Complex Numbers

i

Given that x² = 2 Then x = ± √2 Or that if ax² + bx + c = 0 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

the property that

i

² = -1 ie

i

= √-1.

which has Hence we can solve the equation x² = -2 x = ± √-2 x = ± √2 √-1 x = ± √2

i

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

i

x = -2 ± 2 2

i

x = -1 ±

i

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 + Then a is the real part of z, written Re z.

B is the imaginary part of z, written Im z.

i

b,

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) = -4 + 2

i

● (1,1) = 1 +

i

Re ● (3,-1) = 3 -

i

● (-2,-2) = -2 - 2

i

Arithmetic Operations Addition z 1 z 1 = a + + z 2

i

b , z 2 = c +

i

= (a + c) +

i

d (b + d) e.g. (2 + 3

i

) + (5 – 4

i

) = 7 –

i

Subtraction z 1 z 1 = a + - z 2

i

b , z 2 = c +

i

= (a - c) +

i

d (b - d) e.g. (9 + 2

i

) - (5 – 6

i

) = 4 + 8

i

Arithmetic Operations Multiplication z 1 z 1 z = a + 2

i

b , z 2 = c +

i

= (a +

i

b)(c +

i

d) d = ac +

i

ad +

i

bc +

i

²bd e.g. (3 + 2

i

)(5 = 15 - 3

i

= 15 + 7

i i

+ 10 )

i

- 2

i

² – 2(-1) = 17 + 7

i

Complex Conjugates If z = a +

i

b then the complex conjugate of z is defined to be a –

i

b and denoted z Division e.g. (4 + 2

i

) ÷ = (4 + 2 (2 + 3

i i

) ) x (2 + 3

i

) (2 - 3

i

) (2 - 3

i

) = 8 - 12

i

+ 4

i

– 6

i

² 4 - 6

i

+ 6

i

– 9

i

² = 14 – 8

i

13 = 14 – 8

i

13 13

The Argand Diagram.

The distance of any complex number z from the origin is √(x²+ y²). This is called the modulus of z and is written |z|.

The angle θ is the angle of rotation from the real axis to OZ and is called the argument of z or arg(z).

Im θ x ● (x,y) y Re tan θ = θ = tan Y x Y x -π < θ < π