Download DISTANCE BETWEEN TWO POINTS

Complex Numbers
i
Given that x² = 2
Or that if
ax² + bx + c = 0
Then
Then
x = -b ± √b² - 4ac
2a
x = ±√2
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
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
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.
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 + 2i
● (1,1) = 1 + i
Re
● (3,-1) = 3 - i
●
(-2,-2)
= -2 - 2i
Arithmetic Operations
Addition
z 1 = a + i b , z2 = c + i d
z1 + z2 = (a + c) + i (b + d)
e.g. (2 + 3i) + (5 – 4i) = 7 – i
Subtraction
z 1 = a + i b , z2 = c + i d
z1 - z2 = (a - c) + i (b - d)
e.g. (9 + 2i) - (5 – 6i) = 4 + 8i
Arithmetic Operations
Multiplication
z 1 = a + i b , z2 = c + i d
z1z2 = (a + ib)(c + id)
= ac + iad + ibc + i²bd
e.g. (3 + 2i)(5 - i)
= 15 - 3i + 10i - 2i²
= 15 + 7i – 2(-1)
= 17 + 7i
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) x (2 - 3i) = 8 - 12i + 4i – 6i²
(2 + 3i) (2 - 3i) 4 - 6i + 6i – 9i²
= 14 – 8i
13
= 14 – 8i
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
● (x,y)
θ
x
tan θ = Y
x
θ = tan Y
x
y
-π < θ < π
Re