MTH 324
Complex Analysis
By
Dr. Saqib Hussain
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MTH 324
Lecture # 1
The complex Number System
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Lecture’s outline
• The real number system
• The Complex number system
• Comparison of Real system with Complex
system
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Complex Number System
Set
Collection of well defined and distinct objects
Examples
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Real Number
A number whose square is always non negative is
called a real number.
Examples
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Classification
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Geometry of real numbers
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Properties
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Commutative Laws
Associative Laws
Distributive Laws
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Ordering Property
Intervals
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The Real Number System
The set of real numbers with its properties is called
real number system.
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The Complex Number System
Need of complex numbers
Examples
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Equality
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Complex Number
Terminology
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Arithmetic Operations
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Commutative Laws
Associative Laws
Distributive Laws
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Remark
Ordering property does not hold in the set of complex
numbers.
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Complex Plane
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Conjugate
If z is a complex number then the number obtained by
changing the sign of imaginary part of z is called complex
conjugate.
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Comparison of Real system with Complex
system
• Many of the properties of real number system
hold in complex number system but there are
some remarkable differences as well.
• Ordering property hold in real but not in
complex
• The concept of conjugate is defined in complex
but not in real.
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References
• A First Course in Complex Analysis with
Applications by Dennis G. Zill and Patrick D.
Shanahan.
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