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Exploring the Fascinating World of Complex Numbers

Dive into the intriguing realm of complex numbers, which are defined as the sum of a real part and an imaginary part in the complex plane (z = a + bi). Discover how addition, subtraction, and multiplication with complex numbers can be understood visually like vectors. Learn about Euler's identities and De Moivre's theorem, explore complex conjugates and reciprocals, and see how complex numbers relate to nature and geometry. This guide helps demystify the concepts while showcasing their practical uses and beauty.

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Exploring the Fascinating World of Complex Numbers

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  1. Complex Numbers?What’s So Complex?

  2. Complex numbers are vectors represented in the complex plane as the sum of a Real part and an Imaginary part: z = a + bi Re(z) = a; Im(z) = b

  3. Just like vectors! |z| = (a2 + b2)1/2 is length or magnitude, just like vectors. a = tan-1 (b/a) is direction, just like vectors!

  4. Just like vectors! For two complex numbers a + bi and c + di: Addition/subtraction combines separate components, just like vectors.

  5. Useful identities Euler: eix = cos x + i sin x cos x = (eix + e-ix)/2 sin x = (eix - e-ix)/2i

  6. Things named Euler

  7. Sure, he’s French, but we must give props: DeMoivre: (cos x + i sin x)n = cos (nx) + i sin (nx) cos 2x + i sin 2x = ei2x cos 2x = (1 + cos 2x)/2 sin 2x = (1 - cos 2x)/2

  8. What about multiplication? Just FOIL it!

  9. Scalar multiples of a complex number: a line

  10. Multiplication: the hard way! z1z2= r1 (cosa1 + i sina1) r2 (cosa2 + i sina2) = r1 r2 (cosa1 cosa2 - sina1 sina2) + i r1 r2 (cosa1 sina2 + cosa2 sina1) = r1 r2 [cos(a1 + a2) + i sin(a1 + a2)]

  11. Multiplication: the easy way! “Neither dot nor cross do you multiply complex numbers by.”

  12. Multiplication: by i Rotate by 90o and swap Re and Im

  13. i ‘s all over the Unit Circle! Note i4 = 1 does not mean that 0 = 4

  14. i ‘s all over the Unit Circle! Did you see i½?

  15. Square root of i? Find the square root of 7+24 i. (Hint: it’s another complex number, which we’ll call u+vi). Which can be solved by ordinary means to yield 4+3i and -4 - 3i.

  16. Complex Conjugates

  17. Complex Conjugates Complex conjugates reflect in the Re axis.

  18. Complex Reciprocals The reciprocal of a complex number lies on the same ray as its conjugate!

  19. Powers of z The graph of f(z)=zn for |z|<1 is called an exponential spiral.

  20. This shape is at the heart of the computation of fractals!

  21. The basic geometry of the solar system!

  22. It shows up in nature!

  23. And the decorative arts!

  24. The rotation comes from our old buddy DeMoivre: Raising a unit z to the nth power is multiplying its angle by n. (cos x + i sin x)n = cos (nx) + i sin (nx)

  25. How about a slice of p: Roots of z If z3 = 3+3i = 4.24eip/4then Each successive nth root is another 2p/n around the circle.

  26. Find the roots of the complex equation z2 + 2i z + 24 = 0 Sounds like a job for the quadratic formula!

  27. Was that so complex? And never forget, epi = -1

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