Intro to complex numbers

most of your mathematical lives you've been studying real numbers real numbers include things like zero and one and zero point three repeating and PI and E and I could keep listing real numbers these are the numbers that you're kind of familiar with and then we explored something interesting we explored the notion of well what if there was a number that if I squared it I would get the negative one and we defined that thing that if we squared it we got negative one we define that thing is I and so we defined a whole new class of numbers which you could really view as multiples of the imaginary unit so imaginary numbers would be I and negative I and and PI times I and E times I so this might raise another interesting question what if I combined imaginary in real numbers what if I had numbers that were essentially sums or differences of real and imaginary numbers for example let's say that I had the number let's say I call it Z and Z tends to be what we the most-used variable when we're talking about what I'm about to talk about complex numbers let's say that Z is equal to is equal to the real number five plus the imaginary number three times I so this thing right over here we have a real number plus an imaginary number you might be tempted to add these two things but you can't it won't make any sense these are kind of going and different well we'll think about it visually in a second but you can't simplify this anymore you can't add those real number to this imaginary number a number like this and let me make it clear that's real and this is imaginary imaginary a number like this we call a complex number calm a complex number it has a real part and an imaginary part and sometimes you'll see notation like this well if someone will say well what's the real part what's the real part of our complex number Z well that would be the five right over there and then they might say well what's the imaginary part what's the imaginary part of our complex number Z and then they'll tell and then typically the way that this function is defined they really want to know well what multiple of I is this imaginary part right over here and in this case it is going to be it is going to be 3 and we can visualize this we can visualize this in two dimensions instead of having the traditional two-dimensional Cartesian plane with real numbers on the horizontal and the vertical axis what we do to plot complex numbers is we on the vertical axis we plot the imaginary part so that's the imaginary part and on the horizontal axis we plot the real part we plot the real part just like that we plot the real part so for example Z right over here which is 5 plus 3i the real part is 5 so we would go 1 2 3 4 5 that's 5 right over there the imaginary part is 3 1 2 3 and so on the complex plane on the complex plane we would we would visualize that number right over here this right over here is how we would visualize Z on the complex plane it's 5 positive 5 in the real Direction positive 3 in the imaginary direction we could plot other complex numbers let's say we had the complex number a which is equal to let's say it's negative 2 plus I where would I plot that well the real part is negative 2 negative 2 and the imaginary part is going to be you could imagine this is plus 1 I so we go 1 up it's going to be right over there so that right over there is our complex number our complex number a would be at that point of the complex complex let me write that that point of the complex plane and let me just do one more let's say you had the complex number B which is going to be let's say it is let's say it's 4-3 I where would we plot that well 1 2 3 4 and then let's see - 1 2 3 or negative 3 gets us right over there so that right over there would be the complex number be