Real Numbers Cheat Sheet

Formulae

HCF and LCM of two numbers

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

HCF = highest common factor, LCM = lowest common multiple, for two positive integers a and b.

HCF by prime factorisation

\text{HCF} = \text{product of the smallest power of each common prime factor}

Take only the prime factors common to all the numbers, each to its lowest power.

LCM by prime factorisation

\text{LCM} = \text{product of the greatest power of every prime factor}

Take every prime factor that appears, each to its highest power.

Terminating decimal condition

\frac{p}{q}\ \text{terminates} \iff q = 2^{n}\,5^{m}

Here p/q is in lowest terms; n and m are non-negative integers. Otherwise the decimal is non-terminating repeating.

Theorems

Fundamental Theorem of Arithmetic

Every composite number can be expressed as a product of primes, and this factorisation is unique, apart from the order in which the prime factors occur.

N = p_{1}p_{2}\cdots p_{n}

p₁, p₂, … are primes, not necessarily distinct. This factorisation is the prime factorisation of N.

Divisibility of a square by a prime

Let p be a prime number. If p divides a², then p divides a, where a is a positive integer.

p \mid a^{2} \Rightarrow p \mid a

The symbol | means “divides”. This result is used to prove that √p is irrational for a prime p.

Irrationality of √2

√2 is irrational.

\sqrt{2} \notin \mathbb{Q}

Proved by contradiction: if √2 = a/b with a, b co-prime, then 2 divides both a and b, a contradiction.

Irrationality of √3 and √5

√3 and √5 are irrational numbers.

The same contradiction method used for √2 works for √p, where p is any prime.

Definitions

Natural numbers: The counting numbers 1, 2, 3, … and so on.
Whole numbers: The natural numbers together with 0: 0, 1, 2, 3, …
Integers: Whole numbers together with their negatives: …, −2, −1, 0, 1, 2, …
Rational number: A number that can be written as p/q, where p and q are integers and q ≠ 0.
Irrational number: A number that cannot be written as p/q with integers p and q (q ≠ 0); e.g. √2, √3, π.
Real numbers: The collection of all rational and irrational numbers together.
Prime number: A natural number greater than 1 whose only factors are 1 and itself; e.g. 2, 3, 5, 7.
Composite number: A natural number greater than 1 that has factors other than 1 and itself; e.g. 4, 6, 9.
Co-prime numbers: Two numbers whose HCF (greatest common factor) is 1; e.g. 4 and 9.
HCF: The highest common factor — the largest number that divides each of the given numbers exactly.
LCM: The lowest common multiple — the smallest number that is a multiple of each of the given numbers.
Prime factorisation: Writing a number as a product of its prime factors; e.g. 72 = 2³ × 3².

Key Points

Every composite number has one, and only one, prime factorisation (ignoring the order of factors).
HCF is the product of the smallest power of each common prime factor; LCM is the product of the greatest power of every prime factor.
For any two positive integers, HCF × LCM = product of the numbers. This relation does not hold for three or more numbers.
A rational number p/q (in lowest terms) has a terminating decimal expansion only when q is of the form 2ⁿ5ᵐ; otherwise the expansion is non-terminating and repeating.
Numbers such as √2, √3, √5, √7, 2 + √3 and 3 − √5 are irrational; their decimal expansions are non-terminating non-repeating.
Operations: rational ± irrational = irrational; a non-zero rational × irrational = irrational; irrational ÷ non-zero irrational need not be irrational.
Divisibility tests help find factors: a number is divisible by 2 if its last digit is even, by 3 if the digit sum is divisible by 3, and by 5 if it ends in 0 or 5.