 In this video, I'm going to talk about finding a counter example. So in geometry, you make certain conclusions you use inductive and deductive reasoning. And so in this case, actually, what we're going to do is we're going to find counter examples. So we're going to look at different statements and try to find where these statements are not true. And this is what we call counter examples. Counter example is proving a statement false by coming up with an example. We call it a counter example. All right. So finding a counter example, show that each conjecture is false by finding a counter example. All right. So I got two different questions about this, two different statements. Here we go. For every integer, okay, so hold on a second. For every integer, starting out with the vocabulary, integers are numbers like negative three, negative two, negative one, zero, one, two, three, so on and so forth in both directions. So those numbers. Students usually like to work with your whole numbers that are both positive and negative, usually those numbers. So those are integers. So for every integer, n, so n is these numbers here. n is these numbers here. So usually variables usually represent just one number. In this case, this variable represents a bunch of numbers. So n is all these integers here. All right. So for every integer, n, n to the third is positive. So when I take that number and I cube it, I take it to the third power, it's always going to be positive. That's what this statement says. So for every integer, so take an integer, take any one of these numbers, cube it and it's always going to be positive. Hmm, okay. So let's take a couple, let's do a couple of examples and try to figure this out. So let's take some of these integers, okay, so let's take the number one. What if n is equal to one? Well then n to the third is going to be one to the third, which is going to be one. Okay, that makes sense. One times one times one is one, so n to the third is positive, okay. So that makes sense. That's not a counter example, that's just an example that makes it true. I don't want an example that makes it true, I want an example that makes it false. Okay, so let's try something else. So one is a small number, one's a very small number, so let's try something a little bit larger. n equals 12, too big, too big. I don't know what 12 to the third is. So that was a bad choice, bad choice. So let me do the third power. Let's see, let's try something a little bit better, a little bit simpler. Let's try three. Three's a small number. I can do that very quickly. Okay, so that means n to the third is equal to three to the third. So three to the third, that's three times three times three, which is three times three is nine. Nine times three is 27. So you can see why I chose that one to be a little bit smaller, to make that number a little bit quicker to work with, okay. So 27, okay. So 27 is a positive number, so n to the third is positive. Dang it, I found another example that just makes this true. I don't want something that's true. I want a counter example. I want to find when this is false. But notice the numbers I'm picking so far, n is equal to one, n is equal to three. Let's try something different. I'm just choosing the positive integers here. What about zero? What about these negative numbers? Let's try these. Okay, so what about n equals negative two? Okay, so let's not use three, let's not use one, let's use negative two. So n to the third is equal to, hold on here. When you use negative numbers, you got to remember to use parentheses because you're cubing the entire number. You're cubing not only the two, but you're also cubing the negative. We didn't really have to worry about that over here because they're positive numbers. Positive numbers are easy to work with. Negative numbers on the other hand, you got to use parentheses with this. So what this is equal to is negative two times negative two times negative two. Okay, we do that three times. Okay, so let's just worry about the number part of it. Two times two times two is eight. But then a negative times a negative times a negative, that's three negatives. Two negatives make a positive, but three negatives make a negative all over again. So in fact, negative two to the third is going to be negative eight. And in this case, n to the third is a negative number. So what I just did is I just came up with an example where n to the third is a negative number, not a positive number, but a negative number. So what I just did is I just found a counter example. We're supposed to find a counter example to show that this conjecture is false. This right here is my answer. n is equal to negative two, and here's the math that proves it. That n to the third is a negative number, not a positive number. Okay, so there's my counter example right there. And I'm using a little bit of notation, a little bit of math to prove that. So notation with n's, here's the variable if n is equal to negative two, which is still an integer, it's still an integer, but if I take negative two to the third power, it's going to be a negative number, not a positive number. So there right there is my counter example. Okay, so let's go on to the next, let's go on to the next example, next example. Two complementary, complementary, excuse me, two complementary angles, two complementary angles are not congruent. Okay, so complementary angles, what are complementary angles? Remember, complementary angles are two angles, two angles. That add up to 90 degrees, can shorten it out a little bit. Two angles that add up to 90 degrees. So two complementary angles are not congruent, okay? So they're saying here that if I have two complementary angles, I got two angles that add up to 90 degrees, they are not congruent. They're never gonna be the same, they're never gonna be the same measure. Okay, so let's go over a couple of examples. So I don't know, couple of angles that add up to 90, 10 and 80. Okay, 10 and 80, there's a good example. But that's an example that actually helps, excuse me, that helps this statement. 10 and 80 are two, I should say 10 degrees and 80 degrees. Two complementary angles that are not congruent, well these ones are not congruent. So that really doesn't help me at all. Okay, so let's try something different. Let's try to find something that are congruent, okay? So 40 degrees and 50 degrees, I mean those are pretty close, but they're not congruent, I mean they're complementary, they do add up to 90, okay? But we're getting close, how about 45 degrees and 45 degrees? These two are the same, they are congruent, and they add up to 90 degrees, they are complementary angles. There we go, this two complementary angles, these are two complementary angles that are congruent, are congruent, okay? So they're right there, is my counter example to this statement. There's my counter example. Okay, so in that video just to kind of conclude, kind of wrap things up, this is finding a counter example, finding an example that proves a statement wrong, that proves a conjecture false, okay? By finding a counter example. Now counter examples can be numbers, they can be decimals, or excuse me, degrees, they can be little equations saying the number is equal to negative two, therefore, n to the third is equal to negative eight, okay? And there's lots of different ways to write counter examples. You just got to find a good one and you have to make sure and write it down so that whoever's reading it knows what you're trying to say, knows what you're trying to relay. So anyway, that is finding a counter example. I hope this video was informative.