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Definition of complex numbers

Complex numbers are the sum of real and imaginary numbers (and you thought it ended at combining numbers and letters). A complex number will take the standard polar form: $$a + bi$$ Where $a$ and $b$ are real numbers and $i = \sqrt{-1}$ on the complex plane. It's a slightly different number system.

Here are some examples of complex numbers: $$3 + 4i$$ $$17 – 9i$$ $$\frac{12-4i}{5}$$

Now, there is one interesting fact about complex numbers. That fact is that integers are also complex numbers! For example, $3$ is a complex number because it can be written as $3 + 0i$. You can also have negative complex numbers!

Purely Imaginary vs. Real Numbers

We have become acquainted with real numbers, these are numbers that exist on the real number line. For example, $4$, $\frac{7}{9}$, and $\pi$ are all real numbers because they show up on the real number line.

A purely imaginary number is a multiple of $i$. So, $4i$, $\frac{5}{2}i$ and $\sqrt{17}i$ are all purely imaginary numbers, which are also known as non-real numbers.

Now, real and purely imaginary numbers combine to form complex numbers. So, this means that all imaginary numbers are also complex numbers. This also means that all real numbers are complex numbers!

For example: $3 + 0i$ is equal to $3$, but it is also a complex number because it has an imaginary part. Granted, the imaginary part is equal to $0$, but it is still there. Conversely, $0 + 7i$ is a complex number that is equal to $7i$. The real part is equal to $0$, but it is still there.

Break It Down Now

We now know that complex numbers can be written as $z = a + bi$. We know what $a$ and $b$ are, let's think about what they represent.

$a$ is the real part of the complex number, $z$. We can also write this as $$\mathbf{Re}(z) = a$$ $b$ is the imaginary part of the complex number, $z$. We can also write this as $$\mathbf{Im}(z) = b$$

Note: Both parts of the complex number are real numbers. So if someone asks for the imaginary part of complex number $z = x + yi$, they are asking for the real number $y$, NOT $yi$. However, $yi$ is an imaginary number, but it is not the imaginary part of the complex number.

Simplifying complex numbers often requires understanding powers of imaginary numbers. Non-real roots of polynomials occur in conjugate pairs, as described by the conjugate root theorem.

(Video) Basics of Complex Numbers

by Don't Memorise

In this video by Don't Memorise, you will learn about the basics of complex numbers.

Summary

We previously learned about imaginary numbers, which is represented by $i$.

$$i = \sqrt{-1}$$

We can have instances where we have $bi$, where $b$ represents a real number. That allows us to have terms such as $2i$.

Let's take this a step further. Say we have the expression, $3+7i$. The entire expression is not imaginary, even though there is an $i$ term in the expression. We know that the term $7i$ is imaginary, but $3$ is a real number. That means that the expression $3+7i$ is a combination of a real and imaginary number.

That's what a complex number is! A complex number is an expression with a real number and an imaginary number.

$$\text{real number} + \text{imaginary number} = \text{complex number}$$

Oftentimes, complex numbers take the form: $a+bi$.

Watch the video to learn about some of the cool real-world examples of when we use complex numbers!

Complex Numbers

Complex numbers are a combination of real and imaginary numbers in the standard form, $a+bi$, where $a$ and $b$ are real numbers, and $i$ is an imaginary number of value $\sqrt{-1}$.

Looking Further

In future lessons, you will learn about multiplying complex numbers, dividing complex numbers, complex ordered pairs, and complex conjugates.