Real Numbers

God made the integers; all else is the work of man. — Leopold Kronecker

1.1 Introduction

Real numbers include all rational and irrational numbers on the number line. This chapter revisits prime factorisation, HCF and LCM, and proves that certain numbers are irrational.

1.2 Fundamental Theorem of Arithmetic

Every composite natural number greater than 1 can be expressed as a product of primes, and this factorisation is unique (apart from the order of factors).

HCF of two numbers = product of the smallest powers of common primes.

LCM of two numbers = product of the greatest powers of all primes appearing in either number.

For positive integers $a$ and $b$:

$\text{HCF}(a,b)\times\text{LCM}(a,b)=a\times b$

1.3 Revisiting Irrational Numbers

A number is irrational if it cannot be written as $\dfrac{p}{q}$ with integers $p,q$, $q\mathrel{\htmlClass{course-neq}{\char"2260}} 0$.

Proof by contradiction (idea): assume $\sqrt{2}=\dfrac{p}{q}$ in lowest terms; then $p^2=2q^2$, so $p$ is even; write $p=2k$ and deduce $q$ is even — contradiction.

Similarly, $\sqrt{3}$, $\sqrt{5}$ are irrational.

Chapter Summary

• Fundamental Theorem of Arithmetic: unique prime factorisation
• HCF × LCM = product of the two numbers
• Irrational numbers cannot be written as $\dfrac{p}{q}$
• $\sqrt{2}$, $\sqrt{3}$, $\sqrt{5}$ are irrational (proof by contradiction)

Exercises (NCERT)

• Exercise 1.1 — HCF, LCM, prime factorisation
• Exercise 1.2 — irrationality proofs
• Miscellaneous Exercise