Summary Grade 10-11 AP Maths Notes

AP Mathematics
Complex Numbers
Imaginary Numbers
i : √− 1
i² : − 1 (real number)
i³ : − i
i⁴ : 1 (real number)


Complex Numbers
The combination of a real & imaginary number
x = − 1±√3i
Real: -1
Imaginary: ±√3i

Formula: z = a + bi
NB: all real numbers are complex number with no imaginary part

Basic operations
w = a + bi
z = c + di

Addition:
W+Z
= a + bi + c + di
= (a + c) + (bi + di)
= (a + c) + (b + d)i

Subtraction:
W-Z
= (a + bi) − (c + di)
= a + bi − c − di
= (a − c) + (bi − di)

, = (a − c) + (b − d)i
Multiplication:
WxZ
= (a + bi)(c + di)
= ac + adi + bci + bdi 2
= ac + (ad + bc)i − bd
= (ac − bd) + (ad + bc)i

Division:
w÷Z
= a+bi
c+di x c−di
c−di (multiply by denominator conjugate)
2
ac−adi+bci−bdi
=
c 2 +d 2
ac−(ad−bc)i+bd
=
c 2 +d 2
(ac+bd)−(ad−bc)i
=
c 2 −d 2


Roots ---> Equation
Formula: x 2 − (sum)x + product

Simultaneous Equations to find values
3 steps:
- Expand
- Equate
- Real
- Imaginary
- Solve 2 equations simultaneously