Video transcript
- [Instructor] Let's saythat we have some number a and to that we are goingto add some number b and that sum is going to be equal to c. Let's say that we're also told that both a and b are irrational. Irrational. So based on the informationthat I've given you, a and b are both irrational. Is their sum, c, is that goingto be rational or irrational? I encourage you to pause the video and try to answer that on your own. I'm guessing that you might have struggled with this a little bitbecause the answer is that we actually don't know. It depends on what irrationalnumbers a and b actually are. What do I mean by that? Well, I can pick two irrational numbers where their sum actuallyis going to be rational. What do I mean? Well what if a is equal to pi and b is equal to one minus pi? Now both of these are irrational numbers. Pi is irrational and one minuspi, 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, one minus pi, well these are gonna add up to be equal to one, which is clearly goingto be a rational number. So we were able to find one scenario in which we added two irrationals and the sum gives us a rational. In general you could do this trick with any irrational number. Instead of pi you could've had square root of two plus oneminus the square root of two. Both of these, what we have in thisorange color is irrational, what we have in thisblue 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 as 1/2 minus. You could have done it a bunchof different combinations so that you could end upwith a sum that is rational. But you could also easilyadd two irrational numbers and still end up withan irrational number. For example, if a is pi and b is pi, well then their sum isgoing to be equal to two pi, which is still irrational. Or if you added pi plus the square root of two, this is still going to be irrational. In fact, mathematicallyI would just express this as pi plus the square root of two. This is some number right over here, but this is still going to be irrational. So the big takeaway isif 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. Now let's think about products. Similar exercise, let'ssay we have a times b is equal to c, ab is equal to c, a times b is equal to c. And once again, let'ssay 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. Try to figure out some examples like we just did when we looked at sums. Alright, so let's think about, let's see if we can construct examples where c ends up being rational. Well one thing, as youcan tell I like to use pi, pi might be my favorite irrational number. If a was one over pi and b is pi, well, what's their product going to be? Well, their product is goingto be one over pi times pi, that's just going to be piover pi, which is equal to one. Here we got a situation wherethe product of two irrationals became, or is, rational. But what if I were to multiply, and in general you could this with a lot of irrational numbers, one over square root of twotimes the square of two, that would be one. What if instead I had pi times pi? Pi times pi, that you couldjust write as pi squared, and pi squared is stillgoing to be irrational. This is irrational, irrational. It isn't even always the case that if you multiply thesame irrational number, if you square an irrational number that it's always going to be irrational. For example, if I havesquare root of two times, I think you see where this is going, times the square root of two, I'm taking the product oftwo irrational numbers. In fact, they're thesame irrational number, but the square root of twotimes the square root of two, well, that's just goingto be equal to two, which is clearly a rational number. So once again, whenyou'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 youthe specific numbers. Whether you're taking the product or the sum of irrational numbers, in order to know whetherthe resulting number is irrational or rational,you need to know something about what you're takingthe sum or the product of.
