---
title: "Decimal Numbers"
book: "Primary & Middle School Mathematics"
subject: math
language: en
chapter: 29
exercises: 11
source: https://one-course.com/books/math/1/en/chapter/29-decimal-numbers
---

# Chapter 29 — Decimal Numbers

Tenths and hundredths appeared in [Chapter 25](https://one-course.com/books/math/1/en/chapter/25-tenths-and-hundredths#ch-g4-decimals); 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](https://one-course.com/books/math/1/en/chapter/25-tenths-and-hundredths#def-g4-decimals-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.](https://one-course.com/images/onecourse/chapters/math-1/g5-decimals/fig-7133f9aa0f12.svg)

*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](https://one-course.com/books/math/1/en/chapter/1-counting-to-20#def-g1-counting-zero) changes nothing: $4.362 = 4.3620$. But $4.362 \neq 4.0362$: a [zero](https://one-course.com/books/math/1/en/chapter/1-counting-to-20#def-g1-counting-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.](https://one-course.com/images/onecourse/chapters/math-1/g5-decimals/fig-41d30f23f986.svg)

*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).**

1. Compare the whole parts: $6.1 > 5.987$ .
2. Equal whole parts: compare the decimal parts digit by digit from the point — tenths, then hundredths, then thousandths;
3. padding with final [zeros](https://one-course.com/books/math/1/en/chapter/1-counting-to-20#def-g1-counting-zero) 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](https://one-course.com/books/math/1/en/chapter/21-large-numbers#def-g4-numbers-round) to the unit (or tenth, or hundredth), look at the next digit: $5$ or more [rounds](https://one-course.com/books/math/1/en/chapter/21-large-numbers#def-g4-numbers-round) up, otherwise [round](https://one-course.com/books/math/1/en/chapter/21-large-numbers#def-g4-numbers-round) down. So $4.362$ [rounds](https://one-course.com/books/math/1/en/chapter/21-large-numbers#def-g4-numbers-round) 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](https://one-course.com/books/math/1/en/chapter/21-large-numbers#def-g4-numbers-round) 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 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](#def-g5-decimals-places).

**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 of Exercise 29.3.**

$3.60 = 3.6$: true (a final [zero](https://one-course.com/books/math/1/en/chapter/1-counting-to-20#def-g1-counting-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 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 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 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 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](https://one-course.com/books/math/1/en/chapter/21-large-numbers#def-g4-numbers-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 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 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 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](https://one-course.com/books/math/1/en/chapter/21-large-numbers#def-g4-numbers-round) to $6.4$ when rounded to the tenth. What are the smallest and the biggest?

**Solution of Exercise 29.11.**

[Rounding](https://one-course.com/books/math/1/en/chapter/21-large-numbers#def-g4-numbers-round) to the tenth gives $6.4$ for the two-decimal numbers from $6.35$ (halfway [rounds](https://one-course.com/books/math/1/en/chapter/21-large-numbers#def-g4-numbers-round) up) to $6.44$: the numbers $6.35$, $6.36$, …, $6.44$. Smallest: $6.35$; biggest: $6.44$.