Problem 111 - Calculus for Biology and Medicine

Question

If \(z=a+b i\), find \(z+\bar{z}\) and \(z-\bar{z}\).

Step-by-Step Solution

Answer

\(z + \bar{z} = 2a\) and \(z - \bar{z} = 2b i\).

Step 1: Identify the components

Given a complex number \(z = a + b i\). Here, \(a\) is the real part and \(b i\) is the imaginary part.

Step 2: Compute the conjugate

The conjugate of \(z\), denoted as \(\bar{z}\), is derived by changing the sign of the imaginary part: \(\bar{z} = a - b i\).

Step 3: Sum of the complex number and its conjugate

Add \(z\) and \(\bar{z}\): \(z + \bar{z} = (a + b i) + (a - b i) = a + b i + a - b i\). Simplifying, we get: \(2a\).

Step 4: Difference of the complex number and its conjugate

Subtract \(\bar{z}\) from \(z\): \(z - \bar{z} = (a + b i) - (a - b i) = a + b i - a + b i\). Simplifying, we get: \(2b i\).

Key Concepts

Conjugate

In the realm of complex numbers, the concept of a conjugate plays a crucial role. A complex number, such as \( z = a + bi \), consists of a real part \( a \) and an imaginary part \( bi \). The conjugate of a complex number is obtained by changing the sign of the imaginary component. This means that if you have \( z = a + bi \), the conjugate \( \bar{z} \) is expressed as \( a - bi \).
This switch in the sign is what defines a conjugate. Relying on conjugates can simplify various algebraic operations with complex numbers, especially when working with their sums, differences or even products and quotients.
When a complex number is added to its conjugate, the imaginary parts are canceled out, leaving only the doubled real part. However, when you subtract the conjugate from the original complex number, the real parts cancel out, leaving a doubled imaginary part. This property makes conjugates exceedingly useful in simplifying and understanding operations with complex numbers.

Real Part

The real part of a complex number is fundamental to its structure. In our example, with \( z = a + bi \), "a" is what we call the real part. This is the component that exists on the real number line, independent of any imaginary influences. The real part provides the horizontal dimension in the complex plane, which is the plane on which all complex numbers can be represented.

• If you plot \( z \) on a graph, \( a \) determines how far along the x-axis (the real axis) the number will appear.
• The real part of a complex number impacts how we perceive its proximity to other numbers and to the center, or origin, of the complex plane.

Understanding the real part is crucial when dealing with addition or subtraction of complex numbers, as any changes will be reflected in this component. Thus, when you add a complex number with its conjugate, the result being \( 2a \) (as seen in the exercise) highlights the role of the real part in these operations.

Imaginary Part

In complex numbers, the imaginary part is equally as important as the real part, despite its seemingly abstract nature. The imaginary part consists of \( bi \), where "b" is a real number and \( i \) is the imaginary unit. The imaginary unit \( i \) is defined by the property \( i^2 = -1 \). This fundamental property distinguishes complex numbers from real numbers.

• The imaginary part gives a complex number its vertical position on the complex plane.
• It represents the influence of the imaginary unit \( i \), enabling calculations that extend beyond the real number system.

In the case of \( z = a + bi \), "b" is responsible for the height of the point above or below the real axis, indicating the number's "imaginary" dimension. This is reflected in the outcome of \( z - \bar{z} \), which results in \( 2bi \). This reinforces the imaginary part's role in defining the complex number's interactions. Understanding this makes it easier to manipulate complex mathematical expressions involving complex numbers.