Integers and rational numbers: the number systems

Numbers come in many kinds. The natural numbers $1, 2, 3, \ldots$ count things. Integers add the negatives ($-1, -2, -3, \ldots$) and zero, giving the full number line. Rational numbers add fractions: any number you can write as $\dfrac{a}{b}$ where $a, b$ are integers and $b \neq 0$.

WAEC tests these foundational ideas — ordering numbers on the number line, comparing positives and negatives, working with fractions and decimals. This topic builds the platform for everything that follows.

This lesson introduces each number system, the number line, and basic comparisons. The next lesson covers operations on integers and rational numbers, the order of operations, and the WAEC checklist.

Learning objectives

By the end of this lesson you should be able to:

• distinguish natural numbers, integers, rational and irrational numbers
• place integers and rational numbers on a number line
• order and compare integers and rational numbers
• classify numbers into the correct number systems

The number line

The number line is a visual representation of numbers. Zero is in the middle; positives extend to the right; negatives extend to the left.

... -4   -3   -2   -1    0    1    2    3    4 ...
  ←———|————|————|————|————|————|————|————|———→

Every integer corresponds to a tick mark. Every rational number has a precise location (between tick marks for non-integers). Every real number — including irrationals — has a precise location.

Figure 1. On the number line, numbers to the right are greater, and adding a positive number moves you right. WASSCE Tutor diagram.

Distance on the number line

• Distance between two points is the absolute difference. From $-3$ to $2$: distance is $|2 - (-3)| = 5$.
• Absolute value $|x|$: the distance from $x$ to $0$. So $|-7| = 7$ and $|3| = 3$.

Examples

• $|-5| = 5$
• $|0| = 0$
• $|+8| = 8$
• Distance from $-4$ to $7$: $|7 - (-4)| = 11$.
• Distance from $3$ to $9$: $|9 - 3| = 6$.

Integers — full structure

Integers include positive, negative, and zero.

Examples: $-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5$.

Properties of integers

• Closed under addition: integer + integer = integer.
• Closed under subtraction: integer - integer = integer.
• Closed under multiplication: integer × integer = integer.
• NOT closed under division: integer ÷ integer is not always an integer ($1 ÷ 2 = 0.5$, not an integer).

Ordering integers

On the number line, a number to the right is greater.

• $5 > 3$
• $-2 > -5$ (because $-2$ is to the right of $-5$)
• $0 > -1$
• $-3 < 2$

This matters because students sometimes think $-5 > -2$ (wrong — the more negative, the smaller).

Rational numbers — fractions and integers together

A rational number is any number expressible as $\dfrac{a}{b}$ with integers $a, b$ and $b \neq 0$.

Examples

• $\dfrac{1}{2}$ is rational.
• $\dfrac{-3}{7}$ is rational.
• $\dfrac{5}{1} = 5$ — every integer is rational (denominator $1$).
• $0 = \dfrac{0}{1}$ is rational.
• $0.5 = \dfrac{1}{2}$ is rational.
• $0.\overline{3} = \dfrac{1}{3}$ is rational (recurring decimals are rational).

What's NOT rational?

• $\sqrt{2} \approx 1.414\ldots$ — irrational (cannot be expressed as a fraction).
• $\pi \approx 3.14159\ldots$ — irrational.
• $e \approx 2.71828\ldots$ — irrational.

Irrationals have non-terminating, non-repeating decimal expansions.

Ordering rationals

On the number line, rational numbers have specific positions.

• $\dfrac{1}{2} > \dfrac{1}{3}$ (because $\dfrac{1}{2} = 0.5$ and $\dfrac{1}{3} \approx 0.333$).
• $-\dfrac{2}{3} > -\dfrac{3}{4}$ (the more positive direction).
• $\dfrac{2}{5} > \dfrac{1}{3}$. Cross-multiply: $2 \times 3 = 6$ and $5 \times 1 = 5$. $6 > 5$, so $\dfrac{2}{5} > \dfrac{1}{3}$.

Review the comparison methods from the Fractions topic.

Identifying types

For any given number, identify which sets it belongs to.

Examples

• $7$: natural, whole, integer, rational, real.
• $0$: whole, integer, rational, real (not natural in some conventions).
• $-5$: integer, rational, real.
• $\dfrac{3}{4}$: rational, real.
• $\sqrt{2}$: real (NOT rational, NOT integer).
• $0.\overline{6}$: rational ($= \dfrac{2}{3}$), real.
• $\pi$: real (NOT rational).

Comparing and ordering — examples

Example 1

Arrange in ascending order: $-3, 5, -8, 2, 0, -1$.

Visualise on the number line. From left (smallest) to right (largest): $-8, -3, -1, 0, 2, 5$.

Example 2

Arrange in ascending order: $\dfrac{1}{2}, -\dfrac{3}{4}, \dfrac{2}{5}, -\dfrac{1}{3}$.

Convert to a common denominator or decimal. $\dfrac{1}{2} = 0.5$. $-\dfrac{3}{4} = -0.75$. $\dfrac{2}{5} = 0.4$. $-\dfrac{1}{3} \approx -0.333$.

Ascending: $-0.75, -0.333, 0.4, 0.5$. So $-\dfrac{3}{4} < -\dfrac{1}{3} < \dfrac{2}{5} < \dfrac{1}{2}$.

Example 3

Place each on the number line: $-2.5, 0, \dfrac{1}{4}, -\dfrac{1}{2}, 1.5, 3$.

Left to right: $-2.5, -\dfrac{1}{2}, 0, \dfrac{1}{4}, 1.5, 3$.

Special properties

Closure

A set is closed under an operation if the result is in the same set.

• Integers closed under $+, -, \times$. NOT closed under $\div$.
• Rationals closed under $+, -, \times, \div$ (except by $0$).

Additive identity

$0$ — adding $0$ to any number leaves it unchanged.

Multiplicative identity

$1$ — multiplying any number by $1$ leaves it unchanged.

Additive inverse

For any number $a$, the additive inverse is $-a$ (their sum is $0$).

Multiplicative inverse (reciprocal)

For any nonzero $a$, the multiplicative inverse is $\dfrac{1}{a}$ (their product is $1$).

• The reciprocal of $5$ is $\dfrac{1}{5}$.
• The reciprocal of $\dfrac{2}{3}$ is $\dfrac{3}{2}$.
• The reciprocal of $-7$ is $-\dfrac{1}{7}$.
• $0$ has no reciprocal.

Density of rationals

Between any two rational numbers, there are infinitely many other rationals.

Example: Between $\dfrac{1}{3}$ and $\dfrac{1}{2}$, you can find $\dfrac{2}{5}, \dfrac{3}{8}, \dfrac{5}{12}, \dfrac{7}{18}, \ldots$.

This is in stark contrast to the integers, which are discrete — no integer lies between $2$ and $3$.

WASSCE examination focus

Practice problems

1. Which sets does each number belong to (natural, whole, integer, rational, real)? (a) $7$ (b) $-4$ (c) $\dfrac{5}{8}$ (d) $0$ (e) $\sqrt{5}$

2. Arrange in ascending order: $-2, 4, 0, -8, 3, -1$.

3. Arrange in ascending order: $\dfrac{1}{2}, -\dfrac{2}{3}, \dfrac{3}{4}, -\dfrac{1}{3}, 0$.

4. Find the distance between $-7$ and $5$ on the number line.

5. Find $|-12|$, $|6|$, $|0|$.

6. Find the additive inverse of: (a) $7$, (b) $-3$, (c) $0$, (d) $\dfrac{1}{2}$.

7. Find the reciprocal of: (a) $5$, (b) $\dfrac{3}{4}$, (c) $-2$.

8. Is $0.\overline{6}$ rational or irrational? If rational, give the fraction.

9. Is $\sqrt{9}$ rational? If so, write it as a fraction.

10. Find three rational numbers between $\dfrac{1}{4}$ and $\dfrac{1}{2}$.

Worked solutions

1. (a) Natural, whole, integer, rational, real. (b) Integer, rational, real. (c) Rational, real. (d) Whole, integer, rational, real. (e) Real (not rational).

2. $-8, -2, -1, 0, 3, 4$.

3. Convert: $\dfrac{1}{2} = 0.5$, $-\dfrac{2}{3} \approx -0.667$, $\dfrac{3}{4} = 0.75$, $-\dfrac{1}{3} \approx -0.333$, $0$. Ascending: $-\dfrac{2}{3}, -\dfrac{1}{3}, 0, \dfrac{1}{2}, \dfrac{3}{4}$.

4. $|5 - (-7)| = 12$.

5. $12, 6, 0$.

6. (a) $-7$. (b) $3$. (c) $0$. (d) $-\dfrac{1}{2}$.

7. (a) $\dfrac{1}{5}$. (b) $\dfrac{4}{3}$. (c) $-\dfrac{1}{2}$.

8. Rational: $0.\overline{6} = \dfrac{2}{3}$.

9. $\sqrt{9} = 3 = \dfrac{3}{1}$, so yes.

10. Many possible. For example: $\dfrac{3}{10}, \dfrac{2}{5}, \dfrac{7}{20}$ all lie between $\dfrac{1}{4} = 0.25$ and $\dfrac{1}{2} = 0.5$.

What's next

In the next lesson we cover operations on integers and rational numbers: adding, subtracting, multiplying and dividing positive and negative numbers, and the order of operations. The sign rules need careful attention.