A rational number is any number that can be written as a fraction $\frac{a}{b}$, where $a$ and $b$ are integers and $b \neq 0$.

How to Identify Rational Numbers

A number is rational if it meets any of these conditions:

• It is an integer (like 5, -3, or 0)
• It is a terminating decimal (like 1.75)
• It is a repeating decimal (like 0.333...)
• It can be written as a simple fraction

Examples of Rational Numbers

Observation Activity

Think of cutting a pizza or a strip of paper into equal parts. If you take 3 parts out of 4 equal parts, you have $\frac{3}{4}$ of the whole. This visual shows how rational numbers represent parts of a whole.

An irrational number is a real number that cannot be written as a fraction $\frac{a}{b}$ where $a$ and $b$ are integers. Its decimal expansion never terminates and never repeats.

How to Identify Irrational Numbers

A number is irrational if:

• Its decimal goes on forever without any repeating pattern
• It cannot be expressed as a simple fraction
• It is the square root of a non-perfect square

Examples of Irrational Numbers

Important Note

If you try to calculate $\sqrt{2}$ on a calculator, it will show $1.414213562...$ but the digits continue forever with no pattern. This is the key characteristic of all irrational numbers.

The real number system includes all rational numbers and all irrational numbers together. Every point on a number line represents a real number.

$\text{Real Numbers} = \text{Rational Numbers} \cup \text{Irrational Numbers}$

Visual Representation on a Number Line

<---|---|---|---|---|---|---|---|---|--->
  -2  -1   0   1   2  √2  3   π   4
            ↑
        Rational    Irrational

• Integers like $-1, 0, 1, 2$ are rational
• Fractions like $\frac{1}{2}$ lie between integers
• $\sqrt{2} \approx 1.41$ and $\pi \approx 3.14$ are irrational numbers positioned between rational points

Quick Test: Is it Rational or Irrational?

Ask yourself two questions:

1. Can I write it as a fraction? If yes → Rational
2. If not, does its decimal terminate or repeat? If no → Irrational

Example 1: Classifying Numbers

Classify each number as rational or irrational:

a) $0.25$
b) $\sqrt{5}$
c) $\frac{7}{3}$
d) $0.454545...$

Solutions:

a) $0.25 = \frac{25}{100} = \frac{1}{4}$ → Rational (terminating decimal)

b) $\sqrt{5} = 2.2360679...$ → Irrational (non-perfect square root)

c) $\frac{7}{3} = 2.333...$ → Rational (repeating decimal)

d) $0.454545... = \frac{45}{99}$ → Rational (repeating decimal)

Example 2: Real-World Connection

A student measures a table and finds its length is $1.414$ meters. This value is an approximation of $\sqrt{2}$, which is irrational. The exact length cannot be written as a fraction — it continues forever!

• Rational numbers can be written as fractions; their decimals either terminate or repeat
• Irrational numbers cannot be written as fractions; their decimals never terminate and never repeat
• Real numbers are the combination of all rational and irrational numbers

In Tanzania, when a shopkeeper divides 1,000 TSH among 3 customers equally, each receives 333.33 TSH. The number $333.33...$ (repeating) is a rational number because it can be written as $\frac{1000}{3}$. However, when measuring land using a tape measure, measurements like the diagonal of a rectangular plot often involve $\sqrt{2}$ or similar irrational numbers, which cannot be expressed exactly as simple fractions.