Worked example: rational vs. irrational expressions (unknowns)

- We're to let a and b be rational numbers, and let b be non-zero, they have to say let b be non-zero because we're about to divide by b. Is a over b rational or irrational? Well let's think about it, they're both rational numbers so that means that a, since it's rational, can be expressed as the ratio of two integers, so I can write a is equal to m over n, and same thing about b, I could write b as being equal to p over q, where where m, n, p, and q are integers, are integers, by definition of what a rational number is, they're telling us these numbers are rational so I can express them as these types of ratios. So what is a over b going to be? a over b is going to be m over n over p over q which is equal to m over n, if I divide by a fraction, it's the same thing as multiplying by the reciprocal. q over p, let me write that a little bit, q over p which is equal to mq over np. Well mq is going to be an integer, if the product of two integers is going to be an integer, and np is going to be another integer, the product of two integers is an integer, so I've just shown that a over b can be expressed as the ratio of two integers, so a over b is for sure, in fact I've just proven it to you, a over b is for sure going to be rational. Let's do a few more of these, this is interesting. All right, so now we're saying let a and b be irrational numbers. Is a over b, let a and b be irrational numbers. Is a over b rational or irrational? And, like always, pause the video and try to think this through, and you might want do some examples of some irrational numbers and see if you can get, when you divide them, you can get rational or irrational numbers. Well, let's just imagine a world where, let's say, that a is equal to, I dunno, two square roots of two, and b is equal to the square root of two. Well, in that world, a over b, a over b would be two square roots of two over the square root of two which would be two, which is very much a rational number, I can express that as a ratio of integers, I can write that as two over one, there's actually an infinite number of ways I can express that as a ratio of two integers. So, in this case, I was able to get a over b to be rational, based on a and b being irrational. But, what if, what if instead of, what if a was equal to the square root of two and b is equal to the square root of, let's say, b is equal to the square root of seven. Well, then a over b would be equal to the square root of two over the square root of seven, which is still going to be irrational, I mean another way to think about it, and I'm not proving it here, but you could think about it, this is going to be the square root of two sevens, so we have something that's not a perfect square under the radical, so we're going to end up with an irrational number. So, we can show one example where a over b is rational and we showed one example where it is irrational, so it can be either way. Let's do a few more of these. All right, let a be a non-zero rational number. Is a times the square root of eight rational or irrational? Well, the key here is, if you multiply an irrational number and why is this an irrational number? It has a perfect square in it, but it's not a perfect square in and of itself. The square root of eight is, the square root of eight is equal to the square root of four times two, which is equal to the square root of four times the square root of two, which is equal to two square roots of two. And, this is kind of getting to the punchline of this problem, but if I multiply a rational times an irrational, I am going to get an irrational. So the square root of eight is an irrational, and if I multiply that times a rational number, I'm still going to get an irrational number. So this is going to be, for sure, irrational. Let's do one more of these. So we're to let a be an irrational number. Is negative 24 plus a rational or irrational? And I won't give a formal proof here, but to give you more of an intuitive feel, it's nice to just try out some numbers, and I encourage you to pause the video and try to think through it yourself. Let's just imagine some values, imagine if a is irrational, a is irrational, so what if a was equal to negative pi, which is approximately equal to negative three point one four one five nine and it keeps going on and on forever, never repeating. Well, then we would have negative 24 plus a would be equal to negative 24 minus pi, which would be approximately negative 27 point one for one five nine, the decimal expansion, everything to the right of the decimal, is going to be the exact same thing as pi. So this looks like, at least for this example, is going to be irrational, and let's see, if a was square root of two negative 24 plus the square root of two, well, once again, I'm not doing a proof here, but intuitively, this is going to be a decimal, it's going to have a decimal expansion that's going to go on forever and never repeat, and so this would just change what's to the left of the decimal, but not really change what's, well it would change what's to the right of the decimal because this is negative, but it's still going to go on forever and never repeat, and if, in fact, this was if this was this way, then to the right of the decimal, you would have the same thing as the square root of two to the left of the decimal, you would just have a different value, you would have negative 25 point whatever whatever whatever, and so this is, when you add a rational number to an irrational number, we've proven in other videos, a rational plus an irrational is going to be irrational, irrational. If you want that proof, we have other videos within this tutorial.