Introduction and Revisiting the Number System

Introduction — Why Study Real Numbers?

Let's start with a simple idea. Every number you have ever used — while counting marbles, measuring your height, sharing a pizza, or checking a temperature below zero — belongs to one big family called the real numbers.

In earlier classes you met natural numbers, whole numbers, integers, and rational numbers, and you got a glimpse of irrational numbers like $\sqrt{2}$ and $\pi$. In this chapter we tie all of these together and look more deeply at two powerful ideas:

1. The Fundamental Theorem of Arithmetic, which says every composite number is built from primes in exactly one way — like a unique fingerprint.
2. The behaviour of irrational numbers and the decimal expansions of rational numbers.

Key Point: Real numbers = all rational numbers + all irrational numbers. Every point on the number line is a real number, and every real number sits at exactly one point on the line.

Think of it this way: the number line has no gaps. Wherever you place your pencil tip, you land on a real number.

The Number System — A Quick Recap

Before we go deeper, let's quickly revisit the families of numbers you already know. Each family sits inside the next bigger one.

Natural Numbers (N)

The counting numbers: $1, 2, 3, 4, \dots$ — they start at 1 and never end.

Whole Numbers (W)

Natural numbers together with 0: $0, 1, 2, 3, \dots$

Integers (Z)

Whole numbers together with the negatives: $\dots, -3, -2, -1, 0, 1, 2, 3, \dots$

Rational Numbers (Q)

Any number that can be written as $\dfrac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$. For example $\dfrac{3}{4}$, $-\dfrac{7}{2}$, $5$ (since $5 = \dfrac{5}{1}$), and $0.75$.

Irrational Numbers

Numbers that cannot be written as $\dfrac{p}{q}$. Their decimal expansion is non-terminating and non-repeating. Examples: $\sqrt{2} = 1.41421356\dots$, $\sqrt{3}$, and $\pi$.

Key Point: The chain of inclusion is $N \subset W \subset Z \subset Q \subset R$. Every natural number is a real number, but not every real number is a natural number.

Putting It All Together — The Real Numbers

When we combine all rational numbers and all irrational numbers, we get the set of real numbers, denoted by $R$.

$$R = \{\text{rational numbers}\} \cup \{\text{irrational numbers}\}$$

A neat way to picture the whole system:

Set	Symbol	Includes
Natural numbers	$N$	$1, 2, 3, \dots$
Whole numbers	$W$	$0, 1, 2, 3, \dots$
Integers	$Z$	$\dots, -2, -1, 0, 1, 2, \dots$
Rational numbers	$Q$	fractions $\frac{p}{q}$, $q \neq 0$
Irrational numbers	—	$\sqrt{2}, \pi, \dots$ (non-terminating, non-repeating)
Real numbers	$R$	everything above combined

[Board Important] A common one-mark question asks you to classify a number as rational or irrational. The trick: if it can be written as a ratio of two integers (or has a terminating / repeating decimal), it is rational; otherwise it is irrational.

Common Mistake: $\sqrt{4}$ is not irrational. Since $\sqrt{4} = 2$, it is a natural number. Only the square roots of numbers that are not perfect squares are irrational.

A Glimpse of What's Ahead

This chapter explores real numbers through a few connected ideas. Here's a roadmap so you know where we are heading:

1. Euclid's Division Lemma and Algorithm

A clean, reliable method to find the HCF (Highest Common Factor) of two numbers using repeated division. (Section 2)

2. The Fundamental Theorem of Arithmetic

Every composite number breaks into primes in exactly one way. We'll use this to find HCF and LCM by prime factorisation. (Section 3)

3. Irrational Numbers

We will prove, not just claim, that numbers like $\sqrt{2}$ and $\sqrt{3}$ are irrational. (Section 4)

4. Decimal Expansions

We'll discover a quick rule to tell — without dividing — whether a fraction gives a terminating or a non-terminating repeating decimal. (Section 5)

Exam Tip: Real Numbers is a high-scoring chapter. The proofs of irrationality and the HCF–LCM relation are asked almost every year. Master the standard steps and you secure easy marks.

Solved Examples

Example 1: Classifying numbers

Classify each of the following as rational or irrational: $\dfrac{2}{7}$, $\sqrt{9}$, $\sqrt{5}$, $0.333\dots$

Solution:

1. $\dfrac{2}{7}$ is already in the form $\dfrac{p}{q}$ with integers $p=2$, $q=7$ → rational.
2. $\sqrt{9} = 3 = \dfrac{3}{1}$ → rational (9 is a perfect square).
3. $\sqrt{5} = 2.2360679\dots$ is non-terminating, non-repeating → irrational.
4. $0.333\dots = \dfrac{1}{3}$, a repeating decimal → rational.

Takeaway: A square root is irrational only when the number under the root is not a perfect square.

Example 2: Is the number a whole number?

State whether $-7$, $0$, $\dfrac{5}{1}$ and $\dfrac{3}{2}$ are whole numbers.

Solution:

• $-7$ is an integer but not a whole number (whole numbers are $0, 1, 2, \dots$ — no negatives).
• $0$ is a whole number.
• $\dfrac{5}{1} = 5$ is a whole number.
• $\dfrac{3}{2} = 1.5$ is rational but not a whole number.

Takeaway: Always simplify first. $\dfrac{5}{1}$ hides a whole number; $\dfrac{3}{2}$ does not.

Example 3: Rational number between two numbers

Find a rational number between $\dfrac{1}{4}$ and $\dfrac{1}{2}$.

Solution:

1. A safe method is to take the average of the two numbers.
2. Average $= \dfrac{1}{2}\left(\dfrac{1}{4} + \dfrac{1}{2}\right) = \dfrac{1}{2}\left(\dfrac{1}{4} + \dfrac{2}{4}\right) = \dfrac{1}{2} \times \dfrac{3}{4} = \dfrac{3}{8}$.
3. Check: $\dfrac{1}{4} = 0.25$, $\dfrac{3}{8} = 0.375$, $\dfrac{1}{2} = 0.5$. Indeed $0.25 < 0.375 < 0.5$.

Final Answer: $\dfrac{3}{8}$ lies between $\dfrac{1}{4}$ and $\dfrac{1}{2}$.

Takeaway: Between any two distinct rational numbers there are infinitely many rationals — the average trick always finds one.

Example 4: Identify the smallest set

To which smallest set of numbers does $\sqrt{16}$ belong: natural, whole, integer, or rational?

Solution:

1. Simplify: $\sqrt{16} = 4$.
2. $4$ is a counting number, so it already belongs to the natural numbers $N$ — the smallest set in our chain.

Final Answer: Natural numbers.

Takeaway: Simplify the expression before deciding which set it belongs to — never judge by appearance.

Example 5: Sum of a rational and an irrational number

Is $3 + \sqrt{2}$ rational or irrational?

Solution:

1. Suppose, for contradiction, that $3 + \sqrt{2}$ is rational, say equal to $r$.
2. Then $\sqrt{2} = r - 3$. The right side is a difference of two rationals, hence rational.
3. But $\sqrt{2}$ is known to be irrational — a contradiction.
4. So our assumption is wrong.

Final Answer: $3 + \sqrt{2}$ is irrational.

Takeaway: (rational) + (irrational) is always irrational. This idea returns in Section 4's proofs.

Example 6: Decimal form tells the type

Without a calculator, decide whether $0.101001000100001\dots$ (the gaps between 1's keep growing) is rational or irrational.

Solution:

1. A rational number's decimal either terminates or repeats a fixed block forever.
2. Here the pattern of zeros keeps changing (1 zero, then 2, then 3, …), so no block repeats.
3. The decimal is non-terminating and non-repeating.

Final Answer: Irrational.

Takeaway: 'There is a pattern' is not the same as 'a fixed block repeats'. Only a repeating block makes a decimal rational.

Example 7: Product that turns out rational

Find the product $\sqrt{8} \times \sqrt{2}$ and state whether it is rational.

Solution:

1. Use $\sqrt{a} \times \sqrt{b} = \sqrt{ab}$.
2. $\sqrt{8} \times \sqrt{2} = \sqrt{16} = 4$.
3. $4$ is an integer, hence rational.

Final Answer: The product is $4$, which is rational.

Takeaway: The product of two irrationals can be rational. Don't assume irrational × irrational stays irrational.

Example 8: Two irrational numbers between two integers

Write any two irrational numbers between $2$ and $3$.

Solution:

1. We need non-terminating, non-repeating decimals lying between 2 and 3.
2. $\sqrt{5} \approx 2.236\dots$ lies between 2 and 3 (since $4 < 5 < 9$).
3. $\sqrt{7} \approx 2.645\dots$ also lies between 2 and 3.

Final Answer: $\sqrt{5}$ and $\sqrt{7}$ (other valid answers exist, e.g. $2.1010010001\dots$).

Takeaway: Square roots of non-perfect-squares between $n^2$ and $(n+1)^2$ are handy irrationals to quote.

Example 9: Converting a repeating decimal to a fraction

Express $0.\overline{4}$ (i.e. $0.444\dots$) as a rational number in the form $\dfrac{p}{q}$.

Solution:

1. Let $x = 0.444\dots$
2. Multiply both sides by 10: $10x = 4.444\dots$
3. Subtract the first equation from the second: $10x - x = 4.444\dots - 0.444\dots$, so $9x = 4$.
4. Therefore $x = \dfrac{4}{9}$.

Final Answer: $0.\overline{4} = \dfrac{4}{9}$.

Takeaway: Every repeating decimal is rational. The trick is to multiply by a power of 10 that lines up the repeating block, then subtract.

Example 10: Spotting the odd one out

Three of these are rational and one is irrational. Identify the irrational one: $\dfrac{22}{7}$, $\sqrt{2}$, $1.75$, $0.\overline{27}$.

Solution:

1. $\dfrac{22}{7}$ is a fraction of integers → rational (note: it is an approximation of $\pi$, but the fraction itself is rational).
2. $1.75 = \dfrac{7}{4}$ → terminating → rational.
3. $0.\overline{27} = \dfrac{27}{99} = \dfrac{3}{11}$ → repeating → rational.
4. $\sqrt{2} = 1.41421\dots$ → non-terminating, non-repeating → irrational.

Final Answer: $\sqrt{2}$ is the irrational number.

Takeaway: Don't confuse $\dfrac{22}{7}$ with $\pi$. $\dfrac{22}{7}$ is rational; $\pi$ is irrational.