Problem 96 - 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: $$ \overline{v-w} $$

Step-by-Step Solution

Answer

\( \overline{v-w} = 6i \)

Step 1: Subtract the Complex Numbers

To find \( \overline{v-w} \), we first need to subtract the two complex numbers \( v = 1 - 5i \) and \( w = 1 + i \). Perform the subtraction as follows:\[ v - w = (1 - 5i) - (1 + i) \].Simplify this expression by subtracting the real and imaginary parts separately:\[ v - w = (1 - 1) + (-5i - i) = 0 - 6i \].

Step 2: Find the Conjugate

Now that we have \( v - w = -6i \), the next step is to find the conjugate of this complex number. The conjugate of a complex number \( a + bi \) is given by \( a - bi \).For \( -6i \), the real part is 0 and the imaginary part is -6. Thus, the conjugate is:\[ \overline{v-w} = 0 + 6i = 6i \].

Key Concepts

Complex Conjugate

A complex conjugate is a concept stemming from the realm of complex numbers. Complex numbers consist of a real and an imaginary part, and are typically represented as \(a + bi\), where \(a\) is the real component, and \(bi\) is the imaginary part, involving \(i\) which is the square root of -1.
To find the conjugate of a complex number, you only need to change the sign of the imaginary part. Thus, the conjugate of \(a + bi\) is \(a - bi\).
This operation is particularly useful in complex number algebra, as conjugates are used to simplify division and can help in removing imaginary components from denominators.

• Example: The conjugate of \(3 + 4i\) is \(3 - 4i\).
• If the imaginary part is purely imaginary like \(-6i\), its conjugate is just \(6i\).

Understanding complex conjugates helps in grasping how complex numbers interact and allows for deeper insights into complex number calculations.

Subtraction of Complex Numbers

Subtracting complex numbers is a straightforward process that involves handling the real and imaginary components separately.
Consider two complex numbers \(z_1 = a + bi\) and \(z_2 = c + di\). The subtraction \(z_1 - z_2\) is performed as follows:

• Subtract the real parts: \(a - c\).
• Subtract the imaginary parts: \(bi - di\).

This yields the result \((a - c) + (b - d)i\).
Using this method keeps the operations organized and helps ensure each component is accurately handled separately from the other.
For instance, in the original exercise, the subtraction \(v - w = (1 - 5i) - (1 + i)\) was calculated as \(0 - 6i\) by subtracting the respective real parts and the respective imaginary parts.

Imaginary Numbers

Imaginary numbers are a fascinating component of complex mathematics and are key to understanding complex numbers overall. These numbers are based around the symbol \(i\), defined by the property that \(i^2 = -1\). This allows mathematicians to work with the square roots of negative numbers, which are not possible in the realm of real numbers.
Imaginary numbers are typically expressed in the form \(bi\), where \(b\) is a real number coefficient. These numbers play a crucial role in various fields such as electrical engineering, signal processing, and quantum physics.

• In complex numbers, the imaginary part contributes significantly to the robustness of solutions and their interpretations.
• For instance, in the expression \(-6i\), used in the exercise, the entire value is purely imaginary, meaning it has no real component.

Grasping the significance of imaginary numbers helps students understand more complex systems and provides a foundation for more advanced study in mathematics and engineering.