AFCAT Numerical Ability · Number System

Types of Numbers: Sets, Rationals, Primes, Squares and Cubes

Natural, whole, integer, rational, irrational and real numbers sit inside one another; primes have exactly two factors; perfect squares and cubes are n × n and n × n × n.

All Number System notes

Why this matters

Two thirds of this page is EASY: pick the number of a given kind from four. The marks go on the borderline cases, such as zero, the number 1, 22/7 and the root of a perfect square, so learn where each set begins and ends.

Concept 1 of 4: Natural, whole, integer, rational, irrational and real numbers

Each set is the one before it plus something new. Whole numbers add 0 to the counting numbers, integers add the negatives, rationals add the fractions, and the reals add the numbers no fraction can reach.

Definition

The chain is $\mathbb{N} \subset \mathbb{W} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R}$ : every natural number is whole, every whole number is an integer, every integer is rational, and every rational is real.
Every integer is rational, because $n = \tfrac n1$ . So $-9$ is an integer and a rational number at once.
Zero is whole, an integer, rational and real, and it is even. It is not a natural number, and it is neither prime nor composite.
The irrationals are the reals that are not rational. No irrational number is an integer.

Set	What it holds	In it	Not in it
Natural $\mathbb{N}$	The counting numbers 1, 2, 3, ...	$1,\ 15,\ 200$	$0,\ -4,\ 2.5$
Whole $\mathbb{W}$	The natural numbers and 0	$0,\ 9,\ 41$	$-1,\ \tfrac 34$
Integers $\mathbb{Z}$	The whole numbers and their negatives	$-12,\ 0,\ 8$	$\tfrac 12,\ 0.4$ A negative number is an integer but never a whole or natural number.
Rational $\mathbb{Q}$	$\tfrac pq$ with integers $p, q$ and $q \neq 0$	$\tfrac 34,\ -5,\ 0.25,\ 0.\overline{4}$	$\sqrt 3,\ \sqrt{11}$
Irrational	Decimals that never end and never repeat	$\sqrt 3,\ \sqrt{11},\ 2\sqrt 6$	$\sqrt{25},\ 0.\overline{4}$
Real $\mathbb{R}$	The rationals and irrationals together	Every point on the number line	The square root of a negative number

Practice this conceptself-check · 4 quick reps

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Example 1 · Number System · Types of Numbers

Which is both an integer and a rational number?

Zero is whole, not natural

The natural numbers start at 1. Zero belongs to every other set in the chain, so 'zero is a natural number' is the false statement in an option list.

The chain runs one way

Every natural number is an integer, but not every integer is natural:

-3

is an integer and not natural. A statement that reverses the chain is false.

Drill 7 more on natural, whole, integer, rational, irrational and real numbers

Concept 2 of 4: Rational or irrational, and terminating or repeating

Written as a decimal, a rational number either stops or falls into a repeating block. An irrational number does neither. A fraction stops exactly when its denominator, in lowest terms, is built from 2s and 5s alone, because only then can it be scaled to a power of 10.

Definition

Rational: $\tfrac pq$ with $q \neq 0$ . Its decimal terminates ( $\tfrac{7}{20} = 0.35$ ) or repeats ( $\tfrac{5}{11} = 0.\overline{45}$ ).
In lowest terms, $\tfrac pq$ terminates when $q$ has no prime factor but 2 and 5, and repeats otherwise ( $\tfrac{5}{12} = 0.41\overline{6}$ ).
$\sqrt n$ is rational only when $n$ is a perfect square: $\sqrt{36} = 6$ , but $\sqrt{50} = 5\sqrt 2$ is irrational.
$\pi$ is irrational. $\tfrac{22}{7}$ and 3.14 are rational numbers close to it.
A repeating decimal is rational: $0.\overline{7} = \tfrac 79$ .

Terminating test

\dfrac pq \text{ in lowest terms terminates} \;\text{means}\; q = 2^a \times 5^b

Worked example

Which of

\tfrac{7}{20}

\tfrac{5}{12}

\tfrac{9}{40}

and

\tfrac{4}{15}

have terminating decimals?

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionEasy

Example 2 · Number System · Types of Numbers

Which of the following is an irrational number?

Simplify before you test

\tfrac{6}{15}

looks like a repeater because 15 has a factor 3, but it is

\tfrac 25 = 0.4

. Reduce to lowest terms first.

22/7 is not pi

\tfrac{22}{7}

is a fraction, so it is rational; it only approximates

\pi

. An option list asking for the irrational number wants

\pi

, not

\tfrac{22}{7}

A long decimal can still repeat

0.142857142857\ldots

looks endless, but the block 142857 repeats, so it is

\tfrac 17

and rational.

Drill 5 more on rational or irrational, and terminating or repeating

Concept 3 of 4: Primes, composites, even and odd numbers

A prime has exactly two factors, 1 and itself. To test a number, you need only try the primes up to its square root: if it had a larger factor, the matching factor would be smaller and you would have met it already.

Definition

Prime: exactly two factors. Composite: more than two. 1 has one factor, so it is neither.
2 is the only even prime. Every even number above 2 is composite.
Even numbers end in 0, 2, 4, 6 or 8; odd numbers end in 1, 3, 5, 7 or 9. An odd number need not be prime.
The primes below 50: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47.
To test $n$ , divide by 2, 3, 5, 7, 11, ... up to $\sqrt n$ . For any $n$ below 121 the primes up to 7 are enough. (Quick tests for 3, 7 and 11 come on the next page.)

Prime test

n > 1 \text{ is prime when no prime } p \le \sqrt n \text{ divides it}

Worked example

Which of 111, 113, 117 and 119 is prime?

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionEasy

Example 3 · Number System · Types of Numbers

Which of the following numbers is prime?

1 is not prime

A list that includes 1 is not a list of primes. A prime needs exactly two factors and 1 has only one.

Odd is not prime

51 = 3 \times 17

and

57 = 3 \times 19

are odd and composite. Test each candidate; do not stop at 'it is odd'.

Drill 5 more on primes, composites, even and odd numbers

Concept 4 of 4: Perfect squares and perfect cubes

Know the squares up to

20^2

and the cubes up to

10^3

. A number that falls strictly between two neighbouring squares is not a square, and the same goes for cubes, so one comparison settles each option.

Definition

A perfect square is $n^2$ ; the squares from $11^2$ to $20^2$ are 121, 144, 169, 196, 225, 256, 289, 324, 361 and 400.
A perfect cube is $n^3$ : $6^3 = 216$ , $8^3 = 512$ , $9^3 = 729$ and so on.
A perfect square never ends in 2, 3, 7 or 8, and it ends in an even number of zeros.
A number that is both a square and a cube is a sixth power: 1, then 729 ( $27^2$ and $9^3$ ), then 4096.

Bracketing test

n^3 < N < (n+1)^3 \;\Rightarrow\; N \text{ is not a perfect cube}

Worked example

Which of 289, 324, 350 and 361 is not a perfect square?

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionEasy

Example 4 · Number System · Types of Numbers

Which is NOT a perfect cube?

A cube is not automatically a square

27 = 3^3

is a cube, but it lies between

5^2 = 25

and

6^2 = 36

, so it is not a square. Being both needs a sixth power.

Count the zeros

4,000 ends in three zeros, an odd number, so it cannot be a perfect square. 40,000 ends in four and is

200^2

Drill 3 more on perfect squares and perfect cubes

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 (3)

Rational or irrational, and terminating or repeating
Terminating test
$\dfrac pq \text{ in lowest terms terminates} \;\text{means}\; q = 2^a \times 5^b$
Primes, composites, even and odd numbers
Prime test
$n > 1 \text{ is prime when no prime } p \le \sqrt n \text{ divides it}$
Perfect squares and perfect cubes
Bracketing test
$n^3 < N < (n+1)^3 \;\Rightarrow\; N \text{ is not a perfect cube}$

Reference tables (1)

Natural, whole, integer, rational, irrational and real numbers6 rows

Set	What it holds	In it	Not in it
Natural $\mathbb{N}$	The counting numbers 1, 2, 3, ...	$1,\ 15,\ 200$	$0,\ -4,\ 2.5$
Whole $\mathbb{W}$	The natural numbers and 0	$0,\ 9,\ 41$	$-1,\ \tfrac 34$
Integers $\mathbb{Z}$	The whole numbers and their negatives	$-12,\ 0,\ 8$	$\tfrac 12,\ 0.4$ A negative number is an integer but never a whole or natural number.
Rational $\mathbb{Q}$	$\tfrac pq$ with integers $p, q$ and $q \neq 0$	$\tfrac 34,\ -5,\ 0.25,\ 0.\overline{4}$	$\sqrt 3,\ \sqrt{11}$
Irrational	Decimals that never end and never repeat	$\sqrt 3,\ \sqrt{11},\ 2\sqrt 6$	$\sqrt{25},\ 0.\overline{4}$
Real $\mathbb{R}$	The rationals and irrationals together	Every point on the number line	The square root of a negative number

Watch out for (9)

Zero is whole, not natural
→ Natural, whole, integer, rational, irrational and real numbers
The chain runs one way
→ Natural, whole, integer, rational, irrational and real numbers
Simplify before you test
→ Rational or irrational, and terminating or repeating
22/7 is not pi
→ Rational or irrational, and terminating or repeating
A long decimal can still repeat
→ Rational or irrational, and terminating or repeating
1 is not prime
→ Primes, composites, even and odd numbers
Odd is not prime
→ Primes, composites, even and odd numbers
A cube is not automatically a square
→ Perfect squares and perfect cubes
Count the zeros
→ Perfect squares and perfect cubes

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