Question

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

Short answer

The complex conjugate of \(2i\) is \(-2i\).

Step-by-step solution

Understand Complex Conjugate

The complex conjugate of a complex number is obtained by changing the sign of the imaginary part. For any complex number in the form of \( a + bi \), the complex conjugate is \( a - bi \).

Identify the Real and Imaginary Parts

For the given complex number \( 2i \), the real part \(a\) is 0, and the imaginary part \(bi\) is \(2i\).

Change the Sign of the Imaginary Part

Change the sign of the imaginary part to find the complex conjugate. So, the imaginary part \(2i\) will become \(-2i\).

Write the Complex Conjugate

The complex conjugate of \(2i\) is \(-2i\).

Plot the Complex Number and Its Conjugate

On a complex plane, plot the complex number. For \(2i\), place a point at (0, 2), and for its conjugate \(-2i\), place a point at (0, -2).

Key concepts

Complex Numbers

Complex numbers are numbers that include both a real part and an imaginary part. They are usually written in the form \(a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part. The imaginary unit \(i\) is defined as the square root of \(-1\), so \(i^2 = -1\). Complex numbers are incredibly useful in various fields of mathematics, engineering, and physics because they provide a way to calculate and understand phenomena that cannot be explained using only real numbers. For example, in electrical engineering, complex numbers are used to analyze AC circuits.

Imaginary Unit

The imaginary unit, denoted as \(i\), is a fundamental concept in complex numbers. It is defined as \(i^2 = -1\). This means that \(i\) is the square root of \(-1\), which cannot be represented on the real number line. The imaginary unit allows us to extend the real numbers into the complex plane, where each point represents a unique complex number. In the exercise, the given complex number is \(2i\). Here, the real part is \(0\) (since it is not mentioned) and the imaginary part is \(2i\). The complex conjugate of \(2i\) is found by changing the sign of the imaginary part, resulting in \(-2i\).

Complex Plane Plotting

Plotting complex numbers on the complex plane helps visualize them better. The horizontal axis (x-axis) represents the real part of the complex number, while the vertical axis (y-axis) represents the imaginary part. To plot a complex number like \(2i\), locate the point where the imaginary part intersects the y-axis (0, 2). For its complex conjugate \(-2i\), locate the point where the new imaginary part intersects the y-axis (0, -2). This makes it easier to understand their relationships and perform operations like addition, subtraction, or finding the magnitude. By plotting the complex number and its conjugate, you can see their symmetrical positions relative to the real axis (x-axis).