Decimal numbers and rational numbers – Sets of Numbers (Senior 1 Mathematics) | Ganzaa

Sets of Numbers

Lessons

Senior 1 · Mathematics

Sets of Numbers

5 lessons

Lesson 3 of 5

Decimal numbers and rational numbers

25 min

When division leaves the integers

Four friends share three loaves of bread equally. Each gets $3 \div 4$ of a loaf, and no integer answers that division: each friend gets three quarters of a loaf, $\frac{3}{4}$ , which is also written 0.75. The integers are not closed under division, so we need new numbers: fractions and decimals.

Definition

Decimal number: a number that can be written with a finite number of digits after the decimal point. Every such number is a fraction whose denominator is a power of 10.

\mathbb{D} = \left\{ \frac{a}{10^n} : a \in \mathbb{Z},\ n \in W \right\}

Example: $2.75 = \frac{275}{100} = \frac{275}{10^2}$ and $-0.4 = \frac{-4}{10}$ . Every integer is a decimal number (take $n = 0$ ), so $\mathbb{Z} \subset \mathbb{D}$ .

Operations on decimal numbers

Operation	Method	Example
adding and subtracting	line up the decimal points and fill gaps with zeros	$12.5 + 3.75 = 12.50 + 3.75 = 16.25$
multiplying	multiply as whole numbers, then give the answer as many decimal places as the two factors have together	$2.4 \times 0.35$ : $24 \times 35 = 840$ , 1 + 2 = 3 places, so $0.840 = 0.84$
dividing	multiply both numbers by the same power of 10 to make the divisor a whole number, then divide	$7.2 \div 0.06 = 720 \div 6 = 120$

Why the decimal places add up: $2.4 \times 0.35 = \frac{24}{10} \times \frac{35}{100} = \frac{840}{1000} = 0.84$ . Tenths times hundredths gives thousandths, so 1 + 2 = 3 places. Why the division trick works: multiplying both numbers by 100 does not change their quotient, just as $\frac{7.2}{0.06} = \frac{720}{6}$ .

Worked example 1: decimals at work

A tailor in Nyamirambo has a ribbon 7.2 m long. (a) How many pieces 0.06 m long can she cut from it? (b) She sews them round a tablecloth 1.25 m long and 0.8 m wide. What is the area of the tablecloth?

Solution

(a) $7.2 \div 0.06$ . Multiply both numbers by 100 to make the divisor whole: $720 \div 6 = 120$ pieces.

(b) Area $= 1.25 \times 0.8$ . Multiply as whole numbers: $125 \times 8 = 1000$ . The factors have $2 + 1 = 3$ decimal places, so the answer is $1.000 = 1\ \mathrm{m^2}$ .

Rational numbers

Some divisions, such as $1 \div 3$ , do not give a decimal number: $1 \div 3 = 0.333\ldots$ never ends. To include every quotient of integers we need a larger set.

Definition

Rational number: a number that can be written as a fraction $\frac{a}{b}$ , where $a$ and $b$ are integers and $b \neq 0$ .

\mathbb{Q} = \left\{ \frac{a}{b} : a, b \in \mathbb{Z},\ b \neq 0 \right\}

Notation: $\mathbb{Q}$ , for quotient. Example: $\frac{3}{4}$ , $-\frac{5}{2}$ , $7 = \frac{7}{1}$ , $0.75 = \frac{3}{4}$ and $\frac{1}{3}$ are all rational. Every decimal number is rational, so $\mathbb{Z} \subset \mathbb{D} \subset \mathbb{Q}$ , but $\frac{1}{3} \in \mathbb{Q}$ while $\frac{1}{3} \notin \mathbb{D}$ .

Rational numbers on the number line: $-\frac{3}{2}$ , $-\frac{1}{2}$ , $\frac{1}{4}$ , $\frac{3}{4}$ and $\frac{5}{4}$ . They fill in the gaps between the integers.

Important

Equivalent fractions: for any non-zero integer $k$ ,

\frac{a}{b} = \frac{a \times k}{b \times k}.

Why: multiplying by $\frac{k}{k} = 1$ does not change a number. To write a fraction in simplest form, divide the numerator and the denominator by their highest common factor: $\frac{18}{24} = \frac{18 \div 6}{24 \div 6} = \frac{3}{4}$ .

Operations on rational numbers

For fractions $\frac{a}{b}$ and $\frac{c}{d}$ with non-zero denominators:

\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}

\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}

\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} \quad (c \neq 0)

Adding: rewrite both fractions over a common denominator; $b d$ always works, and the lowest common multiple of $b$ and $d$ keeps the numbers small.
Multiplying: $\frac{2}{3} \times \frac{3}{4}$ means two thirds of three quarters. The area model below shows it is 6 of the 12 equal parts of the square.
Dividing: multiply by the reciprocal of the divisor. This works because $\frac{c}{d} \times \frac{d}{c} = 1$ , so multiplying the dividend and the divisor by $\frac{d}{c}$ turns the divisor into 1.

$\frac{2}{3} \times \frac{3}{4} = \frac{6}{12} = \frac{1}{2}$ : the dark overlap is two thirds of the columns and three quarters of the rows.

Definition

Reciprocal (multiplicative inverse) of a non-zero rational number $\frac{c}{d}$ : the number $\frac{d}{c}$ , because $\frac{c}{d} \times \frac{d}{c} = 1$ .

Example: the reciprocal of $\frac{2}{5}$ is $\frac{5}{2}$ ; the reciprocal of $- 3$ is $-\frac{1}{3}$ . The number 0 has no reciprocal.

Worked example 2: adding fractions

A farmer in Nyagatare plants maize on $\frac{2}{3}$ of a hectare and beans on $\frac{3}{4}$ of a hectare. What area does she plant altogether?

Solution The lowest common multiple of 3 and 4 is 12:

\frac{2}{3} + \frac{3}{4} = \frac{8}{12} + \frac{9}{12} = \frac{17}{12} = 1\frac{5}{12} \text{ hectares.}

Worked example 3: multiplying and dividing with signs

Calculate (a) $\frac{3}{5} \times \left(-\frac{10}{21}\right)$ (b) $-\frac{5}{6} \div \frac{10}{9}$

Solution The sign rules for integers still apply: different signs give a negative answer. Cancelling common factors before multiplying keeps the numbers small.

\text{(a)}\quad \frac{3}{5} \times \left(-\frac{10}{21}\right) = -\frac{3 \times 10}{5 \times 21} = -\frac{1 \times 2}{1 \times 7} = -\frac{2}{7}

In (a), 3 was cancelled against 21 and 5 against 10.

\text{(b)}\quad -\frac{5}{6} \div \frac{10}{9} = -\frac{5}{6} \times \frac{9}{10} = -\frac{45}{60} = -\frac{3}{4}

Important

$\mathbb{Q}$ is closed under all four operations (dividing only by non-zero numbers): the formulas above always give a fraction of integers.

Between any two rational numbers there is another one. Their mean is rational and lies halfway between them: between $\frac{1}{3}$ and $\frac{1}{2}$ lies $\frac{1}{2}\left(\frac{1}{3} + \frac{1}{2}\right) = \frac{5}{12}$ . Repeating this shows there are infinitely many rational numbers between any two of them.

Warning

Do not add the denominators. $\frac{1}{2} + \frac{1}{3} \neq \frac{2}{5}$ . Half a loaf plus a third of a loaf is more than half a loaf, but $\frac{2}{5}$ is less than half. Always rewrite over a common denominator: $\frac{1}{2} + \frac{1}{3} = \frac{3}{6} + \frac{2}{6} = \frac{5}{6}$ .

Tip

To compare fractions quickly, compare them as decimals, or cross-multiply: $\frac{a}{b} \lt \frac{c}{d}$ exactly when $ad \lt bc$ (for positive denominators). For example $\frac{5}{8} \lt \frac{2}{3}$ because $5 \times 3 = 15 \lt 16 = 8 \times 2$ .

Khan Academy: Identifying rational numbers (6 min). Watch how numbers such as 3.66…, −81.25, √3 and π are sorted with one test: can the number be written as a ratio of two integers, or as a terminating or repeating decimal? You will meet the irrational ones in Lesson 5.

✦

Practice

Give fractions in simplest form. The last two are harder.

Write as fractions in simplest form: (a) 0.6 (b) 2.25 (c) $- 0.125$
Simplify $\frac{36}{48}$ and $-\frac{45}{60}$ .
Calculate (a) $4.6 + 12.75$ (b) $3.2 \times 0.15$ (c) $9.6 \div 0.3$
Calculate (a) $\frac{5}{6} - \frac{3}{8}$ (b) $\frac{4}{9} \times \frac{15}{8}$ (c) $\frac{7}{10} \div \frac{14}{25}$
Arrange in increasing order: $\frac{2}{3}$ , $\frac{3}{5}$ , $\frac{5}{8}$ , $0.7$ .
Find a rational number between $\frac{3}{4}$ and $\frac{4}{5}$ .
A sack holds $12\frac{1}{2}$ kg of beans. How many portions of $\frac{5}{8}$ kg can be packed from it?

Show answersHide answers

(a) $\frac{6}{10} = \frac{3}{5}$ (b) $\frac{225}{100} = \frac{9}{4}$ (c) $-\frac{125}{1000} = -\frac{1}{8}$
$\frac{36}{48} = \frac{3}{4}$ (HCF 12) and $-\frac{45}{60} = -\frac{3}{4}$ (HCF 15)
(a) $17.35$ (b) $0.48$ (c) $96 \div 3 = 32$
(a) $\frac{20}{24} - \frac{9}{24} = \frac{11}{24}$ (b) $\frac{60}{72} = \frac{5}{6}$ (c) $\frac{7}{10} \times \frac{25}{14} = \frac{175}{140} = \frac{5}{4}$
As decimals: 0.6, 0.625, 0.666…, 0.7. So $\frac{3}{5} \lt \frac{5}{8} \lt \frac{2}{3} \lt 0.7$ .
Their mean: $\frac{1}{2}\left(\frac{15}{20} + \frac{16}{20}\right) = \frac{31}{40}$ . (Other answers, such as 0.78, are also correct.)
$\frac{25}{2} \div \frac{5}{8} = \frac{25}{2} \times \frac{8}{5} = \frac{200}{10} = 20$ portions.

Sources and credits

3.4: Rational Numbers, from Contemporary Mathematics by Donna Kirk and contributors, OpenStax, provided by LibreTexts, licensed under CC BY 4.0 (licence read on the page, 26 September 2026). Adapted by Ganzaa.
5.3: Decimal Operations (Part 1), from Prealgebra by Lynn Marecek and MaryAnne Anthony-Smith, OpenStax, provided by LibreTexts, licensed under CC BY 4.0 (licence read on the page, 26 September 2026). Adapted by Ganzaa.
Video: Identifying Rational Numbers, Khan Academy, embedded from YouTube.

The Rwandan worked examples, the diagrams, the practice items and the quiz questions were written by Ganzaa.

1When division leaves the integers

2Operations on decimal numbers

3Rational numbers

4Operations on rational numbers

When division leaves the integers

Operations on decimal numbers

Rational numbers

Operations on rational numbers