Studied by 1 person
Sets of number
R = Real numbers (any numbers, e.g. 4.134213)
Q = Rational numbers (numbers that can be written as fractions, e.g. 0.5)
Z = Integers (Numbers that do not include decimals, e.g. -1, 3, etc.)
N = Natural Numbers (positive integers, start from 0)
Q’ or I = Irrational Numbers (numbers that cannot be written as fractions, e.g. Π)
W = Whole Numbers (1,2,3,10, etc.)
C = Complex Numbers (i)
Surds
A surd is an irrational number expressed with a radical sign (√) that cannot be simplified into a rational number.
If a square root simplifies to a whole number, it is not a surd.
Example:
√20 → Surd (cannot simplify to a whole number)
√100 → Not a surd (simplifies to 10, a rational number)
Simplification of Surds
When simplifying surds, look for perfect square factors and then use √a*b = √a * √b.
E.g. √40 = √4×10 = 2√10
Perfect Square Factors (From 1-10)
1, 4, 9, 16, 25, 36, 49, 64, 81, 100 (From 1-10)
+-*/ surds
±: add/substract by combining like terms, e.g. 2√3 + 3√3 -5√7 + 6√7 = 5√3 + √7 (don’t write 1)
*/: multiply and divide like indices:
√x * √y = √xy, √x/√y = √x/y (more generally, a√x/b√y = a/b√x/y)
Use the distributive law (a(b+c) = ab+ac) to expand brackets.
Rationalising the denominators
Not keeping surds in denominator.
Monomial Method: x/z√y = x/z√y * z√y/z√y = zx√y/zy
Binomial Method: 1/√x+√y = 1/√x+√y * √x-√y/√x-√y = √x-√y/1/x-y
Pascal’s Triangle / Binomial Expansion
When expanding a bionomial equation, the coefficients can be found in Pascal's triangle. E.g., (x+y)^8 is 8th row in triangle.
START COUNTING FROM SECOND ROW I.E., 1,1
Formula: (a + b)ⁿ = C₀aⁿ + C₁aⁿ⁻¹b + ... + Cₙbⁿ,
where Cₖ are coefficients from Pascal’s Triangle.
Example: Expand (2x - 5)⁵
Pascal’s Triangle for n = 5: 1, 5, 10, 10, 5, 1
Apply powers of 2x and -5:
32x⁵ - 400x⁴ + 2000x³ - 5000x² + 6250x - 3125
IMPORTANT: For Pascal’s Triangle
ALWAYS + - + - + - if in the form of (a-b)^n
Order matters: use the row of coefficients IN ORDER (or lose marks), e.g., (a+b)³ = 1, 3, 3, 1
Note 
Flashcard (127)