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Sums and products of irrational numbers

CCSS.Math:

Video transcript

let's say that we have some number a and two that we are going to add some number B and that sum is going to be equal to C now let's say that we're also told that both a and B are irrational irrational so based on the information that I've given you a and B are both irrational is there some C is that going to be rational or irrational and encourage you to pause the video and try to answer that on your on your own so I'm guessing that you might have struggled with this a little bit because the answer is that we actually don't know it depends on what irrational numbers a and B actually are what do I mean by that well I can pick two irrational numbers where their sum actually is going to be rational what do I mean well what if a is equal to PI and B B is equal to one minus pi now both of these are irrational numbers pi is irrational and one minus pi whatever this value is this is irrational as well but if we add these two things together if we add PI plus one minus PI 1 minus PI well these are going to add up to be equal to one which is clearly going to be a rational number so we were able to find one scenario in which we added two Irrational's and we the sum gives us a rational and in general you could do this trick with any irrational number instead of Pi you could have had you could have had square root of two plus one minus the square root of two both of these what we have in this orange color is irrational what we have in this blue color is irrational but the sum is going to be rational and you could do this instead of having one minus you could have this is one-half minus you could have done it a bun of different combinations so that you could end up with a sum that is rational but you could also easily add to irrational numbers and still end up with an irrational number so for example if a is PI and B is PI well then you're going to be their sum is going to be equal to 2 pi which is still irrational or if you added PI plus the square root of 2 this is still still going to be irrational in fact just mathematically I would just express this as pi plus the square root of 2 this is some number right over here but this is still going to be irrational so the big takeaway is is if you're taking the sums of two irrational numbers and people don't tell you anything else they don't tell you which specific irrational numbers they are you don't know whether their sum is going to be rational or irrational so now let's think about products so similar exercise let's say we have a times B is equal to C a B is equal to C a times B is equal to C and once again let's say someone tells you that both a and B are irrational pause this video and think about whether C must be rational irrational or whether we just don't know and try to figure out some examples like we just did when we looked at sums all right so let's think about let's see if we can construct examples where C ends up being rational well one thing I as you can tell I like to use pi PI might be my favorite irrational number if a was 1 over PI and B is PI well what's their product going to be well their product is going to be 1 over pi times pi that's just going to be pi over pi which is equal to 1 so here we got a situation where the product of two Irrational's became or is rational but what if I were to multiply what if and in general you could do this with a lot of irrational numbers 1 over square root of 2 times the square root of 2 that would be 1 now what if instead I had pi times pi pi times pi well that you could just write as pi squared + pi squared is still going to be irrational so this is irrational irrational and it isn't even always the case that if you multiply the same the same irrational number if you square in a rational number that's always going to be irrational for example if I have square root of 2 times I think you see where this is going times the square root of 2 I'm taking the product of two irrational numbers in fact they're the same irrational number but the square root of 2 times the square root of 2 well that's just going to be equal to 2 which is clearly a rational number so once again when you're taking the product of two irrational numbers you don't know whether the product is going to be rational or irrational unless someone tells you the specific numbers so whether you're taking the product or the sum of irrational numbers in order to know whether the resulting number is irrational rational you need to know something about what you're taking the sum or the product of