Chapter 1 · Introduction

Chapter 1 — Real Numbers

Primes are the building blocks of every whole number. Split numbers into primes and you can find HCFs and LCMs quickly, and prove that numbers like $\sqrt2$ are not fractions.

Every time you pay with a UPI app, your phone and the bank exchange secret numbers. The lock on that exchange is built from two enormous prime numbers. Multiplying them is easy, but splitting the product back into its two primes would take the fastest computers longer than the age of the universe. Somewhere inside your phone, the simple idea you are about to study keeps your money safe.

That idea is the Fundamental Theorem of Arithmetic: every whole number bigger than $1$ is either a prime or can be built by multiplying primes, and in only one way. In this chapter we use it three times. First we learn to split numbers into primes and see why the answer is unique. Then we use the prime "fingerprints" of numbers to find HCF and LCM, which answer everyday questions about packing, tiling and timetables. Finally we use it to prove that numbers such as $\sqrt2$, $\sqrt3$ and $3+2\sqrt5$ can never be written as fractions.

Primes are the building blocks→Every number splits into primes in one way→HCF and LCM from the prime factors→Proof that $\sqrt2$, $\sqrt3$, $\sqrt5$ are irrational
🎯 By the end of this chapter you can…
  • tell primes from composite numbers and write any number as a product of primes, with a factor tree or a division ladder;
  • state the Fundamental Theorem of Arithmetic and use uniqueness to decide, for example, whether $6^n$ can end in $0$;
  • find the HCF and LCM of two or three numbers from their prime factorisations, and use $\operatorname{HCF}\times\operatorname{LCM}=a\times b$ for two numbers;
  • decide whether a word problem needs the HCF or the LCM, and solve it;
  • prove by contradiction that $\sqrt2$, $\sqrt3$, $\sqrt5$ and numbers such as $5\sqrt2$ and $3+2\sqrt5$ are irrational.
Teaser — Sifting Out the Primes up to 100

This method is over 2000 years old and is called the sieve of Eratosthenes. Circle the first number that is not crossed out (it is prime) and cross out all its other multiples. Repeat. Press Play: after crossing out the multiples of $2, 3, 5$ and $7$, everything left is prime. Why can we stop at $7$? Because $11\times11=121$ is already bigger than $100$.

sifting with—
crossed out so far—
primes found—

Sections in this chapter

1.1

The Fundamental Theorem of Arithmetic

Primes and composites, factor trees and division ladders, and why every number has exactly one prime factorisation.

$360=2^3\times3^2\times5$
Open section →1.2

HCF and LCM by Prime Factorisation

Smallest common powers for the HCF, greatest powers for the LCM, and word problems about packing and timetables.

$\operatorname{HCF}\times\operatorname{LCM}=a\times b$
Open section →1.3

Revisiting Irrational Numbers

Proof by contradiction that $\sqrt2$, $\sqrt3$, $\sqrt5$ and numbers built from them are irrational.

$\sqrt2\notin\mathbb{Q}$
Open section →
✅ Before you start
From earlier classes you need: factors and multiples and the divisibility rules for $2, 3, 5, 9$ and $10$ (Class 6); exponents such as $2^3=8$ and the law $a^m\times a^n=a^{m+n}$ (Class 7–8); and, from Class 9 Number Systems, the meaning of rational numbers (numbers $\tfrac pq$ with $p$, $q$ integers and $q\ne0$) and irrational numbers such as $\sqrt2$ and $\pi$. If HCF and LCM by listing factors feel rusty, try $\operatorname{HCF}(12,18)=6$ and $\operatorname{LCM}(4,6)=12$ by hand before starting Section 1.2.
← Welcome to Class 10 Mathematics1.1 The Fundamental Theorem of Arithmetic →