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which of the following is equivalent to the complex number shown above and then we got this big hairy Messier where we want to take the rational expression 1 plus I over 1 minus I and then add that to 1 over 1 plus I well when you add two fractions like this what you might want to do is find a common denominator and the easiest way to find a common denominator is to take the product of both of these things let me just rewrite everything so we want to take 1 plus I over 1 minus I and then add that and let me switch some colors so we can keep track of things and then add that to 1 plus or so 1 over 1 plus I and actually let me give myself some space here so plus plus 1 over 1 plus I well we're able to add two fractions if we have the same denominator and we can't have the same denominator if we multiply both the numerator and the denominator here by this denominator by 1 plus I and if we multiply both the numerator and the denominator here by this denominator and this is the way that you've always added fractions with unlike denominators we're just finding a least common multiple and that least common multiple the easiest one is just multiplying the denominators so let's do that so let's multiply the numerator denominator here so put some parentheses times 1 plus I and let's multiply the denominator here times 1 plus I so notice 1 plus I over 1 plus I that's just going to be 1 so we aren't changing its value we're just going to find another way of expressing it and this one over here we want to multiply the numerator and the denominator and the denominator by this denominator by 1 minus I 1 minus I so we're going to multiply times we're gonna take this and multiply 1 minus I or multiplied by 1 minus I over 1 minus I and once again this thing right over here is just 1 so we're not changing its value we're just finding a good way to rewrite it so that we're going to have the same denominator notice both of these denominators are going to be are now going to be the same thing they're gonna be 1 plus I times 1 - I let's figure out what this is so this first one right over here so the numerator let's see we're gonna have 1 times 1 so let me do this in different colors so we can keep track of everything so we have this numerator right over here 1 times 1 is 1 1 times I is I times 1 is I and then I times I is negative 1 is negative 1 and then this denominator over here is going to be so we're gonna have divided by 1 times 1 and you could say that this is a difference of squares right over here well I let me just I'll multiply it out you'll see that it's a difference of squares so 1 times 1 is 1 1 times plus I is plus I negative I times 1 is minus I and then negative I times positive I well I times I would be negative 1 but then we have this negative so it's just going to be plus 1 and so what does this simplify to let's see you're gonna have 1 minus 1 so those cancel out and then you're gonna have this I minus I that's just going to be 0 so this simplifies in the numerator you have 2 times I and then in the denominator in the denominator you have 1 plus 1 is equal to 2 and I could simplify this but my goal isn't to simplify this my goal is to have the same this to have a like denominator so I'm just gonna leave this like this just like that right now and now let's move on to this one let's move on to this one so the numerator the numerator here is pretty straightforward 1 times 1 minus I that's just going to be 1 minus I and then the denominator here 1 plus I times 1 minus i we just figure that out that simplifies to 2 so that simplifies to 2 so this has now become 2i over 2 plus 1 minus I over 2 well what's that going to be well our denominator is going to be you could say we have two I haves plus one - I have so how many halves is that going to be well we want to add this has no real parts where you want to add the imaginary parts so we're gonna have the one this one and then we could add take 2i and then subtract I from that 2i minus I is going to be I so it's going to be one plus I over 2 and they don't have exactly that but if we then if we just instead of writing it like this if we viewed this as if we viewed this as one half times one plus I which this is the same thing or if we viewed this as one half if we distribute the 1/2 1/2 plus 1/2 I or plus let me just write it one half plus one half I and when we write it that way when we write it that way we see this choice is exactly that and we are done