Irrational numbers
🎯 In this topic you will
- Understand the difference between rational and irrational numbers
- Use knowledge of square numbers to estimate square roots
- Use knowledge of cube numbers to estimate cube roots
🧠 Key Words
- irrational number
- rational number
- surd
Show Definitions
- irrational number: A number that cannot be written as a simple fraction, with non-repeating and non-terminating decimals.
- rational number: A number that can be expressed as a fraction or ratio of two integers.
- surd: An irrational root that cannot be simplified to a whole number or a simple fraction (e.g., √2).
🔢 Integers and Rational Numbers
Integers are whole numbers. For example, 13, –26 and 100 004 are integers.
You can write rational numbers as fractions. For example, 93/4, –34/15 and 185/11 are rational numbers.
You can write any fraction as a decimal.
93/4 = 9.75 –34/15 = –3.26666666... 185/11 = 18.4545454...
🔎 Reasoning Tip
Rational numbers: The set of rational numbers includes all integers.
🔶 Irrational Numbers and Surds
The fraction either terminates (for example, 9.75) or it has recurring digits (for example, 3.26666666666… and 18.454545454…).
There are many square roots and cube roots that you cannot write as fractions. When you write these fractions as decimals, they do not terminate and there is no recurring pattern. For example, a calculator gives the answer $\sqrt{7} = 2.645751\ldots$
The calculator answer is not exact. The decimal does not terminate and there is no recurring pattern. Therefore, $\sqrt{7}$ is not a rational number.
Numbers that are not rational are called irrational numbers. $\sqrt{7}$, $\sqrt{23}$, $\sqrt[3]{10}$ and $\sqrt[3]{45}$ are irrational numbers.
Irrational numbers that are square roots or cube roots are called surds.
There are also numbers that are irrational but are not square roots or cube roots. One of these irrational numbers is called pi, which is the Greek letter π.
Your calculator will tell you that $\pi = 3.14159\ldots$ You will meet π later in the course.
🔎 Reasoning Tip
Square roots of negative numbers: These do not belong to the set of rational or irrational numbers. You’ll learn more about them if you continue mathematics at a higher level.
❓ EXERCISES
1. Write whether each of these numbers is an integer or an irrational number.
Explain how you know.
a) $\sqrt{9}$
b) $\sqrt{19}$
c) $\sqrt{39}$
d) $\sqrt{49}$
e) $\sqrt{99}$
👀 Show answer
b) Irrational → $\sqrt{19}$ is not a perfect square
c) Irrational → $\sqrt{39}$ is not a perfect square
d) Integer → $\sqrt{49} = 7$
e) Irrational → $\sqrt{99}$ is not a perfect square
2. a) Write the rational numbers in this list:
$\sqrt{1},\quad 7\frac{5}{12},\quad -38,\quad \sqrt{160},\quad -\sqrt{2.25},\quad -\sqrt{35}$
b) Write the irrational numbers in this list:
$0.3333\ldots,\quad -16,\quad \sqrt{200},\quad \sqrt{1.21},\quad \frac{23}{8},\quad \sqrt[3]{343}$
👀 Show answer
b) Irrational: $\sqrt{160}$, $-\sqrt{35}$, $\sqrt{200}$, $\sqrt{1.21}$
Note: $\sqrt[3]{343} = 7$ is rational, and $0.3333\ldots$, $-16$, $\frac{23}{8}$ are all rational
3. Write whether each of these numbers is an integer or a surd.
Explain how you know.
a) $\sqrt{100}$
b) $\sqrt[3]{100}$
c) $\sqrt{1000}$
d) $\sqrt[3]{1000}$
e) $\sqrt{10000}$
f) $\sqrt[3]{10000}$
👀 Show answer
b) Surd → $\sqrt[3]{100}$ is irrational
c) Surd → $\sqrt{1000}$ is irrational
d) Integer → $\sqrt[3]{1000} = 10$
e) Integer → $\sqrt{10000} = 100$
f) Surd → $\sqrt[3]{10000}$ is irrational
4. Is each of these numbers rational or irrational?
Give a reason for each answer.
a) $2 + \sqrt{2}$
b) $\sqrt{2} + 2$
c) $4 + \sqrt[3]{4}$
d) $\sqrt[3]{4} + 4$
👀 Show answer
a) $2 + \sqrt{2}$ → irrational
b) $\sqrt{2} + 2$ → irrational
c) $4 + \sqrt[3]{4}$ → irrational
d) $\sqrt[3]{4} + 4$ → irrational
5. Find:
a) Two irrational numbers that add up to 0
b) Two irrational numbers that add up to 2
👀 Show answer
b) $1 + \sqrt{2}$ and $1 - \sqrt{2}$
🧠 Think like a Mathematician
Question: What happens when you multiply square roots? Can you spot a rule or pattern?
Equipment: Calculator, pencil, paper
- a) Use a calculator to find:
- $\sqrt{8} \times \sqrt{2}$
- $\sqrt{3} \times \sqrt{12}$
- $\sqrt{20} \times \sqrt{5}$
- $\sqrt{2} \times \sqrt{18}$
- b) What do you notice about your answers?
- c) Find another multiplication similar to the ones in part a.
- d) Find similar multiplications using cube roots instead of square roots.
Follow-up Questions:
👀 show answer
- a.i:$\sqrt{8} \times \sqrt{2} = \sqrt{16} = 4$
- a.ii:$\sqrt{3} \times \sqrt{12} = \sqrt{36} = 6$
- a.iii:$\sqrt{20} \times \sqrt{5} = \sqrt{100} = 10$
- a.iv:$\sqrt{2} \times \sqrt{18} = \sqrt{36} = 6$
- Rule:$\sqrt{a} \times \sqrt{b} = \sqrt{ab}$
- Cube Root Version:$\sqrt[3]{2} \times \sqrt[3]{4} = \sqrt[3]{8} = 2$
❓ EXERCISES
6. Without using a calculator, show that:
a) $7 < \sqrt{55} < 8$
b) $4 < \sqrt[3]{100} < 5$
👀 Show answer
b) $4^3 = 64$ and $5^3 = 125$ → so $\sqrt[3]{100}$ is between 64 and 125 → $4 < \sqrt[3]{100} < 5$
7. Without using a calculator, find an irrational number between:
a) 4 and 5
b) 12 and 13
👀 Show answer
b) $\sqrt{150}$, $\sqrt{160}$ — both between 12 and 13
8. Without using a calculator, estimate:
a) $\sqrt{190}$ to the nearest integer
b) $\sqrt[3]{190}$ to the nearest integer
👀 Show answer
b) $\sqrt[3]{190}$ is between $\sqrt[3]{125} = 5$ and $\sqrt[3]{216} = 6$ → estimate: **6**
9. a) Use a calculator to find:
i) $(\sqrt{2} + 1)(\sqrt{2} - 1)$
ii) $(\sqrt{3} + 1)(\sqrt{3} - 1)$
iii) $(\sqrt{4} + 1)(\sqrt{4} - 1)$
b) Continue the pattern of the multiplications in part a.
c) Generalise the results to find $(\sqrt{N} + 1)(\sqrt{N} - 1)$ where $N$ is a positive integer.
d) Check your generalisation with further examples.
👀 Show answer
ii) $(\sqrt{3} + 1)(\sqrt{3} - 1) = 3 - 1 = 2$
iii) $(\sqrt{4} + 1)(\sqrt{4} - 1) = 4 - 1 = 3$
b) The pattern is: result = $N - 1$
c) Generalised form: $(\sqrt{N} + 1)(\sqrt{N} - 1) = N - 1$
This follows the identity: $(a + b)(a - b) = a^2 - b^2$
Here, $a = \sqrt{N}$ and $b = 1$
So: $(\sqrt{N})^2 - 1^2 = N - 1$
d) Example: $(\sqrt{10} + 1)(\sqrt{10} - 1) = 10 - 1 = 9$ ✅
Example: $(\sqrt{7} + 1)(\sqrt{7} - 1) = 7 - 1 = 6$ ✅
10. Here is a decimal: 5.020 020 002 000 020 000 020 000 002…
Arun says:
“There is a regular pattern: one zero, then two zeros, then three zeros, and so on. This is a rational number.”
Is Arun correct? Give a reason for your answer.
👀 Show answer
Even though the decimal has a pattern, it does not repeat in a fixed, finite cycle — the number of zeros increases each time.
This means the decimal is **non-terminating and non-repeating**, so it is **irrational**.
⚠️ Be careful!
Not all square roots are irrational. For example, $\sqrt{9} = 3$ is a rational number. A square root is only irrational if it can’t be written as a fraction or a whole number.