Problem 97 - 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

The value of \( \overline{vw} \) is \( 6 + 4i \).

Step 1: Multiply the Complex Numbers

First, we need to compute the product of the complex numbers \(v\) and \(w\). Given \(v = 1 - 5i\) and \(w = 1 + i\), the product is calculated as follows:\[vw = (1 - 5i)(1 + i) = 1 \cdot 1 + 1 \cdot i - 5i \cdot 1 - 5i^2\].Simplifying this, we get:\[ vw = 1 + i - 5i + 5 = 6 - 4i\].

Step 2: Find the Conjugate

Now, let's determine the conjugate of the product \(vw\). The conjugate of a complex number \(a + bi\) is \(a - bi\). So, for \(6 - 4i\), the conjugate is:\[\overline{vw} = 6 + 4i\].

Key Concepts

Complex Product

When working with complex numbers, a complex product involves multiplying two complex numbers together. Each complex number is in the form of \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary coefficient of the imaginary unit \(i\). The imaginary unit \(i\) is defined such that \(i^2 = -1\). To multiply two complex numbers, you apply the distributive property just like you would with binomials.
For example, consider two complex numbers \(v = 1 - 5i\) and \(w = 1 + i\). Using the distributive property, you multiply each term in \(v\) by each term in \(w\):

• \((1)(1) = 1\)
• \((1)(i) = i\)
• \((-5i)(1) = -5i\)
• \((-5i)(i) = 5i^2 = 5(-1) = -5\)

Combine these values: \(1 + i - 5i - 5\). Simplify by combining like terms, resulting in \(6 - 4i\). Hence, the complex product \(vw = 6 - 4i\).

Calculated this way, multiplying complex numbers transforms them in both modulus (size) and argument (direction in the complex plane).

Complex Conjugate

The complex conjugate of a complex number is formed by changing the sign of its imaginary part. For a complex number \(a + bi\), the conjugate is \(a - bi\). This concept is useful in many areas of mathematics, including simplifying division when dividing by a complex number. Conjugates are used because they help to eliminate the imaginary part when two conjugates are multiplied together.

Taking our product \(vw = 6 - 4i\) as an example, its conjugate is \(\overline{vw} = 6 + 4i\). Notice how only the sign in front of the imaginary unit changes. Conjugates have interesting properties, such as the product of a complex number and its conjugate is always a real number. For instance, \((6 - 4i)(6 + 4i) = 36 + 24i - 24i - 16i^2\) leads to \(36 + 16 = 52\), which is purely real.

Conjugates also help in solving equations and simplifying complex expressions, as we will see when combining them with their original counterparts.

Imaginary Unit

The imaginary unit, denoted by \(i\), is a fundamental part of complex numbers. It is defined such that \(i^2 = -1\). This unique property enables mathematicians to extend the real number system to include solutions to equations that would otherwise have no real solutions, such as \(x^2 + 1 = 0\).

In the context of complex numbers, \(i\) allows us to express numbers in the form \(a + bi\), where \(a\) and \(b\) are real numbers. This format permits a broad range of mathematical operations:

• Adding and subtracting complex numbers involves combining like terms, maintaining the real parts and the imaginary parts separately.
• When multiplying, employ the fact that \(i^2 = -1\) to simplify expressions.

When solving problems involving complex numbers, recognize \(i\) as an essential component. It acts as a bridge between real and imaginary aspects, making the exploration of two-dimensional number spaces possible and intuitive. This understanding of \(i\) plays a significant role in fields such as electrical engineering and quantum physics, where complex numbers model wave behavior and other phenomena.