Complex Numbers

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Complex Numbers
There is nothing complex or hard about them
ei? 1 0
L. Euler
The most beautiful mathematical formula
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Starter
10 min
Solve the following linear equations
a) x 2 3 b) x 5 2 c) 2x 3 0

• Solve the following quadratic equations
• d) x2 9 0
• e) x2 16 0
• f) x2 5x 6 0
• g) x2 - 6x 4 0

Recall the solutions of the quadratic equation
ax2bxc 0 are given by -b?(b2-4ac)
2a and -b-?(b2-4ac) 2a
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Motivation

• Complex numbers are much used in
• Physics and Engineering in wave functions for
  example
• ( very important for mobile phones)
• In Mathematical Research Chaos and Fractals

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God created the counting numbers, everything else
is the invention of man
Man needed to invent new types of numbers to
solve real-life problems he faced.
Problem If you have 5 lambs and a wolf kills 2,
how many would be left?
Since we do not know the number of surviving
lambs, we call this number x (for unknown).
Now, our problem can be formalised as x 5 -
2 x ?
To solve this kind of problem, we just need the
counting numbers 1, 2, 3, 4, 5...
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Negative numbers
Problem If yesterdays temperature was 2C and
we know that the temperature fell by 5C during
the night then what would be the temperature
today?
This can be formalised as x 2 - 5 x ?
Now, the counting number are not enough to solve
this problem. We need to invent a new kind of
numbers, called the negative integers. -1, -2,
-3, -4, ...
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Twist of the tail ZERO
Arabs from India Alkwarizmi Mistake of
Dionysos Exiguus about Jesus age
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Fractions rational numbers
Problem Two boys share equally 3 sweets. How
many should each of them get?
If we call what each of them should get x (the
unknown) then we may write 2x 3 x ?
To solve this problem we need to invent yet
another kind of numbers, called fractions or
rational numbers 2/3, 3/2, 100/133, ...
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Irrational numbers
Problem The sides of a corn field are 1 mile
long. We would like to build a road joining two
opposite corners. How long will the road be? Of
course, we want to know the length of the road
beforehand so that we can plan how much it would
cost.

1 mile
1 mile
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Pythagoras
Using Pythagoras theorem we may write x2 12
12 2 x ?
To solve this real-life problem we need to
introduce yet another kind of numbers, called the
irrational numbers ?2, ?10, ...
Exercise Why are they called irrational? (Hint
fractions are called rational.)
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Exercises Find x a) x2 9 0 b) x2 16
0 c) x2 100 0 d) x2 144 0
Recall the solutions of the quadratic equation
ax2bxc 0 are given by -b?(b2-4ac)
2a and -b-?(b2-4ac) 2a

• Solve these quadratic equations
• x2 5x 6 0
• x2 - 6x 4 0
• x2 7x 3 0
• x2 2x 4 0
• x2 - 2x 8 0

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Complex numbers
x2 - 2x 8 0 x 1 ?-7 or x 1 - ?-7
To solve this equation we have to introduce a new
type of numbers, called the complex numbers.
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x2 -5 What is x ? There is no real number x
such that x2 -5.
Just as mathematicians invented negative numbers
to solve equations like x 3 2 and they
invented rational numbers to solve equations like
3x 1 and they invented irrational numbers
to solve equations like x2 2, they invented
complex numbers to solve equations like x2 -2.
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What is the number i ?
We know that v4 2 because 2 x 2 4. We
know that v9 3 because 3 x 3 9. Also v
1 1 because 1 x 1 1. Now, what is v-1
? That is, what is the number a such that
a x a -1 ? We call that number i.

• Thus
• i x i i2 -1

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• Now
• ?-1 i
• ?-2 ?2 ?-1 i ?2
• ?-4 ?4 ?-1 i ?4 2i

• Exercises What is
• ?-5
• ?-14
• ?-25
• 3 ?-3
• ?-5 ?-9
• 12 - ?-12

A complex number is a number of the form a bi
where a and b are real numbers.
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Addition of complex numbers
(a bi) (c di) (ac) (bd)i
Example (4 3i) (5 10i) 9 13i

• Exercise
• a) (2 6i) (4 7i) ?
• (100 99i) - (55 - 45i) ?
• 33 56i 44 23i 43i ?

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Multiplication of complex numbers
(abi)(cdi) ac bci adi bdi2
ac (bcad)i - bd
(ac-bd) (bcad)i since i2 -1
Example (3 2i)(4 5i) 12 (158)i 10
2 23i

• Exercises
• (2 4i)(4 5i) ?
• (2 4i)2 ?
• (3i)(3 5i) ?
• (3i 4)3 ?

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Division of complex numbers
(a bi)(c di) (c di)(c - di)
ac (bc-ad)i bd c2 d2
(a bi) (c di)
Example (2 3i) / (4 5i) (8 2i 15) /
(16 25) (23 2i) / 41

• Exercise
• (2i) / (2 2i) ?
• (4 5i) / (3i) ?
• (1 - i) / (1 i) ?

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How the different types of numbers are related
complex numbers
reals
rationals
integers
counting numbers
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Geometric representation of numbers
The number line
3/4
-5 -4 -3 -2 -1 0 1
2 3 4 5
A Greek idea consider each number to be a
length. So the number 3 is the distance from
zero to 3. The number 2 is the distance from
zero to 2. What would be the number ¾ ? It
is a quarter of the distance from zero to 3.
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Real numbers cannot actually be represented on
the number line.
They must be approximated.
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Representation of complex numbers on the Argand
diagram
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Exercise represent the following numbers on the
Argand diagram

• 2 - 3i
• -2 - 3i
• i
• 1 - i
• i 1
• 5 4i
• 5 4i

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Complex numbers are vectors in 2-D

• (4i) (22i)
• Exercises add complex numbers using the Argand
  diagram.
• ( 2 i ) ( 3 2i )
• ( i ) ( 2 )
• ( 4i ) ( 3 2i )

(Recall the parallelogram rule of vector
addition.)
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A word on trigonometric representation (if time
permits)
The most beautiful mathematical formula
ei? 1 0
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Consider the number x iy Using
elementary trigonometry we may write x r cos
? y r sin ? Thus xiy r cos ? r isin ?
r ( cos ? isin ?) r is called the
modulus ? is called the argument
r cos ?
r
r sin ?
?
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Example
z 4 3i

• 4 r cos ?
• r sin ?
• Using Pythagoras Theorem, we get
• r 5
• ¾ sin ?/ cos ?
• tan ?
• Thus
• ? tan-1 ( 0.75 ) 37
• Thus z 43i 5( cos 37 i sin 37 )

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Eulers Formula
Theorem e i ? cos ? i sin ?
If ? ? we get e i ? cos ? i sin ?
-1 0
Thus
ei? 1 0