Question

Find and plot the complex conjugate of each number. \(-4 i\)

Short answer

The complex conjugate of (-4i) is (4i). Plot it at (0, 4) on the complex plane.

Step-by-step solution

Understand the Complex Conjugate

The complex conjugate of a number is found by changing the sign of the imaginary part. For a complex number of the form a + bi, its conjugate is a - bi.

Identify the Imaginary Part

In the given number (-4i), there is no real part (a = 0) and the imaginary part is -4 (b = -4).

Find the Complex Conjugate

To find the complex conjugate, change the sign of the imaginary part. For (-4i), the conjugate will be (+4i) or just (4i).

Plot the Original Number

Plot the complex number (-4i) on the complex plane. This is plotted at the point (0, -4) since there is no real part.

Plot the Complex Conjugate

Plot the conjugate (4i) at the point (0, 4) on the complex plane. This corresponds to changing the sign of the imaginary part of the original number.

Key concepts

Imaginary Numbers

Imaginary numbers are numbers that can be written as a real number multiplied by the imaginary unit \(i\), where \(i\) is defined by the property that \(i^2 = -1\).

These numbers allow us to extend the number system beyond the real numbers. For example, the number \(-4i\) is an imaginary number where the imaginary part is \(-4\).

Imaginary numbers are crucial in various fields such as engineering, physics, and computer science. They enable us to solve problems that real numbers cannot.

When combined with real numbers, they form complex numbers, which have a form \(a + bi\), where \(a\) and \(b\) are real numbers.

Complex Plane

The complex plane, also known as the Argand plane, is a way to visualize complex numbers.

In this plane:

• The horizontal axis (x-axis) represents the real part of complex numbers.
• The vertical axis (y-axis) represents the imaginary part of complex numbers.

For example, the complex number \(-4i\) would be represented at the point \((0, -4)\).

The complex plane provides a useful way to visualize operations like addition, subtraction, and especially finding complex conjugates. By plotting a number and its complex conjugate, one can clearly see the symmetry across the real axis.

Plotting Complex Numbers

Plotting complex numbers involves placing them on the complex plane. To plot a complex number:

• Identify its real part and plot it on the x-axis.
• Identify its imaginary part and plot it on the y-axis.

For example, to plot \(-4i\), we:

• Note the real part is \(0\).
• Note the imaginary part is \(-4\).

Thus, it will be plotted at the point \((0, -4)\). To plot its complex conjugate, change the sign of the imaginary part.
For \(-4i\), the conjugate is \(+4i\), which we plot at \((0, 4)\).

This visual representation helps in understanding the nature of complex numbers and their properties.