# What are complex numbers?

As you may know, you can’t add or subtract two different variables to get one. For example, 4a + 3a = 7a but 4a + 3b is still equal to 4a + 3b. In the case of the sum or difference of an imaginary number and a real number, you get a complex number:

but

and, more in general, a complex number is written as follows:

where a and b are real numbers.

Let’s now introduce the complex plane:

As you can see, instead of “x” we have Re (real) and instead of “y” Im (imaginary). Also, the point a+bi has coordinates (a; b), and since b is on the Im axis is an imaginary number, bi. This is why a+bi is between the two axis: it’s got a real part, a, and an imaginary part, b.

Re and Im functions

What is the real part of 4-3i? And what’s the imaginary part? Simply put, the former is the number without the imaginary unit i, the latter is the number that has the i next to it. The function which gives as output the real part of a complex number is Re(), whereas for the imaginary part it is Im().

You can use ( ) instead of { }, but I prefer the latter. It also looks cooler, doesn’t it?

Examples: