Question

Find and plot the complex conjugate of each number. $2i-2$

Short answer

The complex conjugate of $2i-2$ is $-2-2i$. Plot $(-2,2)$ for the original number and $(-2,-2)$ for the conjugate.

Step-by-step solution

Understanding Complex Conjugates

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

Identify Real and Imaginary Parts

In the complex number $2i-2$, identify the real part $a$ and the imaginary part $b$. Here, $a=-2$ and $b=2$.

Find the Complex Conjugate

Change the sign of the imaginary part $b$. The complex conjugate of $2i-2$ is $-2-2i$.

Plot the Original Complex Number

Plot the original complex number $2i-2$ on the complex plane. Place a point at the coordinates $(-2,2)$, where $-2$ is the real part and $2$ is the imaginary part.

Plot the Complex Conjugate

Plot the complex conjugate $-2-2i$ on the complex plane. Place a point at the coordinates $(-2,-2)$, where $-2$ is the real part and $-2$ is the imaginary part.

Key concepts

Complex Conjugate

A complex conjugate of a complex number simply involves changing the sign of the imaginary part. If the complex number is represented as $a+bi$, its complex conjugate is $a-bi$. This concept is very useful in various areas of mathematics and engineering.
Let's break down the key steps with an example.
For the complex number $2i-2$:

• The real part, $a$, is $-2$
• The imaginary part, $b$, is $2$

Finding the conjugate means changing the sign of the imaginary part. So, the complex conjugate of $2i-2$ becomes $-2-2i$. Understanding and applying this process is simple when you remember the key step: change the sign of the imaginary part.

Complex Plane

The complex plane, also known as the Argand plane, is a way to visually represent complex numbers. This helps in understanding their behavior and properties.
Each complex number corresponds to a point on this plane:

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

For the complex number $2i-2$, we plot it at coordinates $(-2,2)$, where $-2$ is the real part and $2$ is the imaginary part. Similarly, for its complex conjugate $-2-2i$, we plot it at coordinates $(-2,-2)$. Plotting these points can provide a clearer understanding of their relationship.

Imaginary Part

The imaginary part of a complex number is the component that involves the imaginary unit $i$. In the expression $a+bi$, $bi$ is the imaginary part. Here, $i$ represents the square root of $-1$.
For example, in the complex number $2i-2$, the imaginary part is $2i$.
To find the complex conjugate, we need to change the sign of the imaginary part. So, as we did earlier, the expression $-2-2i$ highlights that we have switched $2i$ to $-2i$.
Understanding the imaginary part and its behavior is crucial for mastering complex numbers. Always remember, it’s just as important as understanding the real part.