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# Complex numbers org.ppt

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### Complex numbers org.ppt

1. 1. PRESENTATION BY OSAMA TAHIR 09-EE-88
2. 2. COMPLEX NUMBERSA complex number is a number consistingof a Real and Imaginary part.It can be written in the form i  1
3. 3. COMPLEX NUMBERS Why complex numbers are introduced??? Equations like x2=-1 do not have a solution withinthe real numbers x  1 2 x  1 i  1 i  1 2
4. 4. COMPLEX CONJUGATE The COMPLEX CONJUGATE of a complex numberz = x + iy, denoted by z* , is given by z* = x – iy The Modulus or absolute value is defined by z x y 2 2
5. 5. COMPLEX NUMBERSReal numbers and imaginary numbers aresubsets of the set of complex numbers. Real Numbers Imaginary Numbers Complex Numbers
6. 6. COMPLEX NUMBERSEqual complex numbersTwo complex numbers are equal if theirreal parts are equal and their imaginaryparts are equal. If a + bi = c + di, then a = c and b = d
7. 7. ADDITION OF COMPLEX NUMBERS(a  bi)  (c  di)  (a  c)  (b  d )i Imaginary Axis z2EXAMPLE z sum z1(2  3i )  (1  5i ) z2 (2  1)  (3  5)i Real Axis 3  8i
8. 8. SUBTRACTION OF COMPLEX NUMBERS(a  bi)  (c  di)  (a  c)  (b  d )i Imaginary AxisExample z1 z 2( 2  3i )  (1  5i ) z2 z diff ( 2  1)  (3  5)i z 2 Real Axis 1  2i
9. 9. MULTIPLICATION OF COMPLEX NUMBERS(a  bi)(c  di)  (ac  bd)  (ad  bc)iExample( 2  3i )(1  5i ) ( 2  15)  (10  3)i 13  13i
10. 10. DIVISION OF A COMPLEX NUMBERSa  bi  a  bi  c  dic  di c  di c  di ac  adi  bci  bdi 2  c d 2 2 ac  bd  bc  ad i  c d 2 2
11. 11. EXAMPLE6  7i  6  7i   1  2i  1  2i  1  2i  1  2i  6  12i  7i  14i 2 6  14  5i   12  2 2 1 4 20  5i 20 5i     4i 5 5 5
12. 12. COMPLEX PLANEA complex number can be plotted on a plane with two perpendicular coordinate axes  The horizontal x-axis, called the real axis  The vertical y-axis, called the imaginary axis y P x-y plane is known as the z = x + iy complex plane. O x The complex plane Slide 14
13. 13. COMPLEX PLANE Geometrically, |z| is the distance of the point z from the origin while θ is the directed angle from the positive x- axis to OP in the above figure. y   t an1    xθ is called the argument of z and is denoted by arg z. Thus, Im P  y y   arg z  tan   1 z0 z = x + iy  x r |z |= θ O x Re
14. 14. Expressing Complex Number in Polar Formx  r cos  y  r sin So any complex number, x + iy,can be written inpolar form: x  yi  r cos  r sin i
15. 15. Imaginary axis Real axis
16. 16. De Moivre’s Theorem De Moivres Theorem is the theorem which shows us how to take complex numbers to any power easily. Let r(cos F+isin F) be a complex number and n be any real number. Then [r(cos F+isin F]n = rn(cosnF+isin nF)[r(cos F+isin F]n = rn(cosnF+isin nF)
17. 17. Euler FormulaThe polar form of a complex number can be rewritten as z  r (cos  j sin  )  x  jy  re jThis leads to the complex exponentialfunction :z  x  jye z  e x  jy  e x e jy  e x cos y  j sin y 
18. 18. ExampleA complex number, z = 1 - jhas a magnitude | z | (12  12 )  2  1    and argument : z  tan    2n     2n  rad 1  1   4  Hence its principal argument is : Arg z   rad 4 Hence in polar form :  j    z  2e 4  2  cos  j sin   4 4
19. 19. APPLICATIONS Complex numbers has a wide range of applications in Science, Engineering, Statistics etc.Applied mathematics Solving diff eqs with function of complex roots Cauchys integral formula Calculus of residues In Electric circuits to solve electric circuits
20. 20. How complex numbers can be applied to“The Real World”??? Examples of the application of complex numbers:1) Electric field and magnetic field.2) Application in ohms law.3) In the root locus method, it is especially important whether the poles and zeros are in the left or right half planes4) A complex number could be used to represent the position of an object in a two dimensional plane,
21. 21. REFERENCES.. Wikipedia.com Howstuffworks.com Advanced Engineering Mathematics Complex Analysis
22. 22. THANK YOUFOR YOUR ATTENTION..!