Problem 94 - Calculus for Biology and Medicine

Question

Let z \(=3-2 i\), \(u=-4+3 i, v=3+5 i\), and \(w=1-i .\) Compute the following expressions: \(z+u\)

Step-by-Step Solution

Answer

The sum is \(-1 + i\).

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

The given complex numbers are \(z = 3 - 2i\) and \(u = -4 + 3i\). Identify the real and imaginary parts of each. For \(z\), the real part is 3 and the imaginary part is -2. For \(u\), the real part is -4 and the imaginary part is 3.

Step 2: Add the Real Parts

Add the real parts of \(z\) and \(u\). This means adding 3 (real part of \(z\)) to -4 (real part of \(u\)). So, the result is \(3 + (-4) = -1\).

Step 3: Add the Imaginary Parts

Add the imaginary parts of \(z\) and \(u\). This means adding -2 (imaginary part of \(z\)) to 3 (imaginary part of \(u\)). So, the result is \(-2 + 3 = 1\).

Step 4: Write the Result as a Complex Number

Combine the results of Step 2 and Step 3 to form the complex number. The real part is -1 and the imaginary part is 1. Therefore, the sum \(z + u = -1 + 1i\).

Key Concepts

Addition of Complex Numbers

Adding complex numbers is similar to adding simple numerical values, but with the additional step of accounting separately for both real and imaginary components. For any two complex numbers, say, \(a + bi\) and \(c + di\), the procedure is quite straightforward:

• Add the real parts: \(a + c\)
• Add the imaginary parts: \(b + d\)
• Combine these results to form a new complex number: \((a + c) + (b + d)i\)

In our example with \(z = 3 - 2i\) and \(u = -4 + 3i\), the addition is done by separately calculating the sums of the real parts (3 and -4) and the imaginary parts (-2 and 3). This yields \(-1\) as the real part and \(1\) as the imaginary part, giving the final result \(-1 + 1i\). When performing these calculations, always ensure each part is accurately combined to form the complete expression.

Real and Imaginary Parts

Complex numbers consist of two distinct components: the real part and the imaginary part. These components can be visualized in the form \(a + bi\), where \(a\) is the real part, and \(b\) is the imaginary part multiplied by the imaginary unit \(i\), which is defined as \(\sqrt{-1}\).

• Real part (\(a\)): This behaves like any real number you are used to dealing with in standard arithmetic.
• Imaginary part (\(b\)): This introduces a new dimension to numbers since \(i\) cannot be measured on the same axis as real numbers. It represents the imaginary axis in the complex plane.

In the example, with the number \(z = 3 - 2i\):
- The real part is 3.
- The imaginary part is -2.
Distinguishing these parts helps simplify operations like addition, subtraction, or multiplication, allowing each part to be handled separately and ensuring accuracy in final results.

Complex Number Arithmetic

Complex number arithmetic involves operations like addition, subtraction, multiplication, and division. Despite the additional components, the rules are not entirely different from those used with real numbers, just broader to include the imaginary unit.

When it comes to complex arithmetic:

• For **addition and subtraction**, you combine real parts with real parts and imaginary parts with imaginary parts.
• **Multiplication** involves applying the distributive property: multiply each part of one complex number by each part of the other, while remembering that \(i^2 = -1\).
• **Division** might involve more steps, requiring the conjugate of the denominator to simplify expressions.

The key to mastering complex number arithmetic is practice, as it ensures you become familiar with handling the imaginary unit while performing operations. In practical applications, these skills are valuable for solving equations, analyzing signals in engineering, or even interpreting dynamics in physics.