Download Complex Numbers Notes 1. The Imaginary Unit We use the symbol i

Complex Numbers Notes
1. The Imaginary Unit
We use the symbol i to represent the imaginary unit.
i = √-1
i2 = -1
With the imaginary unit, i, we can write the square root of any negative number in terms of i.
E.g.
 16  16(1)  16.  1  4i
Also: i3 = (i2)(i) = (-1)(i) = -I
i4 = (i2)(i2) = (-1)(-1) = 1
2. Complex numbers
A complex number has 2 parts:
1. A real part ( normal number)
2. An imaginary part (with the imaginary unit; i)
E.g. 3 + 2i
5i
-4
real part: 3 imaginary part: 2i
real part: 0 imaginary part: 5i
real part: -4 imaginary part: 0
3. Adding and Subtracting Complex Nos
We do this in 2 steps:
(i) combine the real parts
(ii) combine the imaginary parts
E.g (5+3i) + (1-5i)
(i) 5 + 1 = 6
(ii) 3i – 5i = -2i
(5+3i) + (1-5i) = 6 – 2i
Complex nos. are often referred to as ‘z’, ‘w’ or ‘z1’ and ‘z2’. In such questions we start by
substituting the complex numbers for these letters.
E.g. If z1 = 4+5i and z2 = -1+i find z1 - z2
z1 - z2 = (4+5i) - (-1+i)
4 – (-1) = 4+1 = 5
5i – (i) = 4i
z1 - z2 = 5 + 4i
4. Multiplying Complex Nos
We multiply as we did in algebra, but we must remember that
i2 = -1
Any time we have an i2 in an answer we replace it with -1
E.g (5+3i)(1-5i) = 5 -25i + 3i -15(i2) = 5 – 22i -15(-1) =
5 + 15 - 22i = 20 – 22i
5. Complex Conjugate
When it comes to division of complex numbers we often need to find the complex conjugate.
_
If z is a complex no., then its complex conjugate is written as z
To get the complex conjugate:
1. leave the real part as it is.
2. change the sign of the imaginary part.
_
e.g. if z = 2 + 5i
z = 2 – 5i
6. Division of Complex Nos
To divide a complex no. by a real number we divide each part of the complex no. by the real no.
e.g.
8  7i 8 7
7
  i  2 i
4
4 4
4
To divide one complex no. by another e.g.
2i
1  2i
(i) multiply above and below by the conjugate of the bottom line
2i
(2  i) (1  2i)

1  2i (1  2i) (1  2i)
(ii) multiply out the top and bottom
(2  i ) (1  2i ) 2  4i  i  2(1) 4  3i


(1  2i) (1  2i ) 1  2i  2i  4(1)
5
(iii) divide by the real no to get the answer in the form a+bi
(2  i ) 4  3i 4 3

  i
(1  2i )
5
5 5
7. Equality of Complex Nos
If 2 complex nos. are equal, then the real parts must be equal
and the imaginary parts must be equal.
e.g. (x + 2) + i(y - 1) = 7 + 2i
find the value of x and y
Real parts are equal: (x + 2) = 7 so x = 7 – 2 = 5
Imaginary parts are equal: (y – 1) = 2
so y = 2 + 1 = 3
8. The Argand Diagram
An Argand diagram is a plane that is used to represent complex nos.. It looks like an x,y plane
except the x-axis is now the real axis (Re), and the y-axis is now the imaginary axis (Im).
The complex no. z = a + bi is represented by the point (a,b) on an Argand diagram
If z = 3 - 2i, then the point (3,-2) represents z.
Im
1 2
3
Re
-1
-2
-3
3-2i = (3,-2)
9. Modulus of a Complex No.
The modulus, or length, of a complex no. is the distance from the origin to the point
representing the complex number on an Argand diagram.
We use 2 lines either side of a complex no. to represent the modulus.
E.g the modulus of 3 - 4i is written |3 - 4i|. This is then the distance from (0,0) to (3,-4).
To find this length do the following:
(i) write down a square root sign: √
(ii) write the real part to be squared , and a plus sign:
(iii) write the imaginary part to be squared:
32  ...
32  42
(iv) calculate the value:
32  42  9  16  25  5
In general:
a 2  b2
|a + bi| =
10. Quadratic Equations with complex roots
In complex nos, quadratic equations are ususally written with the variable z (instead of x).
When solving the equation: az2 + bz + c = 0, by using the formula:
 b  b 2  4ac
z
2a
It’s possible we could get the square root of a negative no.. Then the roots will involve i.
E.g. find the roots of: z2 – 8z + 25 = 0
for this equation: a = 1, b = -8, c = 25
So
 (8)  (8)2  4(1)( 25) 8  64  100 8   36 8  6i
z



 4  3i
2(1)
2
2
2
So the roots are:
4 + 3i and
4 - 3i