Question

Consider the complex number $$ z=\frac{a+i b}{c+i d} $$ where \(a, b, c\) and \(d\) are real. Determine \(\operatorname{Re}(z)\) and \(\operatorname{Im}(z)\).

Short answer

Question: Given a complex number in the form \(z=\frac{a+i b}{c+i d}\), where \(a, b, c,\) and \(d\) are real numbers, find the real part (Re\((z)\)) and the imaginary part (Im\((z)\)) of the complex number. Solution: The real part of the complex number is Re\((z)=\frac{ac + b^2}{c^2 + d^2}\) and the imaginary part is Im\((z)=\frac{ad + bc}{c^2 + d^2}\).

Step-by-step solution

Find the complex conjugate of the denominator

First, find the complex conjugate of the denominator, which is in the form \(c + id\). The complex conjugate is given by: $$ c - id $$

Multiply the numerator and the denominator by the complex conjugate

Next, multiply both the numerator and the denominator by the complex conjugate of the denominator: $$ z=\frac{(a+ib)(c-id)}{(c+id)(c-id)} $$

Simplify the expression

Now, we will simplify the expression by expanding the numerator and the denominator: $$ z=\frac{ac + adi + ibc - (ib)^2}{c^2 - (id)^2} $$

Separate real and imaginary terms

Separate the real and imaginary terms in the expression: Real terms: $$ ac - (ib)^2 $$ Imaginary terms: $$ adi + ibc $$ Recall that \(i^2 = -1\). Then, we can rewrite the expression as: $$ z=\frac{(ac + b^2) + i(ad + bc)}{c^2 + d^2} $$

Identify the real and imaginary parts

Now that the expression is in standard form, we can identify the real and imaginary parts: Real part (Re(z)): $$ \operatorname{Re}(z)=\frac{ac + b^2}{c^2 + d^2} $$ Imaginary part (Im(z)): $$ \operatorname{Im}(z)=\frac{ad + bc}{c^2 + d^2} $$

Key concepts

Complex Conjugate

The complex conjugate is a fundamental concept when dealing with complex numbers. It helps in simplifying expressions, especially when a complex number is in the denominator. If you have a complex number of the form
\( c + id \), its complex conjugate would be
\( c - id \). To find the conjugate, simply change the sign of the imaginary part. This operation is incredibly useful because it transforms the denominator into a real number when you multiply it with its conjugate. This makes further calculations easier.
By multiplying a complex number by its conjugate, significant simplifications can be achieved. Generally, using the conjugate to eliminate the imaginary part of the denominator allows us to find meaningful real and imaginary components in the solution.

Real Part and Imaginary Part

Understanding the real and imaginary parts of a complex number is essential. A complex number is usually expressed in the form \( a + ib \), where \( a \) is the real part and \( b \) is the imaginary part.
To extract these parts from a fraction involving complex numbers, you may need to simplify the expression first, as seen in the step-by-step solution.
Once simplified, the expression can be represented as

• Real part: \(\operatorname{Re}(z) = \frac{ac + b^2}{c^2 + d^2}\)
• Imaginary part: \(\operatorname{Im}(z) = \frac{ad + bc}{c^2 + d^2}\)

By breaking the complex expression into these two parts, it becomes easier to understand and handle in mathematical computations. The separation into real and imaginary components is essential in many areas of engineering and physics because it allows one to analyze and interpret the behavior of systems modeled by complex numbers.

Simplification of Complex Fractions

When faced with a complex fraction, simplification is key.
The original expression \( z = \frac{a + ib}{c + id} \) can be unwieldy, but it becomes much simpler after using the conjugate.
First, multiply both the numerator and denominator by the conjugate of the denominator. This step is crucial as it transforms the denominator into a non-complex number \( c^2 + d^2 \), reducing the equation to a simpler format. Then the numerator becomes \( (ac + b^2) + i(ad + bc) \).
Such simplification helps to easily identify the real and imaginary parts of the complex number, and these simpler forms are often much easier to work with in further calculations or applications, such as signal processing in electrical engineering or complex number theory in mathematics.

Mathematical Problem Solving

Problem-solving in mathematics, especially with complex numbers, involves strategic manipulation of expressions. One essential skill is knowing when and how to use tools like the complex conjugate to simplify expressions.
In our problem, we applied the complex conjugate to rid the denominator of its complex nature.
Let's outline the process:

• Identify the problem: Find the real and imaginary parts.
• Determine the solution approach: Multiply by the complex conjugate.
• Simplify step-by-step: Expand and simplify expressions.
• Conclude with clear answers for real and imaginary components.

Mastering these steps in mathematical problem solving, especially when complex numbers are involved, helps in tackling a wide range of problems not only in math but also in various scientific fields. Developing this skill set takes practice and patience but is incredibly rewarding.