Primary & Middle School Mathematics · Grades 1–9

29Decimal Numbers

Tenths and hundredths appeared in Chapter 25; adding one more level, thousandths, completes the decimal system. This chapter consolidates reading, decomposing and — above all — comparing decimal numbers without falling into their famous traps.

29.1 Down to thousandths

Definition 29.1 (Decimal places)

After the decimal point come the tenths, the hundredths, the thousandths — each worth ten times less than the one before:

4.362 = 4 + \frac{3}{10} + \frac{6}{100} + \frac{2}{1000} = \frac{4\,362}{1000}.

The place-value table of 4.362: whole part in blue, decimal part in red, the point between units and tenths. — The place-value table of $4.362$ : whole part in blue, decimal part in red, the point between units and tenths.

Example 29.2 (Many readings, one number)

$4.362$ can be read “four point three six two”, or “four and $362$ thousandths”, or decomposed as $4 + 0.3 + 0.06 + 0.002$ . And padding with final zeros changes nothing: $4.362 = 4.3620$ . But $4.362 \neq 4.0362$ : a zero inside pushes every digit to a smaller place.

Example 29.3 (Where is it on the line?)

To place $4.362$ : between $4$ and $5$ ; zooming, between $4.3$ and $4.4$ ; zooming again, between $4.36$ and $4.37$ , closer to $4.36$ . Each decimal digit is one more level of zoom.

Zooming on the number line: the stretch from 4.3 to 4.4, magnified, is itself cut into ten hundredths — and 4.362 sits just past 4.36. — Zooming on the number line: the stretch from $4.3$ to $4.4$ , magnified, is itself cut into ten hundredths — and $4.362$ sits just past $4.36$ .

29.2 Comparing decimals

Method 29.4 (Comparing two decimals)

Compare the whole parts: $6.1 > 5.987$ .
Equal whole parts: compare the decimal parts digit by digit from the point — tenths, then hundredths, then thousandths;
padding with final zeros makes the comparison fair: $2.5$ vs $2.48$ becomes $2.50$ vs $2.48$ , and $50 > 48$ hundredths.

The trap, one last time: a longer decimal part does not mean a bigger number — $2.48$ has more digits than $2.5$ and is smaller.

Example 29.5

Order $7.3$ ; $7.09$ ; $7.31$ ; $7.299$ . Pad to three decimals: $7.300$ ; $7.090$ ; $7.310$ ; $7.299$ . Then

7.09 < 7.299 < 7.3 < 7.31 .

Note $7.299 < 7.3$ even though $299$ looks big: $299$ thousandths against $300$ thousandths.

Example 29.6 (Squeezing between two decimals)

Is there a number between $5.7$ and $5.8$ ? Yes, plenty: $5.75$ , $5.71$ , $5.799$ … Between $5.79$ and $5.8$ ? Again plenty: $5.795$ , for one. Between two different decimals, one can always squeeze another — there is no “next” decimal number.

29.3 Rounding decimals

Definition 29.7 (Rounding)

To round to the unit (or tenth, or hundredth), look at the next digit: $5$ or more rounds up, otherwise round down. So $4.362$ rounds to $4$ (unit), to $4.4$ (tenth), to $4.36$ (hundredth).

Example 29.8 (Money rounds to cents)

A price computed as $7.4983$ is displayed as $7.50$ : real-life amounts are rounded to the hundredth. Note the cascade: $7.4983 \to 7.50$ , where the $49$ became $50$ — rounding can ripple through several digits.

29.4 Exercises

Exercise 29.1 ★

Write as a decimal number: $5 + \frac{2}{10} + \frac{7}{1000}$ ; $\frac{85}{100}$ ; $\frac{4\,507}{1000}$ ; “twelve and nine thousandths”.

Solution

Solution of Exercise 29.1.

$5.207$ ; $0.85$ ; $4.507$ ; $12.009$ .

Exercise 29.2 ★

In $27.418$ : what does the $4$ count? The $8$ ? Decompose the number as in Definition 29.1.

Solution

Solution of Exercise 29.2.

The $4$ counts the tenths, the $8$ the thousandths:

27.418 = 27 + \frac{4}{10} + \frac{1}{100} + \frac{8}{1000}.

Exercise 29.3 ★

True or false? Explain with places. $3.60 = 3.6$ ; $0.5 = 0.05$ ; $2.070 = 2.07$ ; $1.3 = 1.03$ .

Solution

Solution of Exercise 29.3.

$3.60 = 3.6$ : true (a final zero adds nothing).

$0.5 = 0.05$ : false — $5$ tenths against $5$ hundredths.

$2.070 = 2.07$ : true.

$1.3 = 1.03$ : false — $3$ tenths against $3$ hundredths.

Exercise 29.4 ★

Copy and complete with $<$ , $>$ or $=$ :

4.8 \;?\; 4.79, \qquad 0.302 \;?\; 0.32, \qquad 6.5 \;?\; 6.500, \qquad 9.09 \;?\; 9.1 .

Solution

Solution of Exercise 29.4.

$4.8 > 4.79$ ; $0.302 < 0.32$ (tenths equal, hundredths $0 < 2$ ); $6.5 = 6.500$ ; $9.09 < 9.1$ .

Exercise 29.5 ★

Order from smallest to biggest: $3.14$ ; $3.4$ ; $3.041$ ; $3.104$ ; $3.41$ .

Solution

Solution of Exercise 29.5.

$3.041 < 3.104 < 3.14 < 3.4 < 3.41$ .

Exercise 29.6 ★

Between which two whole numbers does each number lie? Between which two numbers with one decimal? $6.83$ ; $0.492$ ; $12.06$ .

Solution

Solution of Exercise 29.6.

$6.83$ : between $6$ and $7$ ; between $6.8$ and $6.9$ .

$0.492$ : between $0$ and $1$ ; between $0.4$ and $0.5$ .

$12.06$ : between $12$ and $13$ ; between $12.0$ and $12.1$ .

Exercise 29.7 ★

Round $8.276$ : to the unit; to the tenth; to the hundredth. Round $3.95$ to the tenth.

Solution

Solution of Exercise 29.7.

$8.276$ : to the unit $8$ ; to the tenth $8.3$ ; to the hundredth $8.28$ . And $3.95$ to the tenth: the hundredths digit is $5$ , round up: $4.0$ .

Exercise 29.8 ★

Give a number strictly between: $2.6$ and $2.7$ ; $0.41$ and $0.42$ ; $5.99$ and $6$ .

Solution

Solution of Exercise 29.8.

For instance $2.65$ ; $0.415$ ; $5.995$ . (Infinitely many answers each time.)

Exercise 29.9 ★

The heights of four plants are $0.85$ m, $1.2$ m, $0.9$ m and $1.02$ m. Order them from shortest to tallest.

Solution

Solution of Exercise 29.9.

$0.85 < 0.9 < 1.02 < 1.2$ (in meters).

Exercise 29.10 ★★

Lucie says: “ $7.12 > 7.9$ because $12 > 9$ ”. Correct her mistake, comparing tenths first.

Solution

Solution of Exercise 29.10.

Compare the tenths first: $7.12$ has $1$ tenth, $7.9$ has $9$ tenths, so $7.12 < 7.9$ . Lucie compared the decimal parts as whole numbers ( $12$ vs $9$ ), forgetting that $12$ hundredths is far less than $90$ hundredths.

Exercise 29.11 ★★

Find all the numbers with exactly two decimal digits that round to $6.4$ when rounded to the tenth. What are the smallest and the biggest?

Solution

Solution of Exercise 29.11.

Rounding to the tenth gives $6.4$ for the two-decimal numbers from $6.35$ (halfway rounds up) to $6.44$ : the numbers $6.35$ , $6.36$ , …, $6.44$ . Smallest: $6.35$ ; biggest: $6.44$ .