Types Of Numbers

Learn it by playing

Answer these questions to earn energy, then fish and explore. No account needed.

For teachers: ready-to-use lesson slides, revision notes for Types Of Numbers (Maths [CIE], Core) — use them in your lesson, or run the topic as an interactive class activity your students play as a live game.

Lesson notes

Types of Number

Integers are whole numbers (positive, negative, zero), e.g. -3, -2, -1, 0, 1, 2, 3.
Natural numbers are positive integers (counting numbers): 1, 2, 3, 4, ... (0 is not included).
Rational numbers can be written as a fraction a/b where a and b are integers and b ≠ 0; includes terminating and recurring decimals.
Irrational numbers cannot be written as a fraction; they are non-terminating, non-recurring decimals, e.g. π, √2.
All integers are rational (e.g. 5 = 5/1).
A number like 0.3492 recurring is rational because it can be expressed as a fraction.

Multiples

A multiple of a number is formed by multiplying it by a positive integer; e.g. multiples of 3: 3, 6, 9, 12, ...
Every non-zero number has an infinite number of multiples.
A common multiple is a multiple of two or more numbers; e.g. 12 is a common multiple of 4 and 6.
Even numbers are multiples of 2; odd numbers are not multiples of 2.
To list multiples between two values, count up in steps of the number; e.g. multiples of 7 between 10 and 40: 14, 21, 28, 35.

Factors

A factor of a number divides it exactly with no remainder; e.g. 6 is a factor of 18.
Every integer greater than 1 has at least two factors: 1 and itself.
Find factors by listing factor pairs; e.g. for 18: (1,18), (2,9), (3,6).
Use divisibility tests: by 2 (last digit 0,2,4,6,8), by 3 (sum of digits divisible by 3), by 5 (last digit 0 or 5), etc.
If a number is divisible by two coprime integers, it is divisible by their product; e.g. divisible by 2 and 3 → divisible by 6.

Prime Numbers

A prime number has exactly two distinct factors: 1 and itself.
The first ten primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29.
1 is not a prime number (it has only one factor).
2 is the only even prime number.
To show a number is not prime, find a factor other than 1 and itself; e.g. 51 is divisible by 3 (5+1=6), so 51 = 3 × 17.

Squares, Cubes & Roots

A square number is the product of a number multiplied by itself; first 15: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225.
A cube number is the product of a number multiplied by itself twice; first five: 1, 8, 27, 64, 125 (also 1000).
The square root of a number is the value that when squared gives the number; e.g. √49 = 7 (also -7).
The cube root of a number is the value that when cubed gives the number; e.g. ∛64 = 4.
Square roots of non-square numbers are surds (irrational), e.g. √2 is irrational.

Reciprocals

The reciprocal of a number is 1 divided by that number; e.g. reciprocal of 3 is 1/3.
Any number multiplied by its reciprocal equals 1; e.g. 3 × 1/3 = 1.
The reciprocal of a fraction a/b is b/a; e.g. reciprocal of 2/3 is 3/2.
Reciprocals can be written using a power of -1: 1/a = a⁻¹.

Slides

Step through every slide for this topic — plus flashcards and revision notes — with a free account.

Practice questions

Free preview — 8 of 57 questions. Sign up to see them all.

1.Write down the reciprocal of 64.
Easy
- A1/64
- B64
- C0.64
- D64/1
2.Write down the reciprocal of 2.
Easy
- A1/2
- B2
- C0.5
- D2/1
3.Write down a square number greater than 10.
Easy
- A16
- B12
- C9
- D20
4.Write down an irrational number.
Easy
- Aπ
- B0.5
- C2
- D3/4
5.Write down all the factors of 15.
Easy
- A1, 3, 5, 15
- B1, 15
- C3, 5
- D1, 3, 15
6.Write down the reciprocal of 40.
Easy
- A1/40
- B40
- C0.4
- D40/1
7.Write down the reciprocal of 7.
Easy
- A1/7
- B7
- C0.7
- D7/1
8.Write down all the factors of 24.
Easy
- A1, 2, 3, 4, 6, 8, 12, 24
- B1, 2, 3, 4, 6, 12, 24
- C2, 4, 6, 8, 12, 24
- D1, 24

Revision notes

## Types of Numbers (IGCSE Maths Core) ### Integers & Natural Numbers - **Integers**: whole numbers (positive, negative, zero) e.g. -3, -2, -1, 0, 1, 2, 3 - **Natural numbers**: positive integers (1, 2, 3, 4, ...) – 0 is not included. ### Rational & Irrational Numbers - **Rational**: can be written as fraction a/b (a, b integers, b≠0). Includes terminating decimals (0.32 = 32/100) and recurring decimals (0.444... = 4/9). - **Irrational**: cannot be written as a fraction; non-terminating, non-recurring decimals. Examples: π, √2, √3. ### Multiples - A multiple of an integer is formed by multiplying it by another positive integer. Multiples are infinite. - Even numbers: multiples of 2. Odd numbers: not multiples of 2. - Common multiple: multiple of two or more integers (e.g. 12 is a common multiple of 4 and 6). ### Factors - A factor divides a number exactly with no remainder. - Every integer >1 has at least two factors: itself and 1. - Find factors by listing factor pairs (e.g. factors of 18: 1,2,3,6,9,18). - Divisibility tests: 2 (last digit 0,2,4,6,8), 3 (sum of digits divisible by 3), 5 (last digit 0 or 5), 10 (last digit 0). ### Prime Numbers - A prime number has exactly two distinct factors: itself and 1. - First 10 primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29. - 1 is not prime. 2 is the only even prime. ### Squares, Cubes & Roots - **Square numbers**: result of multiplying a number by itself (e.g. 1,4,9,16,25,...). First 15: 1,4,9,16,25,36,49,64,81,100,121,144,169,196,225. - **Cube numbers**: result of multiplying a number by itself twice (e.g. 1,8,27,64,125). - **Square root**: inverse of squaring. √49 = 7 (positive). Surds (e.g. √2) are irrational. - **Cube root**: inverse of cubing. ∛64 = 4. ### Reciprocals - Reciprocal of a number = 1 ÷ that number. A number × its reciprocal = 1. - Reciprocal of 3 is 1/3; reciprocal of 2/3 is 3/2. Also written as a⁻¹. ### Key Tips - On calculator paper, test rationality by seeing if number can be displayed as a fraction. - Memorise first 10 primes, first 15 squares, first 5 cubes. - Use divisibility tests to find factors without a calculator.

Integer: A whole number that can be positive, negative or zero.
Natural number: A positive integer (1, 2, 3, ...). Zero is not included.
Rational number: A number that can be written as a fraction a/b where a and b are integers and b ≠ 0. Includes terminating and recurring decimals.
Irrational number: A number that cannot be written as a fraction; non-terminating, non-recurring decimals (e.g. π, √2).
Multiple: The result of multiplying an integer by another positive integer. Multiples are infinite (e.g. multiples of 3: 3,6,9,...).
Factor: A number that divides another number exactly with no remainder (e.g. factors of 12: 1,2,3,4,6,12).
Prime number: A number that has exactly two distinct factors: itself and 1. The first prime is 2.
Square number: The result of multiplying a number by itself (e.g. 1,4,9,16,...).
Cube number: The result of multiplying a number by itself twice (e.g. 1,8,27,64,125).
Square root: A number that when multiplied by itself gives the original number (e.g. √49 = 7).
Cube root: A number that when multiplied by itself twice gives the original number (e.g. ∛64 = 4).
Reciprocal: The result of dividing 1 by a number. A number multiplied by its reciprocal equals 1 (e.g. reciprocal of 3 is 1/3).

Flashcards

What is a natural number?

Natural numbers are the positive integers (counting numbers): 1, 2, 3, 4, ... Note that 0 is not included.

What is an integer?

An integer is a whole number that can be positive, negative, or zero. Examples: -3, -2, -1, 0, 1, 2, 3.

What is a rational number?

A rational number can be written as a fraction a/b where a and b are integers and b ≠ 0. All terminating and recurring decimals are rational.

What is an irrational number?

An irrational number cannot be written as a fraction. It is a non-terminating, non-recurring decimal. Examples: π, √2.

Write down the reciprocal of 64.

1/64

Write down the reciprocal of 2.

1/2

Write down the reciprocal of 10.

1/10

Write down a square number greater than 10.

16 (or 25, 36, 49, 64, 81, 100, etc.)

Write down an irrational number.

π (or √2, √3, etc.)

Write down all the factors of 15.

1, 3, 5, 15

Write down the reciprocal of 40.

1/40

Write down the reciprocal of 7.

1/7

Write down all the factors of 24.

1, 2, 3, 4, 6, 8, 12, 24

Write down the smallest prime number.

Write down the six factors of 12.

1, 2, 3, 4, 6, 12

Write down all the factors of 16.

1, 2, 4, 8, 16

Write down a prime number between 30 and 40.

31 (or 37)

Write down all the factors of 28.

1, 2, 4, 7, 14, 28

From the list 6, 8, 9, 10, 12, 15, 16, 17, write down a prime number.

From the list 6, 8, 9, 10, 12, 15, 16, 17, write down a square number.

9 (or 16)

From the list 6, 8, 9, 10, 12, 15, 16, 17, write down a cube number.

Write down all the factors of 76.

1, 2, 4, 19, 38, 76

Write down the reciprocal of 5.

1/5

From the list 8, 15, 18, 33, 39, 41, 51, 57, 60, 81, write down a factor of 54.

From the list 8, 15, 18, 33, 39, 41, 51, 57, 60, 81, write down a multiple of 19.

Unlock all 57 questions, flashcards & more

Create a free account to see every question, the slides, flashcards and revision notes for this topic.

Past papers

Past-paper practice for this topic is coming soon.

Coming soon