Complex Numbers Mathmatics N4
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The document covers complex numbers, including definitions, graphical representations (Argand diagrams), and operations such as addition, subtraction, multiplication, and division. It also discusses the transformation of complex numbers into rectangular and polar forms, along with an explanation of De Moivre's theorem. Exercises are provided for practice, illustrating the conversion and calculation of complex numbers in various forms.
![Example 2 continued
(52.998 cis 598.1)4 = (52.998 4 cis 4 X 598.1)
= 7889290. 051 cis 2392.4
7889290. 051 cis 2392.4 / 7.28 cis 74.05
= [7889290. 051 /7.28 ] cis [2392.4 - 74.05 ]
1083693.688Cis 2318.35](https://image.slidesharecdn.com/complexnumbersengineeringsciencen4-180406095458/85/Complex-Numbers-Mathmatics-N4-24-320.jpg)
![Example 4: Prove that the formula
Zn = rn cis (nӨ) is correct for ( Z4 X Z3)4
= ((7 – 2i)(-2 – 7i))4
= (- 28 – 45i)4
= (- 28 – 45i)2 (- 28 – 45i)2
= [((-28) + (-45i))2 ] 2](https://image.slidesharecdn.com/complexnumbersengineeringsciencen4-180406095458/85/Complex-Numbers-Mathmatics-N4-28-320.jpg)
![Example 4 continued
[(-28) + (-45i))2 ] 2
= [(-28)2 + 2(-28)(-45i) + (-45i) 2 ] 2
= [ 784 + 2520i + (2025 X -1) ] 2
= (-1301 + 2520i) 2
= (-1301) 2 + 2( -1301)(2520) + (2520i) 2](https://image.slidesharecdn.com/complexnumbersengineeringsciencen4-180406095458/85/Complex-Numbers-Mathmatics-N4-29-320.jpg)
Complex Numbers Mathmatics N4
- 2. FET College Registrations Engineering N1 – N6 Business N4 – N6 Mechanical Engineering Civil Engineering Electrical Engineering Business Qualifications FET Matric Rewrite New Syllabus Technical Matric N3 Phone: +27 73 090 2954 Fax: +27 11 604 2771 Email: mathsmech206@gmail.com
- 3. Table of Contents 1. Definition of Complex Numbers 2. Argand Diagram 3. Rectangular and Polar Forms of Complex Numbers 4. Addition and Subtraction of Complex Numbers 5. Multiplication and Division 6. De Moivre’s Theorem 7. Applications of Complex Numbers
- 4. Complex numbers are made up real parts and imaginary parts. Complex numbers are written in the form Z = a + bi a= real part bi = imaginary part and i = (-1)1/2 For the complex number z1 = 2 + 7i a= 2 (real) and bi = 7i (imaginary)
- 5. Argand Diagram iy E Z1 = 2 + 7i Z2 = -7 + 2i -X A B O C D X Z4 = 7 – 2i Z3 = -2 – 7i FET College Jhb CBD Ph: 073 090 2954
- 6. Argand Diagrams are the graphical representations of complex numbers. The y axis represents the imaginary axis while the x axis represents the real part of the complex number.
- 7. Rectangular Forms and Polar Forms of Complex Numbers. The complex number Z1 = 2 + 7i can be represented in the form Z1 = r1 cis Ө1 or Z1 = r1 Ө1 Where r1 = modulus of complex number And Ө1 = argument of complex number
- 8. Modulus of a complex number is the length of the line OZ1 Refering to the Argand Diagram and considering the right angle triangle OEC, the angle OEC = Ө1. Ө1 is the angle that the complex number makes with the positive x axis.
- 9. Also considering right angle triangle OEC r1 = (22 + 72) 0.5 = 7.28 Ө1 can be calculated by applying trigonometric ratios Ө1 =Tan -1 (7/2) = 74.05 from (7/2)= tan Ө1 Z1 = 7.28 cis 74.05 The polar form of a complex number can also be written as 7.28 ( cos 74.05 + isin 74.05)
- 10. We are going to write the polar forms of the complex numbers Z2, Z3 and Z4 by considering right angle triangles AOZ2, BOZ3 and DOZ4 respectively. The corresponding right angled triangles that the complex number makes with the x axes is called the reference angle.
- 11. Angle AOZ2 = Ө2ref Angle BOZ3 = Ө3ref Angle DOZ4 = Ө4ref Positive angles are measured from the positive x axis to the corresponding complex number in a counterclockwise direction.
- 12. Negative angles are angles measured in the clockwise direction from the positive x axis. Based on the above definitions Z2 = r2 cis (180 - Ө2ref ) Z2 = 7.28 cis ( 180 - 15.95) = 7.28 cis 164.05 where Ө2ref = tan -1 (2/7) = 7.28 (cos 164. 04 + isin 164.05)
- 13. Similarly, the complex numbers Z3 = r3 cis Ө3 and Z4 = r4cis Ө4 will be calculated Z3 = r3 cis (180 + Ө3ref ) Z3 = 7.28 cis ( 180 +74.05) = 7.28 cis 254.05 where Ө3ref = tan -1 (7/2) = 7.28 (cos 254. 05 + isin 254.05)
- 14. Z4 = r4 cis (360 - Ө4ref ) Z4 = 7.28 cis ( 360 -15.95) = 7.28 cis 344.05 where Ө4ref = tan -1 (2/7) = 7.28 (cos 344. 05 + isin 344.05)
- 15. These complex numbers can also be written in terms of negative angles. Z2 = 7.28 cis (360 – 164.05) =7.28 cis (– 195.95 ) Z3 = 7.28 cis ( 360 - 254.05) = 7.28 cis (-105.95) Z4 = 7.28 cis ( 360 - 344.05) = 7.28 cis (- 15.95)
- 16. Exercise 1 Convert the following complex numbers to rectangular form z = a + ib by applying the formula Z=r(cosӨ + isinӨ). This exercise is based on the argand diagram above. 1. Z1 = 7.28 cis 74.05 2. Z2 = 7.28 cis 164.05 3. Z3 = 7.28 cis 254.05 4. Z4 = 7.28 cis 344.05
- 17. Exercise 1 (continued) 5. Z2 = 7.28 cis (– 195.95 ) 6. Z3 = 7.28 cis (-105.95) 7. Z4 = 7.28 cis (- 15.95) Confirm your answers by comparing with the original complex numbers on the argand diagram.
- 18. Addition and Subtraction of Complex numbers. Real parts are grouped such that the addition and subtraction is performed. Imaginary parts are also grouped. Example 1. z1 + z2 + z3 Based on the argand Diagram 2 + 7i + (-7 + 2i) + (-2 – 7i) (2 -7 -2) + i(7 + 2 – 7) = -7 + 2i
- 19. Multiplication and Division of Complex numbers. Example 1. z1 X z2 (2 + 7i) X (-7 + 2i) 2 (-7 + 2i) + 7i (-7 + 2i) -14 + 4i – 49i + 14i2 where i2 = (-1) 1/2 X 2 = -1 -14 – 45i + 14(-1) = -28 – 45i
- 20. Multiplication and Division of Complex numbers. Example 1. z1 ÷ z2 (2 + 7i) / (-7 + 2i) (2 + 7i) X (-7 – 2i) / (-7 + 2i) X (-7 – 2i) -7 – 2i is the conjugate of -7 + 2i
- 21. Note that (-7 + 2i) X (-7 – 2i) = (-7)2 – (2i)2 = r2 = (modulus)2 (-7)2 – (22 . -1) = 53 (difference of squares) (2 (-7 – 2i) + 7i (-7 – 2i)) ÷ 52.99 ( -14 – 4i – 49i – 14i2) ÷ 51 = (0 -53i ) ÷ 53 0 - i
- 22. De Moivre’s Theorem De Moivre’s Theorem states that Zn = rn cis (nӨ) Example Evaluate ( Z4 )1/4 = (r4 cis 344.05)1/4 7.24 ¼ cis 344.05 X ¼ = 1.64 cis 86.0125
- 23. Example 2: ( Z4 X Z3)4/Z1 (7.28 344.05 X 7.28 254.05 ) 4 / 7.28 74.05 It will be proven that 7.28cis 344.05 X 7.28is 254.05 = 7.28 X 7.28 cis (344.05 + 254.05) = 52.998 cis 598.1
- 24. Example 2 continued (52.998 cis 598.1)4 = (52.998 4 cis 4 X 598.1) = 7889290. 051 cis 2392.4 7889290. 051 cis 2392.4 / 7.28 cis 74.05 = [7889290. 051 /7.28 ] cis [2392.4 - 74.05 ] 1083693.688Cis 2318.35
- 25. Example 3: Prove that continued Z4 X Z3 = 52.998 cis 598.1 (7 – 2i)(-2 – 7i) 7(-2 – 7i) – 2i(-2 -7i) -14 -49i + 4i + 14i2 = - 28 – 45i = 53 cis (180 + 58.11) = 53 cis 238.11
- 26. 53 cis 238.11 = 53 cis (238.11 + 360) 53 cis 598.11 53(cos 598.11 + isin 598) (7 – 2i)(-2 – 7i) 7(-2 – 7i) – 2i(-2 -7i) -14 -49i + 4i + 14i2 = - 28 – 45i
- 27. = 53 cis (180 + 58.11) = 53 cis 238.11 53 cis 238.11 = 53 cis (238.11 + 360) = 53 cis 598.11 = 53(cos 598.11 + isin 598.11) = -27.99 - 45 i
- 28. Example 4: Prove that the formula Zn = rn cis (nӨ) is correct for ( Z4 X Z3)4 = ((7 – 2i)(-2 – 7i))4 = (- 28 – 45i)4 = (- 28 – 45i)2 (- 28 – 45i)2 = [((-28) + (-45i))2 ] 2
- 29. Example 4 continued [(-28) + (-45i))2 ] 2 = [(-28)2 + 2(-28)(-45i) + (-45i) 2 ] 2 = [ 784 + 2520i + (2025 X -1) ] 2 = (-1301 + 2520i) 2 = (-1301) 2 + 2( -1301)(2520) + (2520i) 2
- 30. = 1692601 - 6557040i - 6350400 = -4657799 – 6557040i = 8043001cis 234.612 Convert this complex number to polar form And compare your anwer with the polar complex number below. = 7889290. 051 cis 2392.4
- 31. = (- 28 – 45i)4 = (53 cis 238.11 ) 2 (53 cis 238.11 ) 2 (53 X 53 cis 238.11 + 238.11 ) 2 = 534 cis (238.11 X 4) = 7890481 cis 952.844 = - 4765749.89 - 6288665.87i = 7890481 cis (952.844+ 4(360)) =7890481cis 2392.4
- 32. = 7890481 cis (952.844+ 4(360)) = 7890481cis(952.844 + 4 revolutions) =7890481cis 2392.4