Problem 94 - Calculus for Biology and Medicine

Question

Let \(z=1+2 i, u=2-3 i, v=1-5 i\), and \(w=1+i\). Compute the following expressions: $$ z+u $$

Step-by-Step Solution

Answer

The expression \( z + u \) equals \( 3 - i \).

Step 1: Identify the Real and Imaginary Parts of z and u

The complex number \( z \) is given as \( z = 1 + 2i \), where the real part is 1 and the imaginary part is 2. Similarly, \( u = 2 - 3i \), where the real part is 2 and the imaginary part is -3.

Step 2: Add the Real Parts

Add the real parts of \( z \) and \( u \): \( 1 + 2 = 3 \). This gives the real part of the resulting complex number.

Step 3: Add the Imaginary Parts

Add the imaginary parts of \( z \) and \( u \): \( 2 + (-3) = -1 \). This gives the imaginary part of the resulting complex number.

Step 4: Combine the Results

Combine the results of steps 2 and 3 to form the resulting complex number. Therefore, \( z + u = 3 - i \).

Key Concepts

Real and Imaginary Parts

Complex numbers are composed of two parts: a real part and an imaginary part. For example, in the complex number \( z = 1 + 2i \), the real part is 1 and the imaginary part is 2.
The imaginary part is paired with the imaginary unit \( i \), which represents \( \sqrt{-1} \).

Understanding the distinction between the real and imaginary components of a complex number is crucial for comprehending their usage in mathematical operations such as addition.
These distinctions allow you to independently manipulate the real and imaginary portions, much like how you would handle x and y coordinates in the Cartesian plane.

• The real part of a complex number, \( a + bi \), is \( a \).
• The imaginary part is \( b \), and you write it as \( bi \).

Keeping these parts separate in operations helps maintain clarity and ensures accurate results.

Addition of Complex Numbers

Adding complex numbers follows the principle of combining like terms, which you may remember from adding binomials in algebra. Imagine adding \( z = 1 + 2i \) and \( u = 2 - 3i \).
To do this, you add the real parts together and the imaginary parts together.

In our example:

• Add the real parts: \( 1 + 2 = 3 \).
• Add the imaginary parts: \( 2 + (-3) = -1 \). This result gives us a new imaginary part: \(-1i\).

The final sum of the complex numbers is \( 3 - i \). It's straightforward and involves handling real numbers and imaginary numbers separately during addition.
This separation allows for a simplified calculation process that can easily be applied to any pair of complex numbers.

Step by Step Solution

Let's walk through the example, \( z + u \), with deliberate focus to embrace how step-by-step methods can simplify complex number addition.
The process incorporates breaking down the addition into manageable actions.

Start by examining each number individually:

• Identify \( z = 1 + 2i \). Real: 1, Imaginary: 2.
• Identify \( u = 2 - 3i \). Real: 2, Imaginary: -3.

Next, add the real parts: 1 plus 2 equals 3.
Then, add the imaginary parts: 2 plus -3 equals -1.
Finally, combine the results to form the complex number \( 3 - i \).

Using this step-by-step approach not only avoids confusion but also ensures clarity in every stage of your calculation.

• This methodical approach is valuable when dealing with more complex expressions involving multiple complex numbers.
• It helps highlight errors and maintain focus on each component separately.

This understanding is not only foundational but also strengthens problem-solving skills in more advanced applications.