Math Problem Statement

Given complex numbers x = 6 + i, w = -1 + 5i, z = 4 - 8i, and P = 6 - 8i, Q = -4 + i, solve for various expressions in terms of complex numbers in algebraic form, including 2x + 4z, w * z, x / w, modulus and argument of P and P/Q, and express the division of Z1 and Z2 in Cartesian form.

Solution

Let's go through each part of this problem step-by-step.

Part (a)

Given the complex numbers:

  • x=6+ix = 6 + i
  • w=−1+5iw = -1 + 5i
  • z=4−8iz = 4 - 8i

We are asked to find the following in the form ( a +

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Math Problem Analysis

Mathematical Concepts

Complex Numbers
Modulus and Argument
Multiplication and Division of Complex Numbers

Formulas

Multiplication of Complex Numbers: (a+bi) * (c+di) = (ac-bd) + (ad+bc)i
Division of Complex Numbers: (a+bi) / (c+di) = [(a+bi) * (c-di)] / (c²+d²)
Modulus of Complex Number: |z| = sqrt(a² + b²) for z = a + bi
Argument of Complex Number: arg(z) = atan(b/a)

Theorems

Properties of Complex Numbers in Polar and Cartesian Form

Suitable Grade Level

Undergraduate Level

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