Summary Edexcel a Level Further Mathematics Core Pure Mathematics Years 1 & 2 (books 5 & 10 online) - Core Pure Mathematics (9FM0)

Complex numbers
● A complex number, z (z ∈ ℂ), is the sum of a real number, x (x ∈ ℝ) + an imaginary
number


Imaginary numbers
● Come from equations eg x2 + 1 = 0, ie x2 = -1
○ A solution to this is x = (±)i ⇒ i = (±)√−1, i2 = -1
● A complex number z = x + yi where x = Re(z) {real part), y = Im(z) {imaginary part}
○ ***Re(z), Im(z) ∈ ℝ
● ±yi always come in pairs called complex conjugates, eg z = -1 + 2i, z* = -1 - 2i
○ z + z* = -2 (= 2Re(z)) → real

○ z - z* = 4i (= 2Im(z) × i) → imaginary
○ zz* = 5 (= Re(z) × Re(z*) - Im(z) × Im(z*) = |z|2) → real
■ |z| = √𝑎2 + 𝑏 2 , |z|2 = a2 + b2
zz* = (a + bi)(a - bi) = a2 + b2 ≡ |z|2


The i cycle
● i = √−1, i2 = (√−1)2 = -1, i3 = −√−1 (= -i), i4 = −√−1 × √−1 = 1, i5 = 1 × √−1 = √−1
(= i)...




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Complex roots of polynomials
𝑏 𝑐
● ax2 + bx + c = 0 where 𝛂 + 𝛃 = -𝑎, 𝛂𝛃 = 𝑎
If 𝛂, 𝛃 ∈ ℂ, then 𝛂, 𝛃 are conjugates
𝑏 𝑐
This MUST be so, ∵ otherwise -𝑎, 𝑎 would be complex, ∴ a, b, c MUST be real
○ 𝛂 + 𝛃 + 𝛄 is ALWAYS real



1

,Argand diagram
● On a 2-D graph, x-axis: Re (ℝ), y-axis: Im — both axes form a “complex plane”




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●

Modulus and argument
● Given that z = a + bi, |z| = √𝑎2 + 𝑏 2; arg z = bearing from ℝ axis (in rad)
○ arg z = +ve if Im = +ve, vice versa; -π < arg z ≤ π
● ***Rules:
○ |z2| = (|z|)2
○ |z1 + z2| ≤ |z1| + |z2|
○ |z1z2| = |z1||z2|
𝑧 |𝑧1 |
○ |𝑧1 | = |𝑧2 |
2
○ arg(z2) = 2 arg z
○ arg(z1z2) = arg z1 + arg z2
𝑧
○ 𝑎𝑟𝑔 (𝑧1 ) = 𝑎𝑟𝑔 𝑧1 − 𝑎𝑟𝑔 𝑧2
2




2

,Modulus-argument form




●
● If 𝑧1 = 𝑟(𝑐𝑜𝑠 𝜃 + 𝑖 𝑠𝑖𝑛 𝜃), 𝑧2 = 𝑠(𝑐𝑜𝑠 𝜑 + 𝑖 𝑠𝑖𝑛 𝜑), |z1z2| = rs, arg(z1z2) = 𝜃 + 𝜑
***Derivation: 𝑟(𝑐𝑜𝑠 𝜃 + 𝑖 𝑠𝑖𝑛 𝜃) × 𝑠(𝑐𝑜𝑠 𝜑 + 𝑖 𝑠𝑖𝑛 𝜑)
= 𝑟𝑠 (𝑐𝑜𝑠 𝜃 𝑐𝑜𝑠 𝜑 + 𝑖 𝑐𝑜𝑠 𝜃 𝑠𝑖𝑛 𝜑 + 𝑖 𝑠𝑖𝑛 𝜃 𝑐𝑜𝑠 𝜑 − 𝑠𝑖𝑛 𝜃 𝑠𝑖𝑛 𝜑)
= 𝑟𝑠 (𝑐𝑜𝑠 (𝜃 + 𝜑) + 𝑖 𝑠𝑖𝑛 (𝜃 + 𝜑))
𝜋 𝜋 3√2 3√2
○ Eg z1 = 3 (𝑐𝑜𝑠 + 𝑖 𝑠𝑖𝑛 ) (= + 𝑖)
4 4 2 2
𝜋 𝜋
z2 = 2 (𝑐𝑜𝑠 6
+ 𝑖 𝑠𝑖𝑛 6 ) (= √3 + 𝑖)
5𝜋 5𝜋 √6 − √2 √6 + √2 3
z1z2 = 6 (𝑐𝑜𝑠 + 𝑖 𝑠𝑖𝑛 ) =6( + 𝑖 ) = (√6 − √2) +
12 12 4 4 2
3
2
𝑖 (√6 + √2)
𝑧1 3 𝜋 𝜋
𝑧2
= 2 (𝑐𝑜𝑠 12 + 𝑖 𝑠𝑖𝑛 12)
𝜋 𝜋
● Eg z = 1 + i = √2 (𝑐𝑜𝑠 4 + 𝑖 𝑠𝑖𝑛 4 )
𝜋 𝜋
z2 = 2 (𝑐𝑜𝑠 + 𝑖 𝑠𝑖𝑛 )
2 2
3 3𝜋 3𝜋
z = 2√2 (𝑐𝑜𝑠 4 + 𝑖 𝑠𝑖𝑛 4 )
z4 = 4 (𝑐𝑜𝑠 𝜋 + 𝑖 𝑠𝑖𝑛 𝜋)
3𝜋 3𝜋
***z5 = 4√2 (𝑐𝑜𝑠 − 4
+ 𝑖 𝑠𝑖𝑛 − 4
)
𝜋 𝜋
z6 = 8 (𝑐𝑜𝑠 − 2 + 𝑖 𝑠𝑖𝑛 − 2 )
...
○ Powers of z form a spiral enlargement (where |z| > 1); |z| < 1: spiral inwards,
|z| = 1: forms a circle




3

, Equations on the Argand diagram
● ***For 2 complex numbers z1 = x1 + y1i, z2 = x2 + y2i, |z2 - z1| represents the distance
between the points z1 and z2




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● ***Given z1 = x1 + y1i, the locus of point z such that |z - z1| = r OR |z - (x1 + y1i)| = r is
a circle with centre (x, y) and radius r




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