Can someone help me solve these problems? they're the only ones i'm struggling with FRACTAL GEOMETRY Many fractal images are generated by substituting an initial value into a complex function, calcu- lating the output, and then using the output as the next value to substitute into the function. The process is then repeated indefinitely. This recycling of outputs is called iter- ation, and each output is called an iterate. In Exercises 63 to 66, we will use the letter z to symbolize a complex number. The initial value is called the seed and is denoted by z0. Suc- cessive iterates are denoted by z1, z2, z3, ... 65) Let f(z) = iz. Begin with the intial value z0 = 1 - i. Determine z1, z2, z3, and z4. Find a pattern and use it to predict z5, z6, z7, and z8. The property that the product of conjugates of the form (a + bi)(a - bi) is equal to a2 + b2 can be used to factor the sum of two perfect squares over the set of complex num- bers. For example, x2 + y2 = (x + yi)(x - yi). In Exercises 73 to 78, factor the binomial over the set of complex numbers. 73) x2 + 16 75) z2 + 25 79) Show that if x = 1 + 2i, then x2 - 2x + 5 = 0.
Can someone help me solve these problems? they're the only ones i'm struggling with
FRACTAL GEOMETRY Many fractal images are generated by substituting an initial value into a complex function, calcu- lating the output, and then using the output as the next value to substitute into the function. The process is then repeated indefinitely. This recycling of outputs is called iter- ation, and each output is called an iterate. In Exercises 63 to 66, we will use the letter z to symbolize a complex number. The initial value is called the seed and is denoted by z0. Suc- cessive iterates are denoted by z1, z2, z3, ...
65) Let f(z) = iz. Begin with the intial value z0 = 1 - i. Determine z1, z2, z3, and z4. Find a pattern and use it to predict z5, z6, z7, and z8.
The property that the product of conjugates of the form (a + bi)(a - bi) is equal to a2 + b2 can be used to factor the sum of two perfect squares over the set of complex num- bers. For example, x2 + y2 = (x + yi)(x - yi). In Exercises 73 to 78, factor the binomial over the set of
73) x2 + 16
75) z2 + 25
79) Show that if x = 1 + 2i, then x2 - 2x + 5 = 0.
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