Goodday guys and gals! In a previous post I defined what complex numbers actually are. Make sure to check that one out before you dive into this one. In this article I will explain how to do arithmetic on complex numbers. This means: addition, subtraction, multiplication and division. In particular, I will talk about addition and subtraction here, and in part two of this post the other ones. But it’s all pretty easy actually, addition and subtraction is pretty straightforward, and we have formulas for the other guys. So, I am sure that you will get it, otherwise it is always a good idea to reread stuff or look up concepts that you don’t understand full yet. Here’s the video, and a summary underneath. Enjoy my friend!

## Summary: Arithmetic on Complex numbers

### Addition

To begin, we have $i + i$, which is of course $2i$, Furthermore, $i + i +1 = 3i$. And so on. Subtraction works in a similar fashion, you know how this works. But until now, we have only been doing math on the purely imaginative part of a complex number. Complex numbers do not have to be purely imaginative. They can also have a real part!

So, complex numbers typically have a real part and an imaginary part. We write this as $a + bi$ where $a$ and $b$ are real numbers. So it is a real number plus a real number multiplied by the imaginary unit $i$. The left part is called the real part, and the right part is called the imaginary part. Of course, if $a = 0$ and $b = 1$ , we have $0 + 1i = i$. That was the special case we were considering before, but now we know that complex numbers can be much more diverse than this.

Let’s return to addition. Let’s say we have one complex number $z$, which consists of a real part and an imaginary part, let’s call it $x + yi$. I just changed the letters a bit, but the principle is the same as explained before. We now have a complex number, we call it $z$, and we say that its components are $x + yi$ . So this thing is $z$. Let’s take another complex number, call it $w$. Then $w = u + vi$. Okay, if we have these two complex numbers, we can define addition as follows.

$z+w = (x+yi)+(u+vi)=(x+u)+(y+v)i$

### Practice Examples

Let’s say that $z=2+3i$ and that $w=1+2i$. What is the answer to $z+w = ?$. If you want you can try this out yourself before I provide the answer. In fact, I strongly recommend you to do this. Mathematics is not a spectator sport, if you really want to become skilled in this you should try to build your intuition by solving problems yourself. This will train your brain in active problem solving, and this will improve your ability to solve problems in all aspects of your life. Doing math is lots of fun and really good for you. It’s like playing around with solving problems. And when you face a problem for real in life, then you will have had enough practice. But solving this problem would be really easy, you just need to plug in the values into the formula and compute the answer.

Okay, I will now provide the answer. $z+w = (2+3i)+(1+2i)=(2+1)+(3+2)i=5+5i$. Easy right? Let’s try another one.

If $z=20+9i$ and $w=5+17i$, what is $z+w?$

Answer: $z+w = (20+9i)+(5+17i)=(20+5)+(9+17)i=25+26i$.

### Subtraction

Great, now we understand addition, let’s look at subtraction. The principle is similar. In general, we have the formula:

$z-w = (x+yi)-(u+vi)=(x-u)+(y-v)i$

The place of the letters is the same, only we replaced some plusses with some minuses. So, similarly, if we subtract one complex number from another, we just plug in the values in this formula, and then compute the calculation and then we are done. Let’s try some examples again.

If $z=4+3i$ and $w=3+1i$, what is $z-w?$

Answer: $z-w = (4+3i)-(3+1i)=(4-3)+(3-1)i=1+2i$.

Let’s try another one. What is $z-w$ if $z=8+11i$ and $w=10+15i?$.

Answer: $z-w = (8+11i)-(10+15i)=(8-10)+(11-15)i=-2-4i$.

Hope you learned how to do this, stay tuned for our next post on how to do multiplication and division with complex numbers. See ya!