Hey guys, we have published yet another cool video on our YouTube channel to help you out with your studies. The topic of today is an introduction to complex numbers. These numbers were shrouded in mystery in the past, many people thought they were too weird to even exist! But now we know that they do exist, and they are just as real as the real numbers. Find out how these numbers work, and some neat properties of it in the video below. You can of course find a summary of the theory underneth it. Enjoy!

## Summary Complex Numbers: Introduction

What are complex numbers? It is a number that involves $\sqrt { -1 }$. But wait, isn’t it impossible to take the square root of $-1$? What number $x$ can we square to yield ${ x }^{ 2 }=-1$?

Well, there is no number in the real number line $\mathbb{R}$ that we can square to get $-1$, but we can extend the real number line by adding some numbers to it. Illustration of the real number line, it has all comma numbers that you can think of.

Let us return to the equation ${ x }^{ 2 }=-1$ So, we cannot find any real number to replace for $x$ , but let us for a second pretend that we can. We replace $x$ by the symbol $i$. We then get ${ i }^{ 2 }=-1$. Here, $i$ represents the imaginary number that when squared, produces $-1$. But then this opens up an entire new arena of numbers, like $2i$, $i+1$, ${ i }^{ 3 }$, etcetera. By adding this imaginary number to the real number line, we extend the numbers by adding all these complex numbers to it. Now, the equation ${ x }^{ 2 }=-1$ suddenly does have solutions. In the real number system it had no solutions.

Now, we know complex numbers exist, let us spend some time to figure out how they work exactly. We know that ${ i }^{ 2 }=-1$. What if we increase the exponent by 1? We get

${ i }^{ 3 }=i\times i\times i= { i }^{ 2 }\times 1=-1\times i=-i$.

What if we add another 1 to the exponent? We get

${ i }^{ 4 }=i\times i\times i\times i= { i }^{ 2 }\times { i }^{ 2 }=-1\times -1=1$.

But wait, one property of taking powers is that any number to the power of 0 is 1. thus, for any ${ x }^{ 0 }$, we have ${ x }^{ 0 }=1$. Hence, ${ i }^{ 0 }=1$. But, we also have that ${ i }^{ 4 }=1$. Therefore, ${ i }^{ 4 }= { i }^{ 0 }=1$.

And then, if we take ${ i }^{ 5 }$, we have ${ i }^{ 4 }\times i=1\times i=i$, and the circle is round! So, we have that ${ i }^{ 0 }= { i }^{ 4 }= { i }^{ 8 }...$. And, ${ i }^{ 1 }= { i }^{ 5 }= { i }^{ 9 }...$

Similarly for ${ i }^{ 2 }$ and ${ i }^{ 3 }$. So basically, if we take this number i to some power, and then add 4 to that power, we end up with exactly the same number all the time. That is a pretty cool property. Illustration of how the powers of i work.

### Further Reading

Find out more about the different number systems we have in this post.

More information: https://www.purplemath.com/modules/complex.htm