Chapter 1: Real Numbers

You already know how to count, add, multiply, and find common factors. This chapter zooms in and asks a sharper question: what is the deep structure of the integers, and how does that structure spill over into the rest of the real number system? The headline result is the Fundamental Theorem of Arithmetic , every integer greater than $1$ is a product of primes, in essentially one way. From this single fact flow many powerful tools.

For the board exam this is a short but high-yield chapter. Expect crisp $1$ - and $2$ -mark questions on HCF and LCM via prime factorisation, $3$ -mark applications to word problems (synchronised lights, racing tracks, packing boxes), and a $3$ - to $5$ -mark proof that some surd like $\sqrt{2}$ , $\sqrt{3}$ , $\sqrt{5}$ or $3 + 2\sqrt{5}$ is irrational. Memorising the steps is easy, but examiners reward students who understand why the proof works.

The real-world hook is everywhere. Cryptography on the phone in your pocket rests on the difficulty of prime factorisation. Music and gears use LCM-style synchronisation. The very definition of length, hidden inside the Pythagorean theorem, forced the ancient Greeks to admit irrational numbers exist. So although this chapter looks short, it is the chapter where you first see number theory breathe.

We will keep the toolkit small and the reasoning tight: prime factorisation, the relation $\operatorname{HCF}(a,b) \cdot \operatorname{LCM}(a,b) = a \cdot b$ , and proof by contradiction. By the end you should be comfortable proving that a specific number is irrational and that two specific irrational numbers add to an irrational result , without copying a template.

What's inside

Fundamental Theorem of Arithmetic , every integer $> 1$ has a unique prime factorisation.
HCF and LCM by prime factorisation , a clean alternative to long division.
The product relation $\operatorname{HCF} \cdot \operatorname{LCM} = a \cdot b$ , and why it fails for three numbers.
Revisiting irrational numbers , $\sqrt{p}$ for prime $p$ is irrational.
Operations with rationals and irrationals , sums, products, and traps.

Key results / Formula card

Result	Statement
Fundamental Theorem	Every $n \ge 2$ factors uniquely as $p_1^{a_1} p_2^{a_2} \cdots p_k^{a_k}$ (primes ascending).
HCF (prime fact.)	Take the smallest power of each common prime.
LCM (prime fact.)	Take the largest power of each prime that appears.
Product rule	$\operatorname{HCF}(a,b) \cdot \operatorname{LCM}(a,b) = a \cdot b$ (two numbers only).
Irrationality lemma	If $p$ is prime and $p \mid a^2$ , then $p \mid a$ .
Standard surds	$\sqrt{2},\ \sqrt{3},\ \sqrt{5},\ \sqrt{p}$ for any prime $p$ , are irrational.
Closure	(rational) $\pm$ (irrational) = irrational; (non-zero rational) $\times$ (irrational) = irrational.

Keep this card visible , almost every question in the chapter is solved by one or two of these lines.

Chapter 1: Real Numbers

What's inside

Key results / Formula card

Sub-topics

Practice quiz