Types of number (OCR GCSE Maths): Revision Notes

Types of number

1. Integers

Definition: Integers are whole numbers. They can be positive, negative, or zero.

• Important Note: Numbers like $2.5$ or $\frac{1}{2}$ are not integers because they are not whole.

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2. Rational Numbers

Definition: A rational number is any number that can be written as a fraction $\frac{a}{b}$, where a and b are integers, and $b≠0$.

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3. Irrational Numbers

Definition: An irrational number cannot be written as a simple fraction. These numbers have non-repeating, non-terminating decimal expansions.

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4. Square Numbers

Definition: A Square Number is the result of multiplying any whole number (integer) by itself.

Visual Representation:

You can visualise square numbers by arranging dots in square patterns:

• 1 dot forms a $1×1$ square.
• $4$ dots form a $2×2$ square, and so on.

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5. Triangle Numbers

Definition: A Triangle Number is a number that can be arranged in the shape of an equilateral triangle. You can obtain these by starting with $1$ and then adding the next whole number each time.

Formula:

• The $n$th Triangle Number can be calculated using the formula: ${Triangle Number} = \frac{n \times (n+1)}{2}$

Visual Representation:

Triangle numbers can be represented by dots arranged in a triangular shape:

$$1,3,6,10,15,21,28,36,45,55$$

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6. Cube Numbers

Definition: A Cube Number is a number obtained by multiplying a whole number (integer) by itself twice. For example, $1^3 = 1, 2^3 = 8, 3^3 = 27,$ and so on.

Formula:

• If $n$ is an integer, then the cube number is given by:

                                                               ${Cube Number} = n \times n \times n = n^3$
  

Visual Representation:

• Cube numbers can be visualised by arranging dots in a cubic pattern, where each dimension of the cube corresponds to the integer being cubed.

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7. Factors

Definition: The Factors of a number are all the whole numbers (integers) that divide exactly into the given number without leaving a remainder.

Important Points:

• $1$ is a factor of all numbers, and any number is a factor of itself.

Observation:

• Notice that square numbers (like $4, 9, 16$) have an odd number of factors. Can you think about why that might be?

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8. Multiples

Definition: The Multiples of a number are all the numbers you get by multiplying that number by the whole numbers (integers).

Important Points:

• You must include the number itself as its first multiple.

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9. Prime Numbers

Definition: A prime number is a number that has exactly two distinct factors: $1$ and the number itself. This means that it can only be divided by $1$ and itself without leaving a remainder.

Important Notes:

• $1$ is NOT a prime number because it only has one factor (itself).
• $2$ is the ONLY even prime number because it has two factors ($1$ and $2$). All other even numbers are divisible by $2$ and at least one other number, making them not prime.

List of Prime Numbers Between 1 and 100**:** $2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97$