1. Complex Numbers – Part 1 By Dr. Samer Awad
Assistant professor of biomedical engineering
The Hashemite University, Zarqa, Jordan
Last update: 15 November 2018

2. Numbers • Historical succession of discovering classes of numbers:2
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Numbers
• Historical succession of discovering classes of numbers:
Natural numbers: counting.
Integers: we added zero and the –ve numbers.
Rational numbers: fraction of two integers m/n.
Real numbers: rational and non-rational (𝜋, 2 ).
Complex numbers: −1 .

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Numbers
• One of the greatest advances in mathematics, was the introduction of decimal numbers and the decimal point by Indians.
• Al-Khwarizmi (c. 780 – c. 850), documented this work and introduced the Arabic numerals.
• Before that the Roman numerals were used: I = 1, V = 5, X = 10, L = 50, C = 100, D = 500, M = 1,000. So many symbols were used to represent numbers instead of only 10 symbols.

4. Square Roots of Negative Numbers?4
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Square Roots of Negative Numbers?
• The most commonly occurring application problems that require people to take square roots of numbers are problems which result in a quadratic equation.
• e.g. Determine the dimensions of a square with an area of 9 cm2, 25 cm2.
• Thus, having to take the square root of a negative number in this context means that such a rectangle does not exist.

5. Motivation: The Cubic Equation5
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Motivation: The Cubic Equation
• The solution of a general cubic equation that contains the square roots of negative numbers led to the introduction of complex numbers.
• This was mainly done by the Italian mathematician Gerolamo Cardano in around 1545 after several attempts by other mathematicians before him in the 16th century.
• The term “complex number” was introduced by Carl Friedrich Gauss who also paved the way for a general use of complex numbers.

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Imaginary Numbers

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Complex Numbers
• By definition, a complex number z is an ordered pair (x, y) of real numbers x and y, written as:
𝑧=(𝑥,𝑦)
• x is called the real part and y the imaginary part of z, written as:
𝑥=𝑅𝑒 𝑧, 𝑦=𝐼𝑚 𝑧
• (0, 1) is called the imaginary unit and is denoted by 𝑖:
𝑖=(0,1)

8. Complex Numbers Notations8
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Complex Numbers Notations
• Ordered pair notation: 𝑧=(𝑥,𝑦)
• Some references use the notation:
– 𝑧=𝑥+𝑖𝑦
– 𝑧=𝑥+𝑦𝑖
• In some disciplines, in particular electrical engineering, 𝑗 is used instead of 𝑖, since 𝑖 is frequently used for electric current:
– 𝑧=𝑥+𝑗𝑦
– 𝑧=𝑥+𝑦𝑗

9. Addition & Multiplication of Complex Numbers9
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Addition & Multiplication of Complex Numbers
Notation: 𝑧=𝑥+𝑖𝑦
• Addition of two complex numbers :
𝑥1+𝑖 𝑦1 + 𝑥2+𝑖 𝑦2 = 𝑥1+𝑥2 +𝑖(𝑦1+𝑦2)
• Multiplication is defined by:
𝑥1+𝑖 𝑦1 𝑥2+𝑖 𝑦2 =(𝑥1𝑥2+𝑖𝑥1𝑦2+𝑖𝑥2𝑦1+𝑖2 𝑦1𝑦2).
= 𝑥1𝑥2 − 𝑦1𝑦2 +𝑖( 𝑥1𝑦2+𝑥2𝑦1).

10. Examples: Addition & Multiplication10
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Examples: Addition & Multiplication
• Add and multiply z1 and z2:
• Multiply the following two complex numbers:

11. Subtraction & Division of Complex Numbers11
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Subtraction & Division of Complex Numbers
• Subtraction of two complex numbers :
• Division is defined by:

12. Examples: Subtraction & Division12
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Examples: Subtraction & Division
• Subtract and divide z1 and z2:

13. Example: Division • Divide the following two complex numbers: 13
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Example: Division
• Divide the following two complex numbers:

14. Complex Plane • Remember: a complex number z can be written as:14
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Complex Plane
• Remember: a complex number z can be written as:
𝑧=(𝑥,𝑦)
• x is called the real part and y the imaginary part of z, written as:
𝑥=𝑅𝑒 𝑧, 𝑦=𝐼𝑚 𝑧
• Hence, it is possible to
present z on an xy-plane
called the complex plane.
• This is called the Cartesian
coordinate system
(as opposed to the polar coordinate system that will be explained later)

15. Example: Complex Plane15
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Example: Complex Plane
• Plot 4−3𝑖 on the complex plane

16. Complex Plane: Addition & Subtraction16
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Complex Plane: Addition & Subtraction

17. Complex Plane: Addition Example17
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Complex Plane: Addition Example

18. Complex Conjugate Numbers18
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Complex Conjugate Numbers
• The complex conjugate 𝑧 of a complex number 𝑧= 𝑥+𝑖𝑦 is defined by:
𝑧 =𝑥−𝑖𝑦
• Mathematically, replace 𝑖 with −𝑖.
• Graphically, flip z around the x-axis (real axis):
• Some references use the notation 𝑧∗ for complex conjugate.

19. Complex Conjugate Numbers19
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Complex Conjugate Numbers
• Prove the following:
– 𝑧 𝑧 =𝑥2+𝑦2
–𝑧+ 𝑧 =2𝑥
–𝑧− 𝑧 =2𝑖𝑦

20. Conjugation Properties20
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Conjugation Properties
• Re 𝑧=𝑥= 𝑧+ 𝑧
• 𝐼𝑚 𝑧=𝑦= 1 2𝑖 𝑧− 𝑧
• If z is real →𝑧=𝑥→𝑧= 𝑧
• If z is imaginary→𝑧=𝑖𝑦→𝑧=− 𝑧
• Working with conjugates is easy, since we have:

21. Examples: Conjugation Properties21
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Examples: Conjugation Properties
• Find 𝐼𝑚 𝑧1 𝑎𝑛𝑑 𝑧 1 𝑧 2

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Coordinate Systems
• The Cartesian coordinate system is commonly used to determine the location of a point in two or three dimensional space.
• The cylindrical and spherical coordinate systems – that will be addressed later – are also used to determine the location of a point in two or three dimensional space.
• The polar coordinate system is used to determine the location of a point in two dimensional space.
• The polar coordinate system is a special case of the cylindrical and the spherical coordinate systems.

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Polar Form
• So far, we’ve adopted the Cartesian coordinate system to represent a complex number.
• Complex numbers can also be represented in polar form, that is, in terms of magnitude and angle.
where:
𝑥=𝑟 cos 𝜃 𝑎𝑛𝑑 𝑦=𝑟 sin 𝜃

24. Polar Form: Absolute Value24
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Polar Form: Absolute Value
• Hence, a complex number can be written as:
• By using Euler’s formula:
• The absolute value (aka:
modulus, magnitude or
amplitude) of a complex
number in polar form are:

25. Polar Form: Absolute Value25
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Polar Form: Absolute Value
• |z| is the distance between point z and the origin. The letter “r” stands for radius.
• In class: explain relationship to |-5| = 5
• |z| = constant  circle

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Polar Form: Argument
• The argument (or angle) of a complex number in polar form is:
• Here, all angles are measured in radians and positive in the counter clockwise sense.
• For z=0, arg z is undefined
(why?).

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Warning!!! b -a a -b

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Polar Form: Argument
• For z ≠ 0, arg z corresponds to the same value every 2𝜋.
• Principal value Arg z: −𝜋<𝐴𝑟𝑔 𝑧≤𝜋.
• arg z=Arg z ±𝑛2𝜋 (𝑛=0,±1,±2, …)

29. Polar Form: Conjugation29
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Polar Form: Conjugation
• For complex conjugates:
𝑧=𝑟 𝑒 𝑗𝜃
𝑧 =𝑟 𝑒 −𝑗𝜃
• | 𝑧 |=|𝑧|
• arg 𝑧 =− arg 𝑧

30. Example: Polar Form • Find the polar form of 𝑧=1+𝑖 𝑎𝑛𝑑 𝑧=3+𝑖 3 3 : 30
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Example: Polar Form
• Find the polar form of 𝑧=1+𝑖 𝑎𝑛𝑑 𝑧=3+𝑖 3 3 :

31. Triangle Inequality 𝑧1+𝑧2 ≤ 𝑧1 +|𝑧2|31
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Triangle Inequality
𝑧1+𝑧2 ≤ 𝑧1 +|𝑧2|
• 𝑧1+𝑧2 = 𝑧1 +|𝑧2| when z1 and z2 lie on the same straight line through the origin.

32. Example: Triangle Inequality32
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Example: Triangle Inequality
• Let 𝑧1=1+𝑖, 𝑧1=−2+𝑖3
• 𝑧1+𝑧2 = −1+𝑖4 = 17 =4.12
• 𝑧1 + 𝑧2 = =5.02
• 4.12 < 5.02

33. Polar Form: Multiplication33
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Polar Form: Multiplication
• Multiplying two complex numbers gives a complex number whose modulus is the product of the two moduli and whose argument is the sum of the two arguments.
Proof: see textbook.
Prove it in class using polar form.

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Polar Form: Division
• Dividing two complex numbers gives a complex number whose modulus is the quotient of the two moduli and whose argument is the difference of the two arguments.
Proof: see textbook.

35. Example: Polar Multiplication & Division35
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Example: Polar Multiplication & Division
• Multiply and divide z1 and z2 in polar form:

36. Integer Powers of Complex Numbers36
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Integer Powers of Complex Numbers
• De Moivre’s Formula: If a complex number is raised to the power n the result is a complex number whose modulus is the original modulus raised to the power n and whose argument is the original argument multiplied by n.

37. Example: Integer Powers of Complex Numbers37
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Example: Integer Powers of Complex Numbers
• Find 𝑖

38. Integer Roots of Complex Numbers38
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Integer Roots of Complex Numbers
• Need to calculate 𝑤= 𝑘 𝑧  𝑤𝑘=𝑧
• Let 𝑧 = 𝑟 𝑒 𝑖𝜃 and
𝑤 =𝑅 𝑒 𝑖∅
 𝑤𝑘=𝑅𝑘 𝑒 𝑖𝑘∅ = 𝑟 𝑒 𝑖𝜃 =𝑧
• Then, 𝑅= 𝑘 𝑟 .

39. Integer Roots of Complex Numbers39
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Integer Roots of Complex Numbers
•𝑤𝑘=𝑅𝑘 𝑒 𝑖𝑘∅ = 𝑟 𝑒 𝑖𝜃 =𝑧
• Does ∅= 𝜃 𝑘 ? The answer is NO!
• Since 𝜃 is determined only up to integer multiples of 2𝜋 (i.e. 𝜃≡𝜃+𝑛2𝜋), then ∅= 𝜃+𝑛2𝜋 𝑘 = 𝜃 𝑘 +𝑛 2𝜋 𝑘 .
• For 𝑛=0, 1, 2, 𝑘−1 we get k distinct values of w. Further integers of n would give values already obtained. n=k→𝑛2𝜋/k=2𝜋 ≡ n=0→𝑛2𝜋/k=0.

40. Integer Roots of Complex Numbers40
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Integer Roots of Complex Numbers
 𝑤= 𝑘 𝑧 = 𝑘 𝑟 exp 𝜃 𝑘 + 𝑛2𝜋 𝑘 ,
𝑤ℎ𝑒𝑟𝑒 𝑛=0, 1, 2, 𝑘−1
• Hence, there are k distinct roots of a complex number z.
• These k roots lie on a circle of radius 𝑘 𝑟 and are separated by 2𝜋/k from their neighbouring root.

41. Examples: Integer Roots of Complex Numbers41
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Examples: Integer Roots of Complex Numbers
• Find the roots of: 3 1 , :

42. Examples: Integer Roots of Complex Numbers42
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Examples: Integer Roots of Complex Numbers
• Find the roots of:

43. Other Topics • 13.3 Derivative. Analytic Function43
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Other Topics
• 13.3 Derivative. Analytic Function
• 13.4 Cauchy–Riemann Equations. Laplace’s Equation