Because place value was never about whole numbers specifically — it is about each place being ten times the next. Going left, places grow ten times; going right past the point, they shrink ten times: tenths, hundredths, thousandths. The decimal point just marks where “one” sits so you know which way each place leans.
Common mistake
A common mistake is reading 0.7 as smaller than 0.25 because “25 is bigger than 7”.
But 0.7 is 7 tenths and 0.25 is 25 hundredths — different-sized places. 0.7
is 0.70, which is 70 hundredths, clearly more than 25. Compare decimals place by
place from the point, not by the raw digit strings.
0.7 equals the fraction ?/10. Type the numerator.
Write 0.6 as a fraction over 10. Type the numerator.
0.50 — how many hundredths is that? Type the number.
Write the fraction 1/2 as a decimal. Type it using a dot.
0.25 equals the fraction ?/100. Type the numerator.
Why are 0.5 and 0.50 the same number?
A decimal writes a fraction using place value, continuing the pattern below one. The decimal point separates whole units from the part under one; the places after it are tenths, hundredths, and so on, each ten times smaller than the last. A decimal and a fraction are two names for one amount — rewrite a fraction over 10 or 100 to read it as a decimal. A zero on the right end of a decimal adds nothing.