1. Complex Numbers i

2. Given that x² = 2
Or that if ax² + bx + c = 0
Then x = ±√2
Then x = -b ± √b² - 4ac 2a
How do we solve x² = -2 or a quadratic with
b² - 4ac < 0 ?
We can introduce a new number called i which has
the property that i² = -1 ie i = √-1.
Hence we can solve the equation x² = -2
x = ±√-2
x = ±√2 √-1
x = ±√2 i

3. Example
Solve x² + 2x + 2 = 0
x = -b ± √b² - 4ac 2a
x = -2 ± √2² - 4x1x2
2x1
x = -2 ± √4 - 8 2
x = -2 ± √-4 2
x = -2 ± √4 √-1 2
x = -2 ± √4 i 2
x = -2 ± 2 i 2
x = -1 ± i

4. i is an imaginary number.
The set of complex numbers, C, are the family of numbers of the form, a + i b, where a and b are real numbers.
If z is a complex numbers, C, where z = a + i b,
Then a is the real part of z, written Re z.
B is the imaginary part of z, written Im z.

5. The Argand Diagram.
The Argand Diagram gives a geometric representation of complex numbers as points in the plane R².
ie z = x + i y can be expressed by the point (x,y). Im
● (-4,2)
= i
● (1,1) = 1 + i Re
● (3,-1) = 3 - i ●
(-2,-2)
= i

6. Arithmetic Operations
Addition
z1 = a + i b , z2 = c + i d
z1 + z2 = (a + c) + i (b + d)
e.g. (2 + 3i) + (5 – 4i)
= 7 – i
Subtraction
z1 = a + i b , z2 = c + i d
z1 - z2 = (a - c) + i (b - d)
e.g. (9 + 2i) - (5 – 6i)
= 4 + 8i

7. Arithmetic Operations
Multiplication
z1 = a + i b , z2 = c + i d
z1z2 = (a + ib)(c + id)
= ac + iad + ibc + i²bd
e.g. (3 + 2i)(5 - i)
= i + 10i - 2i²
= i – 2(-1)
= i

8. Complex Conjugates
If z = a + i b then the complex conjugate of z is defined to be a – ib and denoted z
Division
e.g. (4 + 2i) ÷ (2 + 3i)
= (4 + 2i)
(2 + 3i)
(2 - 3i)
= i + 4i – 6i²
4 - 6i + 6i – 9i² x
= 14 – 8i 13
= 14 – 8i

9. The Argand Diagram. Im
The distance of any complex number z from the origin is √(x²+ y²). This is called the modulus of z and is written |z|.
● (x,y) y θ Re x
The angle θ is the angle of rotation from the real axis to OZ and is called the argument of z or arg(z).
tan θ = Y x
-π < θ < π Y x
θ = tan