---
title: "Decimal Numbers"
book: "Primary & Middle School Mathematics"
subject: math
language: en
chapter: 38
exercises: 12
source: https://one-course.com/books/math/1/en/chapter/38-decimal-numbers
---

# Chapter 38 — Decimal Numbers

Between $2$ and $3$ there are plenty of numbers: $2.5$, $2.71$, $2.999$ … [Decimal numbers](#def-g6-decimals-places) extend the place-value system to the right of the units, and they follow the same rules as whole numbers — with a few traps this chapter will teach you to avoid.

## 38.1 Decimal writing

**Definition 38.1 (Decimal places).**

A *decimal number* has a whole part and a decimal part, separated by the [decimal point](https://one-course.com/books/math/1/en/chapter/25-tenths-and-hundredths#def-g4-decimals-point). Each place to the right of the point is worth ten times less than the one before: tenths, hundredths, thousandths. For instance

$$
13.407 = 13 + \frac{4}{10} + \frac{0}{100} + \frac{7}{1000}
= 1 \times 10 + 3 + 4 \times \frac{1}{10} + 7 \times \frac{1}{1000}.
$$

![The place-value table of 13.407: the decimal point sits between the units and the tenths.](https://one-course.com/images/onecourse/chapters/math-1/g6-decimals/fig-39917da6f6a8.svg)

*The place-value table of $13.407$: the [decimal point](https://one-course.com/books/math/1/en/chapter/25-tenths-and-hundredths#def-g4-decimals-point) sits between the units and the tenths.*

**Remark 38.2 (Zeros that matter, zeros that don’t).**

Adding [zeros](https://one-course.com/books/math/1/en/chapter/1-counting-to-20#def-g1-counting-zero) at the *end* of the decimal part changes nothing: $2.5 = 2.50 = 2.500$. But [zeros](https://one-course.com/books/math/1/en/chapter/1-counting-to-20#def-g1-counting-zero) *between* digits are essential: $13.407 \neq 13.47$. And $0.5$ is very different from $0.05$!

**Method 38.3 (Comparing decimal numbers).**

1. Compare the whole parts first: $7.2 > 5.99$ because $7 > 5$ .
2. If the whole parts are equal, compare the decimal parts digit by digit from the left — padding with final [zeros](https://one-course.com/books/math/1/en/chapter/1-counting-to-20#def-g1-counting-zero) helps: to compare $3.4$ and $3.15$ , write $3.40$ and $3.15$ ; since $40 > 15$ hundredths, $3.4 > 3.15$ .

Beware: *longer does not mean larger*. $3.15$ has more digits than $3.4$ but is smaller.

![Zooming on the number line between 3 and 4: each tick is one tenth. The point 3.15 sits halfway between 3.1 and 3.2.](https://one-course.com/images/onecourse/chapters/math-1/g6-decimals/fig-d3c0370768d1.svg)

*Zooming on the number line between $3$ and $4$: each tick is one tenth. The point $3.15$ sits halfway between $3.1$ and $3.2$.*

## 38.2 Adding, subtracting, multiplying

**Method 38.4 (Column computations with decimals).**

To add or subtract [decimal numbers](#def-g6-decimals-places), align the *[decimal points](https://one-course.com/books/math/1/en/chapter/25-tenths-and-hundredths#def-g4-decimals-point)* (so units sit under units, tenths under tenths), padding with final [zeros](https://one-course.com/books/math/1/en/chapter/1-counting-to-20#def-g1-counting-zero) if needed; then compute as with whole numbers, and place the point of the result under the others.

**Example 38.5.**

Compute $13.7 + 2.85$, aligning the points ($13.7 = 13.70$):

$$
\begin{array}{r}
1\,3.7\,0 \\
+\ \ \,2.8\,5 \\
\hline
1\,6.5\,5
\end{array}
$$

Compute $6 - 2.35$ (write $6 = 6.00$): $6.00 - 2.35 = 3.65$. Check: $3.65 + 2.35 = 6$.

**Example 38.6 (Multiplying decimals).**

To compute $2.3 \times 1.4$: multiply as whole numbers, $23 \times 14 = 322$; then count the decimal digits of the factors ($1 + 1 = 2$), and place the point so the result has that many: $2.3 \times 1.4 = 3.22$. Sanity check on sizes: $2.3 \times 1.4$ should be a bit more than $2.3 \times 1 = 2.3$ — and $3.22$ is.

## 38.3 Multiplying and dividing by 10, 100, 1000

**Proposition 38.7 (Shifting the point).**

Multiplying a [decimal number](#def-g6-decimals-places) by $10$, $100$, $1000$ moves its [decimal point](https://one-course.com/books/math/1/en/chapter/25-tenths-and-hundredths#def-g4-decimals-point) $1$, $2$, $3$ places to the *right*; dividing moves it to the *left* (padding with [zeros](https://one-course.com/books/math/1/en/chapter/1-counting-to-20#def-g1-counting-zero) when needed):

$$
3.75 \times 100 = 375, \qquad
42.1 \div 1000 = 0.0421 .
$$

**Proof.** Multiplying by $10$ makes each digit worth ten times more: tenths become units, units become tens, and so on — every digit moves one column to the left in the place-value table, which is the same as moving the point one place to the right. Dividing reverses this. ∎

**Example 38.8 (Units of measurement).**

Converting units is exactly this game: since $1$ m $= 100$ cm,

$$
3.75 \text{ m} = 3.75 \times 100 \text{ cm} = 375 \text{ cm},
\qquad
42 \text{ mm} = 42 \div 10 \text{ cm} = 4.2 \text{ cm}.
$$

## 38.4 Rounding

**Definition 38.9 (Rounding).**

The *rounding* of a number to the unit (or tenth, hundredth, …) is the closest number with that precision. Look at the next digit: if it is $0,1,2,3,4$, [round](https://one-course.com/books/math/1/en/chapter/21-large-numbers#def-g4-numbers-round) down (keep); if it is $5,6,7,8,9$, [round](https://one-course.com/books/math/1/en/chapter/21-large-numbers#def-g4-numbers-round) up.

**Example 38.10.**

$7.38$ rounded to the unit is $7$ (next digit $3$: keep); rounded to the tenth it is $7.4$ (next digit $8$: [round](https://one-course.com/books/math/1/en/chapter/21-large-numbers#def-g4-numbers-round) up). The price $4.996$ rounded to the hundredth is $5.00$ — [rounding](#def-g6-decimals-rounding) can change every digit!

## 38.5 Exercises

**Exercise 38.1 ★.**

Write as a [decimal number](#def-g6-decimals-places): $5 + \dfrac{3}{10} + \dfrac{7}{100}$; $\dfrac{9}{10}$; $12 + \dfrac{4}{1000}$; “eight units and five hundredths”.

**Solution of Exercise 38.1.**

$5.37$; $0.9$; $12.004$; $8.05$.

**Exercise 38.2 ★.**

In $86.354$: what is the digit of the tenths? Of the hundredths? What does the digit $8$ count? Write this number as in [Definition 38.1](#def-g6-decimals-places).

**Solution of Exercise 38.2.**

Tenths digit: $3$; hundredths digit: $5$; the $8$ counts the tens.

$$
86.354 = 8 \times 10 + 6 + 3 \times \frac{1}{10} + 5 \times
\frac{1}{100} + 4 \times \frac{1}{1000}.
$$

**Exercise 38.3 ★.**

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

$$
5.3 \;?\; 5.29, \qquad
0.7 \;?\; 0.70, \qquad
2.09 \;?\; 2.9, \qquad
14.5 \;?\; 14.49 .
$$

**Solution of Exercise 38.3.**

$5.3 > 5.29$ (compare $5.30$ with $5.29$); $0.7 = 0.70$ (final [zeros](https://one-course.com/books/math/1/en/chapter/1-counting-to-20#def-g1-counting-zero) change nothing); $2.09 < 2.9$ (tenths: $0 < 9$); $14.5 > 14.49$.

**Exercise 38.4 ★.**

Order from smallest to largest: $4.2$; $4.05$; $4.51$; $4.15$; $4.5$.

**Solution of Exercise 38.4.**

Pad to two decimals: $4.20$; $4.05$; $4.51$; $4.15$; $4.50$. Order:

$$
4.05 < 4.15 < 4.2 < 4.5 < 4.51 .
$$

**Exercise 38.5 ★.**

Which [decimal numbers](#def-g6-decimals-places) correspond to the points $A$, $B$, $C$ on a number line graduated in tenths, if $A$ is $3$ ticks after $6$, $B$ is $7$ ticks after $6$, and $C$ is $2$ ticks after $7$?

**Solution of Exercise 38.5.**

Each tick is one tenth. $A$: $6.3$; $B$: $6.7$; $C$: $7.2$.

**Exercise 38.6 ★.**

Compute in columns: $45.8 + 7.65$; $23.4 - 8.72$; $5.6 \times 2.4$ (count the decimal digits!).

**Solution of Exercise 38.6.**

$45.80 + 7.65 = 53.45$.

$23.40 - 8.72 = 14.68$ (check: $14.68 + 8.72 = 23.4$).

$56 \times 24 = 1344$, and two decimal digits in the factors: $5.6 \times 2.4 = 13.44$.

**Exercise 38.7 ★.**

Compute without any written work:

$$
6.42 \times 10, \qquad
0.35 \times 1000, \qquad
78.1 \div 100, \qquad
5 \div 1000 .
$$

**Solution of Exercise 38.7.**

$6.42 \times 10 = 64.2$; $0.35 \times 1000 = 350$; $78.1 \div 100 = 0.781$; $5 \div 1000 = 0.005$.

**Exercise 38.8 ★.**

Convert: $2.4$ m into cm; $370$ g into kg; $0.85$ km into m; $56$ mm into m.

**Solution of Exercise 38.8.**

$2.4$ m $= 240$ cm; $370$ g $= 0.37$ kg (divide by $1000$); $0.85$ km $= 850$ m; $56$ mm $= 0.056$ m.

**Exercise 38.9 ★.**

Round $23.867$: to the unit; to the tenth; to the hundredth. Round $9.97$ to the tenth.

**Solution of Exercise 38.9.**

To the unit: $24$ (next digit $8$: [round](https://one-course.com/books/math/1/en/chapter/21-large-numbers#def-g4-numbers-round) up). To the tenth: $23.9$. To the hundredth: $23.87$. And $9.97$ to the tenth: the next digit is $7$, so [round](https://one-course.com/books/math/1/en/chapter/21-large-numbers#def-g4-numbers-round) up the $9$ tenths — $10.0$.

**Exercise 38.10 ★★.**

A baguette costs $1.15$. Lena buys three baguettes and pays with a $5$ bill. Write the two computations needed, and give her change.

**Solution of Exercise 38.10.**

Price of the baguettes: $3 \times 1.15 = 3.45$. Change: $5 - 3.45 = 1.55$.

**Exercise 38.11 ★★.**

Find a [decimal number](#def-g6-decimals-places) strictly between $7.4$ and $7.5$; then between $3.99$ and $4$. How many such numbers are there in each case?

**Solution of Exercise 38.11.**

Between $7.4$ and $7.5$: for instance $7.45$ (or $7.41$, $7.499$, …). Between $3.99$ and $4$: for instance $3.995$. In both cases there are *infinitely many* such numbers: one can always add more decimal places.

**Exercise 38.12 ★★★.**

Using each of the digits $2$, $5$, $8$ exactly once and one [decimal point](https://one-course.com/books/math/1/en/chapter/25-tenths-and-hundredths#def-g4-decimals-point), write the largest possible number, then the smallest possible one. (Numbers like $.58$ are not allowed: the whole part must contain at least one digit.)

**Solution of Exercise 38.12.**

Largest: put the biggest digits first and the point as late as possible: $85.2$. Smallest: smallest digits first and the point as early as possible: $2.58$.

## 38.6 Problem: The number just after 3 does not exist

**Problem 38.1.**

Weekend problem — between any two decimal numbers there is always another: zooming on the number line

On the ladder of whole numbers, every number has a next-door neighbour: right after $7$ comes $8$, and nothing lives in between. [Decimal numbers](#def-g6-decimals-places) are a completely different world. What is the number just after $3$? Is it $3.1$? $3.01$? $3.001$? This problem develops a *zooming* technique on the number line (continuing [Exercise 38.11](#exo-g6-decimals-11)) and reaches a famous conclusion: the number just after $3$ *does not exist* — between any two [decimal numbers](#def-g6-decimals-places), however close, there is always room for more.

**Part I — Zooming in.**

1. Which is larger, $2.999$ or $3$ ? Compute the [difference](https://one-course.com/books/math/1/en/chapter/3-subtraction-first-steps#ex-g1-subtraction-difference) between them.
2. Draw a number line from $7.4$ to $7.5$ , graduated in hundredths, and place $7.42$ , $7.45$ and $7.48$ on it. List *all* the numbers with two decimal digits that lie strictly between $7.4$ and $7.5$ . How many are there?
3. Zoom again: between $7.44$ and $7.45$ , list all the numbers with three decimal digits. How many are there this time?
4. Using the same idea, explain how to count the numbers with exactly three decimal digits lying strictly between $7.4$ and $7.5$ , and give that count.
5. Describe the recipe hiding behind questions 2–4 (the *zoom* ): given two numbers that look like neighbours, such as $5.67$ and $5.68$ , how does writing one more decimal place always reveal a number strictly between them? Apply your recipe to $5.67$ and $5.68$ , then to $0.1999$ and $0.2$ .

**Part II — The missing neighbour.**

6. Tom claims: “the number just after $3$ is $3.1$ .” Prove him wrong by naming a number strictly between $3$ and $3.1$ . Tom retreats to $3.01$ , then to $3.001$ . Beat each of his candidates.
7. Explain why *nobody* can win this game against you: whatever number strictly greater than $3$ Tom proposes, the zoom recipe of question 5 produces a number strictly between $3$ and his proposal. What does this prove about “the number just after $3$ ”?
8. Now the other side: Tom hunts for the number just *before* $3$ and tries $2.9$ , then $2.99$ , then $2.999$ . Beat his three candidates. Then compute $3 - 2.999$ , and describe (without computing in columns) the [difference](https://one-course.com/books/math/1/en/chapter/3-subtraction-first-steps#ex-g1-subtraction-difference) between $3$ and the number written with a $2$ , a point, and twenty digits $9$ .
9. Why does none of this work for *whole* numbers? Explain in one or two sentences why there is no whole number strictly between $7$ and $8$ , even though there are plenty of [decimal numbers](#def-g6-decimals-places) there.
10. A length is announced as “ $3.7$ cm, rounded to the tenth” ( [Definition 38.9](#def-g6-decimals-rounding) ). Give the smallest length that [rounds](https://one-course.com/books/math/1/en/chapter/21-large-numbers#def-g4-numbers-round) to $3.7$ cm, and explain why a length of $3.75$ cm does *not* [round](https://one-course.com/books/math/1/en/chapter/21-large-numbers#def-g4-numbers-round) to $3.7$ — so the true length is at least $3.65$ cm and strictly below $3.75$ cm.

**Part III — Games with digits and points.**

11. Order from smallest to largest: $0.6$ ; $0.58$ ; $0.123$ ; $0.0999$ . Then explain to a classmate, in one sentence, why $0.0999 < 0.6$ even though it is written with more digits ( [Method 38.3](#met-g6-decimals-compare) ).
12. Prices in a shop always have exactly two decimal digits. Is there a *price* strictly between $4.99$ and $5$ ? Compare with question 9: what do prices and whole numbers have in common?
13. Using each of the digits $2$ , $5$ , $8$ exactly once and one [decimal point](https://one-course.com/books/math/1/en/chapter/25-tenths-and-hundredths#def-g4-decimals-point) (whole part not empty, as in [Exercise 38.12](#exo-g6-decimals-12) ), find the number closest to $6$ . Justify by computing the distance of your best candidates to $6$ .
14. The most famous number of mathematics, $\pi$ , satisfies $3.141 < \pi < 3.142$ . Name a [decimal number](#def-g6-decimals-places) strictly between $3.141$ and $3.142$ ; then one strictly between $3.1415$ and $3.1416$ . (Mathematicians squeeze $\pi$ exactly this way — each new decimal digit is one more zoom. You will meet $\pi$ at work in [Chapter 43](https://one-course.com/books/math/1/en/chapter/43-perimeter-area-volume#ch-g6-measure) .)
15. The finale: explain why there are *more [decimal numbers](#def-g6-decimals-places) between $0$ and $1$ than any number you can name* . (How many does one zoom produce? Can the zooming ever stop?)

**Solution of Problem 38.1.**

**1.** $3$ is larger: padding, $3.000 > 2.999$. The [difference](https://one-course.com/books/math/1/en/chapter/3-subtraction-first-steps#ex-g1-subtraction-difference) is $3.000 - 2.999 = 0.001$, one thousandth.

**2.** Strictly between $7.4 = 7.40$ and $7.5 = 7.50$ lie

$$
7.41,\ 7.42,\ 7.43,\ 7.44,\ 7.45,\ 7.46,\ 7.47,\ 7.48,\ 7.49 :
$$

nine numbers, one per new graduation mark.

**3.** The same picture, ten times smaller: between $7.440$ and $7.450$ lie $7.441, 7.442, \dots, 7.449$ — nine numbers again.

**4.** Between $7.400$ and $7.500$, the numbers with three decimal digits are $7.401, 7.402, \dots, 7.499$: all the thousandths from $401$ to $499$, that is $99$ numbers.

**5.** The zoom recipe: write both numbers with the same number of decimal places, then *add one more decimal place* — the smaller number followed by a digit from $1$ to $9$ lands strictly between the two. Indeed $5.67 = 5.670$ and $5.68 = 5.680$, and

$$
5.670 < 5.675 < 5.680 ;
$$

likewise $0.1999 = 0.19990 < 0.19995 < 0.20000 = 0.2$. Between two neighbouring graduations there is always a whole new level of nine finer graduations.

**6.** $3 < 3.05 < 3.1$; then $3 < 3.005 < 3.01$; then $3 < 3.0005 < 3.001$. Each candidate is beaten by one more zoom.

**7.** Whatever number Tom proposes — call it his candidate, strictly greater than $3$ — question 5 produces a number strictly between $3$ and the candidate. So the candidate was *not* the closest number to $3$: something [even](https://one-course.com/books/math/1/en/chapter/14-numbers-up-to-10-000#def-g3-numbers-evenodd) closer exists. Since this happens to every candidate without exception, no number can be “the number just after $3$”: it simply does not exist.

**8.** $2.9 < 2.95 < 3$, $2.99 < 2.995 < 3$, $2.999 < 2.9995 < 3$. And $3 - 2.999 = 0.001$. With twenty nines, the [difference](https://one-course.com/books/math/1/en/chapter/3-subtraction-first-steps#ex-g1-subtraction-difference) is a [decimal point](https://one-course.com/books/math/1/en/chapter/25-tenths-and-hundredths#def-g4-decimals-point) followed by nineteen [zeros](https://one-course.com/books/math/1/en/chapter/1-counting-to-20#def-g1-counting-zero) and a $1$ — one unit in the twentieth decimal place: tiny, but not [zero](https://one-course.com/books/math/1/en/chapter/1-counting-to-20#def-g1-counting-zero). However many nines Tom writes, he never reaches $3$.

**9.** Whole numbers climb in steps of $1$: more than $7$ but less than $8$ would mean a whole number of units strictly between $7$ and $8$ units, and there is none. The zoom escapes this only by writing digits *after the point* — exactly what whole numbers do not have.

**10.** The smallest length rounding to $3.7$ is $3.65$ cm: its next digit is $5$, which [rounds](https://one-course.com/books/math/1/en/chapter/21-large-numbers#def-g4-numbers-round) *up* ([Definition 38.9](#def-g6-decimals-rounding)). And $3.75$ [rounds](https://one-course.com/books/math/1/en/chapter/21-large-numbers#def-g4-numbers-round) up to $3.8$ for the same reason. So “$3.7$ cm to the nearest tenth” means: at least $3.65$ cm, and strictly less than $3.75$ cm — a whole zoomed-in segment of possible true lengths hides behind one rounded value.

**11.** $0.0999 < 0.123 < 0.58 < 0.6$. One sentence: comparing digit by digit from the left, $0.0999$ has $0$ tenths while $0.6$ has $6$, and the comparison is settled there — the number of digits written says nothing about size ([Method 38.3](#met-g6-decimals-compare)).

**12.** No: in cents, $4.99$ euros is $499$ cents and $5$ euros is $500$ cents — consecutive *whole* numbers, with nothing between them. Prices, having exactly two decimal places, are really whole numbers of cents in disguise: like the whole numbers of question 9, they do have next-door neighbours. The endless zoom needs the right to write ever more decimal places.

**13.** The candidates near $6$ are $5.82$ and $8.25$ (a number starting with $2$, or with $58$, $25$, $82$, $85$, is far from $6$). Distances: $6 - 5.82 = 0.18$ and $8.25 - 6 = 2.25$. The closest is $5.82$.

**14.** For instance $3.1415$, since $3.1410 < 3.1415 < 3.1420$; then $3.14159$, since $3.14150 < 3.14159 < 3.14160$. (Both are real steps in the actual hunt for $\pi = 3.14159\dots$)

**15.** Suppose someone names a number, as large as they like. One zoom turns every pair of neighbouring graduations between $0$ and $1$ into nine new numbers, and the zoom can be repeated forever — questions 2, 3 and 4 were only the first two levels. Repeating it enough times produces more [decimal numbers](#def-g6-decimals-places) between $0$ and $1$ than the named number. So no number is large enough to count them: there are infinitely many.