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Complex Conjugates and Rationalizing the Denominator

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why we've been learning about complex numbers and imaginary numbers we learned about complex conjugates and so I want to talk briefly about what complex conjugates are and why do we care so complex tickets are pairs of complex numbers where the form is a + bi and a minus bi and so the legendary part of the complex number has the opposite signs in the two conjugates so let's look at an example of the complex conjugate of that would be 5-3 I noticed that the egg doesn't signs only the coefficient changes signs so if I had something like -4 + 2i the complex conjugate of that would be -4 - 2 so again the coefficient of a doesn't change if I had something like 3-7 I then the complex conjugate of that would be 3 + 7 I and if I had something where the complex number was not written in that standard form so that the I was at the front let's I had I plus 5 so really that I part of this expression should change signs and become negative I and the five would remain positive so if I rewrote this in standard form it would be 5 + I and if I rewrote this one in it would be 5-5 which makes it a little more obvious so why do we even care about complex conjugates in the first place while we learned that if we multiply complex conjugates together something cool all of the eyes go away so let's just take away all of these except for the top one and let's see what happens when you multiply these complex conjugates together so I'm going to actually multiply these together using foil so here goes I'm going to multiply 5 x 5 which is 25 and then 5 x -3 I which is -15 I 3 i x 5 which is positive 15 I a negative 3i x 3i which is -9 pi squared now we see that the two terms have the same purpose it was opposite sign so they basically are going to eliminate eliminate each other we have negative 59 and positive 15 on I-90 so now we just have the 25 and the negative 9 I squared remember that the value of I squared is -1 so we can replace the I squared with negative one and ends up getting us a positive 9 I was going to add together 25 and 9 to get 34 so it's nice when we multiply complex conjugates because the imaginary part of the number goes away and we get a real number so we will use complex conjugates to rationalize the denominator of a fraction that has an eye on the bottom so let's look at that now have a rational expression something that has a fraction in it and it has an eye on the bottom I don't want that because that's like having a radical on the bottom so let's look at 4 / 3 - 2i again I represent the square root of -1 so I basically have something that's like a square root on the bottom of my fraction and that's not allowed so I can eliminate I by multiplying by the complex time to get up the denominator so the complex conjugate of 3 - 2i is going to be 3 + 2 I and since I did that to the bottom I also have to multiply it the top and so we will multiply each the top and the bottom separately so on the top I have 4 x 3 which is 12 and then four times do I was just positive 8i that's my new writer and then on the bottom I'm going to have to either boil or use a box I have 3-2 I3 Plus to I that is 9 - 6 y + 6 y - 4 Square so again the negative 6 I and the positive 6 I will eliminate giving us know is there and then we also have this I squared which were going to substitute a negative one so that's going to change this to a positive for so now I have 9 + 4 which is 13 semi denominator is 13 and this is in simplest form we can't divide all of these by any common factor because 13 is prime so that's the answer

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Complex Conjugates and Rationalizing the Denominator

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