1.1 Types of Numbers

Quick Notes & Formulae

Essential concepts for solving types of numbers problems.

Natural Numbers

N

: 1, 2, 3, 4, ... (positive counting numbers, excluding 0)

Integers

Z

: ..., -2, -1, 0, 1, 2, ... (whole numbers including negatives and zero)

Prime Numbers

Numbers with exactly two distinct factors: 1 and itself. 2 is the only even prime.

Square Numbers

1, 4, 9, 16, 25, . . .

formed by

n^{2}

. Also called perfect squares.

Cube Numbers

1, 8, 27, 64, 125, . . .

formed by

n^{3}

Rational Numbers

Q

: Can be written as

\frac{a}{b}

where

a, b

are integers,

b \neq 0

Irrational Numbers

Cannot be written as a fraction. Non-terminating, non-recurring decimals like

\sqrt{2}, π

Triangle Numbers

1, 3, 6, 10, 15, . . .

where

T_{n} = \frac{n (n + 1)}{2}

. Can form equilateral triangles.

Past Paper Style Questions

Question 1: Number Classification

Consider the following list of numbers:

- 8, - 3, 0, \frac{2}{5}, 1, 4, \sqrt{7}, 9, 16, π

(a) Write down all the natural numbers from the list. [1]

(b) Write down all the integers from the list. [2]

(d) State which numbers are irrational. [2]

(e) Explain why

\frac{2}{5}

is a rational number. [1]

Question 2: Prime Numbers

Consider the numbers:

17, 21, 29, 33, 37, 49, 51, 59

(a) Write down all the prime numbers from the list. [2]

(b) Show that 51 is not a prime number. [2]

(d) Explain why 1 is not considered a prime number. [1]

(e) Write down the next prime number after 59. [1]

Question 3: Square and Cube Numbers

(a) Write down the first eight square numbers. [2]

(b) Write down the first six cube numbers. [2]

(d) Explain why your answer to part (c) is both a square and a cube number. [2]

(e) Calculate

\sqrt{144} + \sqrt[3]{64}

. [2]

Question 4: Rational and Irrational Numbers

Consider:

\sqrt{9}, \sqrt{10}, 0. \dot{3}, \frac{5}{8}, \sqrt[3]{27}, \sqrt[3]{30}, 0.1010010001 . . .

(a) Write down all the rational numbers from the list. [2]

(b) Write down all the irrational numbers from the list. [2]

0. \dot{3} = \frac{1}{3}

. [2]

(d) Convert

\frac{5}{8}

to a decimal. [1]

(e) Explain why

0.1010010001 . . .

is irrational. [1]

Question 5: Triangle Numbers

The nth triangle number is given by the formula:

T_{n} = \frac{n (n + 1)}{2}

(a) Calculate the 5th and 6th triangle numbers. [2]

(b) Verify that 21 is a triangle number. [2]

(d) Show that the sum of two consecutive triangle numbers is a square number. [2]

(e) Explain the name "triangle number" with a diagram or description. [1]

Question 6: Factors and Multiples

(a) Write down all the factors of 48. [2]

(b) Write down the first six multiples of 8. [1]

(d) Find the Lowest Common Multiple (LCM) of 12 and 18. [2]

(e) A number

N

has exactly 4 factors and is less than 20. List all possible values of

N

. [3]

Question 7: Real Number Classification

Given the set:

A = {- 5, - \frac{3}{4}, 0, \sqrt{4}, \sqrt{5}, 2.75, π, 10}

(a) List the elements of set

A

that are integers. [2]

(b) List the elements of set

A

that are natural numbers. [1]

A

that are irrational. [2]

(d) Express

2.75

as a fraction in its simplest form. [2]

(e) Place the following numbers in ascending order:

- \frac{3}{4}, 0, \sqrt{5}, 2.75

. [2]

Question 8: Number Properties

(a) Find the smallest number that is both a square number and a multiple of 6. [2]

(b) Explain why the product of two consecutive integers is always even. [2]

(d) Find the value of

k

2^{k} = 64

. [1]

(e) Write

72

as a product of its prime factors in index form. [2]

Question 9: Surds and Exact Values

(a) Simplify

\sqrt{50}

in the form

a \sqrt{b}

where

a

and

b

are integers and

b

is as small as possible. [2]

(b) Calculate the exact value of

(\sqrt{3})^{4}

. [2]

\sqrt{8} + \sqrt{18} = 5 \sqrt{2}

. [3]

(d) A square has an area of

72 {cm}^{2}

. Find the exact length of one side. [2]

(e) Rationalize the denominator of

\frac{3}{\sqrt{2}}

. [2]

Question 10: Mixed Problem Solving

(a) Find two prime numbers that have a sum of 40 and a product that is as large as possible. State this product. [3]

(b) A rectangle has a length of

(2 x + 3)

cm and a width of

(x - 1)

cm. Its area is 44 cm². Find the value of

x

. [3]

\sqrt{20} \times \sqrt{5} = 10

. [2]

(d) Find the smallest positive integer

k

such that

\sqrt{108 k}

is an integer. [2]

(e) The reciprocal of a number

n

\frac{1}{4}

. Find

n

. [1]

Question 11: Natural Numbers Between 5 and 15

(a) Write down all the natural numbers between 5 and 15. [1]

(b) From these numbers, identify the prime numbers. [2]

(d) How many integers are there between 5 and 15 inclusive? [1]

Question 12: Classification of Various Forms

The numbers below are written in different forms:

\sqrt{16}, \frac{7}{3}, 0.25, \sqrt{7}, 8

(a) Write down all the integers. [1]

(b) Write down all the rational numbers. [2]

(d) Explain why

\sqrt{7}

is irrational. [1]

Question 13: Triangle Numbers

(a) Write down the first five triangle numbers. [2]

(b) Identify which of these triangle numbers are square numbers. [1]

(d) Write down one triangle number greater than 10. [1]

Question 14: Square and Cube Numbers

Consider the set of numbers:

1, 2, 4, 8, 9, 27

(a) Write down all the cube numbers. [1]

(b) Write down all the square numbers. [1]

(d) State whether 27 is a prime number. Give a reason. [1]

Question 15: Rational and Irrational

(a) Write down three irrational numbers. [2]

(b) Write down two rational numbers that are not integers. [1]

(d) Is

\sqrt{25}

rational or irrational? Give a reason. [1]

Question 16: Classifying the Number 36

The number 36 can be classified in different ways.

(a) Write 36 as a square number. [1]

(b) Write down all the factors of 36. [2]

(d) Explain why 36 is a rational number. [1]

Question 17: Prime Numbers Less Than 20

(a) Write down all the prime numbers less than 20. [2]

(b) Identify which of these prime numbers are one more than a square number. [2]

(d) Write down the smallest composite number. [1]

Question 18: Square Roots Classification

Consider the numbers:

\sqrt{2}, \sqrt{4}, \sqrt{9}, \sqrt{11}

(a) Write down the integers. [1]

(b) Write down the irrational numbers. [1]

(d) Explain why

\sqrt{11}

is not an integer. [1]

Question 19: Cube and Square Numbers Comparison

(a) Write down the first four cube numbers. [1]

(b) Write down the first four square numbers. [1]

(d) Explain why this number appears in both lists. [2]

Question 20: Set Classification

The following numbers are given:

0, 1, 2, 3, 5, 10

(a) Write down all the natural numbers. [1]

(b) Write down all the prime numbers. [1]

(d) Explain whether 0 is a natural number. [1]