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# Intro to complex number conjugates

## Video transcript

I want to make a quick clarification and then add more tools in our complex number toolkit in the first video I said that if I had a complex number Z and it's equal to a plus bi I use the word and I have to be careful about that word because it you know I used it in kind of the everyday sense but it also has a formal reality to it so clearly the real part of this of this complex number is a clearly that is the real part and clearly this complex number is made up of a real number plus an imaginary number so I just kind of talking in everyday terms I called this the imaginary part I called this imaginary number this imaginary number the imaginary part but I want to I want to just be careful there I mean I did make it clear that if you were to see the function the real part of Z this would spit out the a and the function the imaginary part of Z this would spit out and we talked about this in the first video it would spit out the number that's scaling the I so it would spit out the B so if someone is talking in the formal sense about the imaginary part they're really talking about the number that is scaling the eye but in my brain when I think of a complex number I think of it having a real number and an imaginary number and if someone would say well what part of that is the imaginary number I would have given this whole thing but if someone says just what's the imaginary part or they give you this function just give them the B hopefully that clarifies things frankly I think the word imaginary part is badly named because clearly this is this whole thing is an imaginary number this right here is not an imaginary number it's just a real number it's the real number scaling scaling the I so they should call this the number scaling the imaginary part of Z anyway with that said what I want to introduce you to is the idea of a complex numbers conjugate so if this is Z the conjugate of Z the conjugate of Z it'd be denoted with Z with a bar over it sometimes it's Z with a little asterisk right over there that would just be equal to a a minus bi so let's look how let's see how they look on an Argand diagram so that's my real axis my real axis and then that is my imaginary axis and then if I have Z this is Z over here this height over here is B this base or this length right here is a that's Z the conjugate of Z is a minus bi so it comes out a and the real axis but it has minus B as its imaginary part so just like this so just like this so this is the conjugate of Z so just to visualize it it really is the conjugate of a complex number is really that the mirror image of that complex number reflected over the x-axis you can imagine if this was a pool of water we're seeing its reflection over here and so we can actually look at this to visually add the two cottony the number to add the complex number and its conjugate so we said these are just like position vectors so if we were to add Z and its conjugate we could essentially just take this vector shift it up here do heads to tails shift it up here and do heads to tails so this right here we are adding Z to its conjugate and so this point right here this point right here or the vector that specifies that point is Z plus Z's conjugate and you can see right here just visually this is going to be 2a this is going to be 2a and we could do that algebraically if we were to add Z if we were to add Z that's a plus bi and add that to its conjugate so plus a minus bi what are we going to get these two guys cancel out we're just going to have to a or another way to think about it and really we're just playing around with math if I at take any complex number if I take any complex number and to it I add its conjugate I am going to get 2 times 2 times the real part of the complex number and this is also going to be 2 times the real part of the conjugate because they have the exact same real part now with that said let's think about where the conjugate could be useful where it could be useful so let's say I had something like 1 plus one plus 2 I divided by 4 minus 5i 4 minus 5x so it's no real obvious way to simplify this expression maybe I don't like having this I in the denominator maybe I just want to write this as one complex number if I divide one complex number by another I should get another complex numbers but how do I do that well one thing to do is to multiply the numerator and the denominator by the conjugate of the denominator so 4 plus 5i over 4 plus 5i and clearly I'm just multiplying by one because this is the same number of the same number but the reason why this is valuable is if I multiply a number times its conjugate I'm going to get a real number so let me just show you that here so let me let's just multiply this out so we're going to get 1 times 4 plus 5i is 4 plus 5i and then 2i times 4 is plus 8i and then 2i times 5i that would be 10 I squared or negative 10 and then we that will be over now this has the form a minus B times a plus B so that's that's the product of well a plus B times a minus B is a squared minus B squared so it's going to be equal to 4 squared which is 16 minus 4 squared I wanted to have 4 plus 4i here this would be 4 plus 5i what am i doing 4 plus 5i the same number over the same number and this was a this was a 10 right over there this is the conjugate I don't know my brain must have been thinking in 4 so obviously I don't want to change the number 4 plus 5i over 4 plus 5i so let's multiply this is a minus B times a plus B so 4 times 4 so it's going to be 4 squared minus 5i squared 4 squared minus 5i squared and so this is going to be equal to 4 minus 10 let's add the real parts 4 minus 10 is negative 6 5i plus 8i is 13 I add the imaginary parts and then you have 16 minus 5i weird well five I squared I squared is negative one five squares this would be negative twenty five the negative and the negative cancel out so you have 16 plus 25 so that is 41 so we can write this as a complex number this is negative six over 41 plus 13 over 41 I we were able to divide these two complex numbers so the useful thing here is the property that if I take any if I take any complex number and I multiply it any complex number and I multiply it by its conjugate and obviously the conjugate of the conjugate is the number is the original number but I take any complex number and I multiply it by its conjugate so this would be a plus bi times a minus bi I'm going to get a real number it's going to be a squared a squared minus bi squared difference of squares which is equal to a squared now this is going to be negative B squared but we have a negative sign out here so they cancel out a squared plus B squared and just out of curiosity this is the same thing as the magnitude of our complex number squared so this is the neat property this is what makes conjugates really useful especially when you want to simplify division of complex numbers anyway hopefully you found that useful