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Answers to Exercises for Chapter 1
Multiplication by i causes a rotation through 90 degrees (counter-clockwise)
Let z be any complex number not zero. Show that points z, 0, zi are verticles of a right triangle by showing that
|z - zi|² = |z|² + |zi|² (Pythagorean theorem)
Let z = x + yi. Then zi = (x + yi)i = -y + xi. The (length of the hypotenuse)² =
|z - zi|² = |(x+y) + (y-x)i|² = (x+y)² + (y-x)² =
x² + 2xy + y² + y² - 2xy + x² = 2(x²
+y²)
The sum of (the lengths)² of the legs =
|x+yi|² + |(x+yi)i|² =