Question

Find and plot the complex conjugate of each number. \(2-2 t\)

Short answer

The complex conjugate of 2 - 2t is 2 - 2t; it's purely real.

Step-by-step solution

Identify the complex form

The given expression is in terms of a variable, but it can still be written in a form resembling a complex number: a = 2 - 2t + 0i

Compute the complex conjugate

The complex conjugate of a complex number a + bi is a - bi. Here, since there's no imaginary part, we treat the expression in the form a + bi.Therefore, the conjugate is: 2 - 2t + 0i ⇒ 2 - 2t - 0i which simplifies to: 2 - 2t

Plot the complex conjugate

Since the expression does not have an imaginary part (it remains purely real), the plot lies along the real axis in the complex plane.To plot this, mark points for various values of t on the real number line where the real part is 2 - 2t.

Key concepts

complex numbers

Complex numbers are numbers that have both a real part and an imaginary part. They are usually written in the form of \(a + bi\), where \(a\) is the real part, and \(bi\) is the imaginary part. The imaginary part involves the unit imaginary number \(i\), which has the property that \(i^2 = -1\). For example,

• \(3 + 4i\)
• \(-2 - 5i\)
• \(7 + 0i\) (which is just 7)

In the context of the exercise, the expression \(2 - 2t\) can be thought of as a complex number where the imaginary part is zero (\(0i\)). This simplifies our work as we focus mainly on the real part.

real axis

The real axis is a critical concept in understanding complex numbers and their plots. In the complex plane, which has a horizontal axis for real numbers and a vertical axis for imaginary numbers, the real axis is simply the horizontal line where the imaginary part is zero. Everything plotted on this axis is purely real. For example,

• The number 3 on the real axis is \(3 + 0i\).
• The number -2 on the real axis is \(-2 + 0i\).

In our exercise, since the complex conjugate of \(2 - 2t\) remains \(2 - 2t\), it lies purely on the real axis regardless of the value of \(t\). This allows us to focus entirely on the horizontal line while plotting.

plotting complex functions

Plotting complex functions is a way to visualize complex numbers and to understand their behavior better. When we plot a complex number like \(a + bi\), \(a\) is placed on the real axis (horizontal), and \(bi\) is placed on the imaginary axis (vertical).Here’s a simple guide to help you plot complex numbers:

• Identify the real part (\(a\)).
• Identify the imaginary part (\(bi\)).
• Plot the point using \(a\) on the horizontal axis and \(bi\) on the vertical axis.

In our specific exercise, because the numbers have no imaginary part, we plot them purely on the real axis. So, if \(t = 1\), \(2 - 2 \times 1 = 0\). Therefore, for \(t = 1\), plot the point at zero on the real axis.Similarly, try other values for \(t\) and plot the real values on the horizontal line. This helps in visualizing the function’s behavior explicitly.