If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

Main content

# Classifying complex numbers

CCSS.Math:

## Video transcript

now that we know a little bit about the imaginary unit I let's see if we can if we can simplify more involved expressions like this one right over here 2 plus 3i plus 7i squared plus 5i to the 3rd power plus 9i to the 4th power I encourage you to pause the video right now and try to simplify this on your own so as you can see here we have various powers of I you could view this is I to the first power we have I squared here and we already know that I squared I squared is defined to be negative 1 then we have I to the third power I to the third power just be I times this or negative I we already reviewed this when we first introduced the imaginary unit I but I'll do it again I to the fourth power it's just going to be I times this which is the same thing as negative 1 times I that's I to the third power times I again I times I is negative 1 so that's negative 1 times negative 1 which is equal to 1 again so we can rewrite this whole thing we could rewrite it as 2 plus 3i 7i squared is going to be the same thing so I squared is negative 1 so this is the same thing as 7 times negative 1 so that's just going to be minus 7 and then we have 5i to the third power I to the third power is negative I so this could be rewritten as negative I so this term right over here we could write as minus 5i or negative 5i depending on how you want to think about it and then finally I to the fourth power I to the fourth power is just 1 so this is just equal to 1 so this whole term just simplifies to 9 so how could we simplify this more well we have several terms that that are not not imaginary that they are real numbers for example we have this 2 is a real number negative 7 is a real number and 9 is a real number so we could just add those up so 2 plus negative 7 would be nay five negative five plus nine would be four so the real numbers add up to four and now we have these imaginary numbers so three times I minus five times I so if you have three of something and then I were to subtract five of something of that same something from it now you're going to have negative two of that something so or another way of thinking about it is the coefficients 3 minus 5 is negative 2 so 3 is minus 5 is that's going to give you negative 2i now you might say well can we simplify this any further well no you really can't this right over here this right over here is a real number 4 is a number that we've been dealing with throughout our mathematical careers and negative 2i negative 2i that's an imaginary number an imaginary number and so what we really consider this is this this 4 minus 2i this is we can now consider this entire expression to really be a number so this is a number that has a real part and an imaginary part and numbers like this we call complex numbers come it is a complex it is a complex number why is it complex well it has a real part and an imaginary part and you might say well you know gee well it wouldn't get any real number be considered a complex number for example if I have the real number if I have the real number 3 can't I just write the real number 3 is 3 plus 0i and you would be correct this any real number is a complex number you could view this you could view this right over here as a complex number and actually the real numbers are a subset of the complex numbers likewise imaginary numbers are a subset of the complex numbers for example you could rewrite I as a real part 0 as a real number 0 plus plus I so the imaginary ZAR subset of complex numbers real numbers are subsets of complex numbers and then complex numbers also have all of the the sums and differences or all the numbers that have both real and imaginary parts