Numbers : Gate Academy

Natural numbers are used for counting purposes. They are made up of all possible rational and irrational numbers.
An integer is a whole number.
A prime number is a number that is divisible only by itself and one. 1 is not a prime number.
A rational number is a number that can be written as a fraction.
An irrational number is a number that cannot be written as a fraction. E.g. Any recurring decimals; π, 3.3333333⋅⋅⋅

Highest Common Factor and Lowest Common Multiple:

From the Venn diagram above:
- HCF = Product of common factors of x and y.
- LCM = Product of all items in the Venn diagram.

Prime Factorization
- This is just finding prime numbers that multiply together to make the original number.
- For example:
- ∴ 48 = 2 × 2 × 2 × 2 × 3

Definition of sets. For example:
- A = {x: x is a natural number}
- B = {(x, y): y = mx + c}
- C = {x: a ≤ x ≤ b}
- D = {a, b, c, …}
Notations:
- n(A) = Number of elements in A
- ∈ = is an element of
- ∉ = is not an element of
- A’ = Compliment set of A
- Ø or {} = Empty set
- ξ or U = Universal set
- A ∪ B = Union of A and B
- A ∩ B = Intersection of A and B
- A ⊆ B = A is a subset of B
- A ⊂ B = A is a proper subset of B
- A ⊄ B = A is not a subset of B
Set Representations:
- A ∩ B (the darker shade)
- A ∪ B
- ⊂ ‘is a subset of’
- b ∈ X
- ξ = {a, b, c, d, e}
- A’ is shaded

Ratio is used to describe a fraction. E.g. 3 : 1
Foreign exchange: Money changed from one currency to another using proportion. For example:
- Convert $22.50 to Dinars
- $1 : 0.30KD
- $22.50 : 6.75KD
Map Scales: Using proportion to work out map scales. For example:
- 1 km = 1000 m
- 1 m = 100 cm
- 1 cm = 10 mm
Direct Variation
- y is proportional to x
- y ∝ x
- ∴ y = kx
Inverse Variation
- y is inversely proportional to x
- y ∝ ¹/_x
- ∴ y = ^k/_x

Percentage
- This is a convenient way to express fractions.
- Per cent means per 100
- Percentage increase/decrease = ^{Actual increase/decrease}/_{Original Amount}