Problem 107 - Algebra and Trigonometry

Question

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I multiplied two complex numbers in polar form by first multiplying the moduli and then multiplying the arguments.

Step-by-Step Solution

Answer

The given statement doesn't make sense because while moduli of complex numbers in the polar form are multiplied during the multiplication operation, the arguments are not multiplied but added.

Step 1: Identify the Property of Complex Numbers

The multiplication of two complex numbers in polar form has a fundamental property. Namely, when two complex numbers \(z_1=r_1(\cos{\theta_1}+i\sin{\theta_1})\) and \(z_2=r_2(\cos{\theta_2}+i\sin{\theta_2})\) are multiplied, the resulting complex number \(z\), given by \(z=z_1 \times z_2\), is also in polar form \(r(\cos{\theta}+i\sin{\theta})\), where \(r=r_1 \times r_2\) is the modulus and \(\theta=\theta_1 + \theta_2\) is the argument.

Step 2: Analyze the Given Statement

The statement 'I multiplied two complex numbers in polar form by first multiplying the moduli and then multiplying the arguments' is partially correct. The multiplication of the moduli is the correct part of the operation, however, the multiplication of the arguments isn't correct as per the above explained property of complex numbers.

Step 3: Explain the Reasoning

The operation in the statement doesn't completely make sense because while the moduli of complex numbers in polar form are multiplied during multiplication, the arguments are not multiplied but added. Therefore, the statement is not a valid operation for multiplication of complex numbers in polar form.