Exploring the Beauty of Complex Numbers and Their Real-World Applications
Dive into the elegance of complex numbers and their practical significance in fields such as orbital dynamics, computer graphics, and more. Discover the fundamental theorem of algebra, Euler's neat formula, and the polar form of complex numbers. Uncover why complex numbers matter and how they offer a unique perspective on mathematical concepts and real-life scenarios.
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Presentation Transcript
The Simplicity of Complex Numbers Teacher Quality Workshop 9/21/2016
What are complex numbers and why does anyone care? Beauty and neatness Elegant theory Real, real world applications
Elegant theory Fundamental Theorem of Algebra Every (non-constant) polynomial of degree n has exactly n roots (counting multiplicities).
Real, real world applications Orbital dynamics, Kepler s laws Computer graphics Navigation 2ndorder differential equations: Electric circuits Oscillating mechanical systems a y + b y + c y = 0 a r2+ b r + c =0
Eulers Neat Formula 2 4 6 3 5 7 = + + = + + cos 1 ... sin ... 2! 4! 6! 3! 5! 7! = = 0 1 i 2 3 4 e = + + + + + 1 ... 1! 2! 3! 4! 1 i i ( ) ( ) 1! i i + ( ) 3! 3 ( ) 4! 4 = = 2 2 3 4 1 i i i i i = + + + + + i 1 ... e 2! 2 = 3 2 i i i i 2 3 4 5 5 6 6 i i i i = + + + + + 1 ... = = = = 4 3 1 i i i 1! 2! 3! 4! 5! 6! 5 4 i i i i i 2 3 4 5 6 i i = + + + 1 . .. 1! 2! 3 ! 4! 5! 6! Cycle continues
Eulers Neat Formula i 2 3 4 5 6 i i = + + + i 1 ... e 1! 2! 3! 4 ! 5 ! 6 ! 2 4 6 3 5 = + + + + + ie 1 ... ... i 2! 4! 6! 1! 3! 5! 2 4 6 3 5 7 = + + = + + cos 1 ... sin ... 2! = 4! 6! 3! 5! 7! ( ) + ie co s s in i
The Polar Form of a Complex Number a r = = cos cos a r b r = = sin sin b r ( ) = = a bi + = + cos sin z r + r i ( ) cos sin r re i = i
Useful Resources https://www.math.toronto.edu/mathnet/ques tionCorner/complexinlife.html https://betterexplained.com/articles/a-visual- intuitive-guide-to-imaginary-numbers/
