IMAT Mathematics · Numbers, Powers and Logarithms

Number Sets, Primes and Ordering

The real numbers are built in layers (natural, integer, rational, irrational); primes are the building blocks of whole numbers, and fractions and powers can be put in order without a calculator.

All Numbers, Powers and Logarithms notes

Why this matters

The 2019 paper asked for a highest common factor written as a product of prime powers, and the 2011 paper asked which expression is largest for a number between 0 and 1. The symbols ℝ and ∅ also appear in the answer options of recent inequality questions.

Concept 1 of 3: The number sets ℕ, ℤ, ℚ and ℝ

Each set fixes a problem of the one before it. You cannot subtract freely in the natural numbers, so the integers add negatives. You cannot always divide in the integers, so the rationals add fractions. Some lengths, like the diagonal of a unit square, are not fractions at all, and the irrationals fill those gaps to make the real number line.

Definition

Each set sits inside the next: $\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R}$ .

A rational number can be written $\frac{p}{q}$ with integers $p, q$ and $q \ne 0$ . Its decimal either ends or repeats.
An irrational number is real but not rational. Its decimal never ends and never repeats.
$\sqrt{n}$ for a whole number $n$ is irrational unless $n$ is a perfect square.
Rational with rational (add, subtract, multiply, divide by non-zero) stays rational. Rational plus irrational is always irrational. Two irrationals can give a rational: $\sqrt{2} \times \sqrt{8} = 4$ .

Set	Symbol	What it contains	Examples
Natural numbers	$\mathbb{N}$	The counting numbers (Italian books usually include 0)	0, 1, 2, 3, ...
Integers	$\mathbb{Z}$	Whole numbers: positive, negative and zero	$-4,\ 0,\ 7$
Rational numbers	$\mathbb{Q}$	Fractions of integers; decimals that end or repeat	$\frac{3}{8} = 0.375,\ 0.\overline{3} = \frac{1}{3},\ -5$
Irrational numbers	$\mathbb{R} \setminus \mathbb{Q}$	Decimals that never end and never repeat	$\sqrt{2},\ \pi,\ e,\ \sqrt{12}$
Real numbers	$\mathbb{R}$	Rationals and irrationals together: every point of the number line	All of the above

Each row's numbers also belong to every larger set: 7 is natural, integer, rational and real.

Practice this conceptself-check · 4 quick reps

22/7 is not π

\frac{22}{7}

is a fraction of two integers, so it is rational. It is only an approximation of

\pi

, which is irrational. An option offering

\frac{22}{7}

as the irrational number is wrong.

Concept 2 of 3: Prime factorisation, highest common factor and lowest common multiple

Every whole number above 1 breaks into primes in exactly one way, like a molecule into atoms. Once two numbers are written as products of primes, what they share (the HCF) and the smallest number they both divide into (the LCM) can be read straight off the powers.

Definition

A prime has exactly two divisors, 1 and itself. 1 is not prime; 2 is the only even prime.

Divisibility tests: by 2 if the last digit is even; by 3 if the digit sum is divisible by 3; by 4 if the last two digits form a multiple of 4; by 5 if it ends in 0 or 5; by 9 if the digit sum is divisible by 9.
HCF (highest common factor, also GCD): take the primes common to all the numbers, each to its lowest power.
LCM (lowest common multiple): take every prime that appears, each to its highest power.
For two numbers only, $\text{HCF} \times \text{LCM}$ equals their product.

HCF and LCM of two numbers

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

$a, b$ two positive whole numbers

Worked example

Find the HCF and the LCM of 84 and 90.

Practice this conceptself-check · 3 quick reps

HCF uses the lowest powers, LCM the highest

Swapping the rule turns the HCF into the LCM, and IMAT puts that number among the options. The HCF can never be larger than the smallest of the numbers; the LCM can never be smaller than the largest.

Concept 3 of 3: Fractions, recurring decimals and putting numbers in order

To compare fractions, give them a common denominator or turn them into decimals. Powers behave differently on each side of 1: multiplying by a number between 0 and 1 shrinks, so squaring such a number makes it smaller and taking its square root makes it bigger. Above 1 everything reverses.

Definition

Comparing fractions: for positive fractions, $\frac{a}{b} < \frac{c}{d}$ exactly when $ad < bc$ (cross-multiply).
Recurring decimal to fraction: if $x = 0.\overline{45}$ , then $100x = 45.\overline{45}$ ; subtract to get $99x = 45$ .
For $0 < x < 1$ : $x^2 < x < \sqrt{x} < 1$ , and the reciprocals are bigger than 1, with $\frac{1}{\sqrt{x}} < \frac{1}{x}$ .
For $x > 1$ the order reverses: $\frac{1}{x} < \frac{1}{\sqrt{x}} < 1 < \sqrt{x} < x < x^2$ .
Quick test in a question with letters: pick a value ( $x = \frac{1}{4}$ or $x = 4$ ) and compute every option.

Order for 0 < x < 1

0 < x^2 < x < \sqrt{x} < 1 < \frac{1}{\sqrt{x}} < \frac{1}{x}

Worked example

(a) Write

0.\overline{45}

as a fraction in lowest terms. (b) Put

\frac{3}{7}

, 0.43 and

\frac{4}{9}

in increasing order.

Practice this conceptself-check · 4 quick reps

Squaring does not always make a number bigger

For a number between 0 and 1, the square is smaller and the square root is larger:

0.5^2 = 0.25

but

\sqrt{0.25} = 0.5

. The intuition that powers grow is only true above 1.

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (2)

Prime factorisation, highest common factor and lowest common multiple
HCF and LCM of two numbers
$\text{HCF}(a,b) \times \text{LCM}(a,b) = a \times b$
Fractions, recurring decimals and putting numbers in order
Order for 0 < x < 1
$0 < x^2 < x < \sqrt{x} < 1 < \frac{1}{\sqrt{x}} < \frac{1}{x}$

Reference tables (1)

The number sets ℕ, ℤ, ℚ and ℝ5 rows

Set	Symbol	What it contains	Examples
Natural numbers	$\mathbb{N}$	The counting numbers (Italian books usually include 0)	0, 1, 2, 3, ...
Integers	$\mathbb{Z}$	Whole numbers: positive, negative and zero	$-4,\ 0,\ 7$
Rational numbers	$\mathbb{Q}$	Fractions of integers; decimals that end or repeat	$\frac{3}{8} = 0.375,\ 0.\overline{3} = \frac{1}{3},\ -5$
Irrational numbers	$\mathbb{R} \setminus \mathbb{Q}$	Decimals that never end and never repeat	$\sqrt{2},\ \pi,\ e,\ \sqrt{12}$
Real numbers	$\mathbb{R}$	Rationals and irrationals together: every point of the number line	All of the above

Each row's numbers also belong to every larger set: 7 is natural, integer, rational and real.

Watch out for (3)

22/7 is not π
→ The number sets ℕ, ℤ, ℚ and ℝ
HCF uses the lowest powers, LCM the highest
→ Prime factorisation, highest common factor and lowest common multiple
Squaring does not always make a number bigger
→ Fractions, recurring decimals and putting numbers in order

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