Worked example: rational vs. irrational expressions (unknowns) | High School Math | Khan Academy

We're told let A and B be rational numbers and let B be non-zero. They had 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 could 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 M, N, P, and Q 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. 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 proven 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 rational or irrational? And like always, pause the video and try to think this through. You might want to 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 don't know, two square roots of two and B is equal to the square root of two. Well, in that world, 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 could write that as 2 over 1, and 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 instead, what if A was equal to the square root of two and B is equal to the square root of, well, let's say B is equal to the square root of 7. Well then, A over B would be equal to the square root of 2 over the square root of 7, which is still going to be irrational.

I mean, another way to think about it, I'm not proving it here, but you could think about this as going to be the square root of 27ths. 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 could 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 nonzero rational number. Is A times the square root of 8 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 8 is equal to the square root of 4 times 2, which is equal to the square root of 4 times the square root of 2, 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 8 is 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 told let A be an irrational number. Is -4 plus A rational or irrational? I won't give a formal proof here, but I'll 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.

Well, 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 -3.1415 and it keeps going on and on forever, never repeating? Well then we would have -4 + A would be equal to -4 minus pi, which would be approximately -7.14159. 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, it's going to be irrational. And let's see if A was the square root of 2. -4 + the square root of 2. Well, once again, I'm not doing a proof here, but intuitively you know this is going to be a decimal. It's going to have a decimal expansion that's going to go on forever, never repeat.

This would just change what's to the left of the decimal but not really change what's... Well, it will 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.

If, in fact, this was this way, then to the right of the decimal you would have the same thing as the square root of 2. To the left of the decimal you would just have a different value; you would have what, negative 2.55 point whatever, whatever, whatever.

So this is when you add a rational number to an irrational number. We prove it in other videos: a rational plus an irrational is going to be irrational. If you want that proof, we have other videos within this tutorial.