Types Of Numbers

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 $b \ne$ 0; includes terminating and recurring decimals.
• **Irrational numbers** cannot be written as a fraction; they are non-terminating, non-recurring decimals, e.g. $\pi , \sqrt{2}$.
• All integers are rational (e.g. $5 = \frac{5}{1})$.
• 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 $51 = 3 \times 17$.

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. $\sqrt{49} = 7 (also -7)$.
• 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. $\sqrt{2}$ 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. $3 \times \frac{1}{3} = 1$.
• The reciprocal of a fraction a/b is b/a; e.g. reciprocal of $\frac{2}{3}$ is 3/2.
• Reciprocals can be written using a power $of -1: \frac{1}{a} =$ a⁻¹.

Number Classification Venn Diagram

Multiples and Factors Example

Square and Cube Numbers

Reciprocals Illustration