1. 1 By Dr. Saqib Hussain Complex Analysis MTH 324

2. 2 The complex Number System Lecture # 1 MTH 324

3. 3 Lecture’s outline The real number system The Complex number system Comparison of Real system with Complex system

4. 4 Complex Number System Set Collection of well defined and distinct objects Examples

5. 5 Real Number A number whose square is always non negative is called a real number. Examples

6. 6 Classification

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9. 9 Geometry of real numbers

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11. 11 Properties

12. 12 Commutative Laws Associative Laws Distributive Laws

13. 13 Ordering Property Intervals

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15. 15 The Real Number System The set of real numbers with its properties is called real number system.

16. 16 The Complex Number System Need of complex numbers Examples

17. 17 Equality

18. 18 Complex Number Terminology

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20. 20 Arithmetic Operations

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22. 22 Commutative Laws Associative Laws Distributive Laws

23. 23 Remark Ordering property does not hold in the set of complex numbers.

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25. 25 Complex Plane

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27. 27 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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30. 30 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.

31. 31 References A First Course in Complex Analysis with Applications by Dennis G. Zill and Patrick D. Shanahan.