Counterexamples | Worked example

- [Narrator] We are asked, which of the following numbers can be used to show that if the product of two integers is positive, their sum is not always positive? So pause this video and see if you can figure this out, and you just have to pick one of these choices. Okay, now lets do it together. So we wanna show that if the product of two integers is positive, that their sum is not always positive. So what we wanna do is actually find two numbers that would be a counter example to someone who's saying if the product is positive, then the sum is positive as well. And so what we wanna do is look at these choices, and first of all, if these choices don't meet this first condition, that the product of the two integers is positive, then it's not going to be useful for showing the counter example, so we can rule that out. And then we have to see if it meets the first condition, that it also shows that the sum is not positive. So lets do this. So five and five, they're both integers, and so five times five is equal to 25, so that's positive, the product of the two integers is positive so it meets that first condition. Their sum is not always positive. The sum here, five plus five, is equal to 10. So this is a situation where the sum is positive. So this wouldn't be counter example, if someone thought that hey, maybe when the product is positive the sum is always positive. So this wouldn't help us for showing that the sum is not always positive, so rule that out. So five and negative five. Let's multiply them first, five times negative five is negative 25. So that doesn't meet our first condition. The first condition says if the product of the two integers is positive. Here the product of the two integers are not positive, so it's not going to help us to try to show the counter example that the sum is not always positive. So rule that out. One and 25, one times 25 is indeed positive, so it meets the first condition, but the sum one plus 25, well that's positive as well, that's going to be 26. And so this won't be able to show that the sum is not always positive. Alright, negative one and 25. Well if we multiply negative one times 25, you get negative 25. So it doesn't even meet the first condition, that if the product of the two integers is positive. Here the product isn't positive so let's rule that out. And so it's probably going to be E, but let's test it out. So on the first condition, we take the product negative five times negative five, that is indeed equal to 25. So that shows that this is a situation where the product of the two integers is positive, and now lets look at the sum. Negative five plus negative five is going to be equal to negative 10, and so this is the counter example. This shows that the sum is not always positive, even if the product is positive. So this is doing exactly what we set out to do.