Problem 1. Let z1 = 2 + 3i and 22-1-21. Compute 2. (21 2) 3. (*i - 2)-2 Problem

Question

Problem 1. Let z1 = 2 + 3i and 22-1-21. Compute 2. (21 2) 3. (*i - 2)-2 Problem 2. Find a complex number z so that (1-5i): -1 + 3i = Problem 3, Let z1 = 1 + V3i and z2 =-1-i 1. Express and 22 using exponential notation, i.e., z = re0i. 2. Compute 121 . zal and arg(·22); 3. Compute | | and arg( ) Problem 4. Express the following complex numbers using exponential notation, i.e., z re" 1, cos a-isin ; sin + cos ; sin cos 2. 3, (Hint: first use trigonometric identities to write each one in the form of cos + i sin .) Problem 5. Consider the following polynomial equation 1. Find all real valued solutions of this equation; 2. Find all (real and complex) solutions of this equations

Explanation / Answer

1) z1/z2 = 2+3i / 1-2i x 1+2i /1+2i

= -5/5 +7i/5

= -1 + 7i/5

2) (z1+z2)3 = (2+3i +1-2i)3

= (3+i)3 = (27 -i +27i -9)

= 18 + 26i

3) (z1 - z'2)-2 = (2+3i -1-2i)-2

= (1+i)-2 = 1/2i = -i/2

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