Problem 95 - 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: $$ \overline{z+v} $$

Step-by-Step Solution

Answer

\( \overline{z+v} = 2 + 3i \)

Step 1: Find the sum of z and v

To find the sum of \( z \) and \( v \), we need to add their real parts and imaginary parts separately.\( z = 1 + 2i \) and \( v = 1 - 5i \).\[ z + v = (1 + 2i) + (1 - 5i) = (1 + 1) + (2i - 5i) = 2 - 3i \]So, \( z + v = 2 - 3i \).

Step 2: Compute the conjugate of the sum

The conjugate of a complex number \( a + bi \) is \( a - bi \). We need to find the conjugate of \( 2 - 3i \), which we obtained from Step 1.\[ \overline{2 - 3i} = 2 + 3i \]Thus, the conjugate of \( z + v \) is \( 2 + 3i \).

Key Concepts

Complex Conjugate

The complex conjugate is a fundamental concept in complex numbers. If you have a complex number given by \( a + bi \), its complex conjugate is simply \( a - bi \). This means you change the sign of the imaginary part while keeping the real part unchanged. For instance, the complex conjugate of \( 2 - 3i \) is \( 2 + 3i \). This transformation helps in simplifying calculations, especially when dividing complex numbers, as multiplying a complex number by its conjugate yields a real number.

Using the conjugate in practical problems is simple but important. To verify, you can imagine subtracting the imaginary part from itself in multiplication, which cancels out the imaginary components. It's a vital tool to grasp when performing operations with complex numbers, ensuring results can be expressed without imaginary units.

Addition of Complex Numbers

Adding complex numbers involves combining their real parts and imaginary parts separately. This is a straightforward operation. Let’s consider two complex numbers, say \( x + yi \) and \( u + vi \). When you add them, you simply combine like terms:

• Combine the real parts: \( x + u \)
• Combine the imaginary parts: \( yi + vi \)

So the result of addition will be \( (x + u) + (y + v)i \).

In our exercise, adding \( z = 1 + 2i \) and \( v = 1 - 5i \) means we add the real parts \(1 + 1\) to get \(2\), and then add the imaginary parts \( 2i - 5i \) to get \(-3i\). The sum \( 2 - 3i \) is obtained as a direct result of this addition process.

Real and Imaginary Parts

Understanding real and imaginary parts of complex numbers is key to working with them effectively. In a complex number \( a + bi \):

• The real part is \( a \)
• The imaginary part is \( b \)

These parts are added to form a complex number, which can be visualized on the complex plane.

The real part corresponds to the horizontal axis, and the imaginary part represents the vertical axis. Complex numbers like \( 1 + 2i \) have a real part of 1 and an imaginary part of 2.

In computation, these parts help in operations such as addition and subtraction, where matching components are dealt with separately. This division into real and imaginary parts simplifies handling the intricacies of computations, making it easier to follow and manage complex arithmetic.