Patterns in arithmetic
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- What I'm going to introduce you to in this video is a way to classify numbers as either being even, or being odd. So what does it mean to be even? Well an even number is one where if you had that many doughnuts, you could split it evenly between two people. So even numbers are numbers that are multiples of two. So if you have two doughnuts, you could split them. If you had two people, you could give one doughnut to each person. Four doughnuts, you could split that evenly with two people. They could each get two doughnuts. And then we could keep going. All of the multiples of two, these are even numbers. 8, 10, 12, and of course we could keep going on, and on, and on, and on. An easy way to spot an even number is that it's ones place is going to be even. So, for example, the number 32 is an even number because in the ones place you have a two. The number 5,977,354, 5,977,354, well that's an even number, because the ones place is even. Now there's one number that's an interesting one, why some people sometimes say, "Well is this one really even?" And that's the number zero. And the number zero is even because it is a multiple of two. How is it a multiple of two? Well, zero times two is equal to zero. So zero is a multiple of two, and so zero, for sure, is an even number. And then that actually makes our looking at the ones place idea hold up. Because a number like... a number like 150 is an even number, and we can look at the ones place of it and see, we have a zero there. And a zero is an even number. So if, in the ones place, you have an even number, you are looking, the whole thing is going to be an even number. So what are odd numbers? Well one way to think about them is they're the numbers that aren't even. So not multiples of two. Not multiples... not multiples of two. So what are some examples of odd numbers? Well one, three, five, seven, nine, and of course you can go on, and on, and on. And just like we could look at the ones place to spot an even number, you can also look at the ones place to spot an odd number. The number 59, well I have a nine over here in the ones place. Nine is odd, so this is going to be an odd number. The number 1,441 has a one in the ones place. That is an odd number. So this whole thing is going to be odd. And so what's another way of... why is it called odd? Well, it'd be hard to split 1,441 doughnuts. You can't split it evenly between two people without breaking up a doughnut. If you wanted to leave the doughnuts whole, one person would have to get one extra doughnut than the other person. You can't split it evenly. The easiest way I think about odd, it's not a multiple of two, it's not even. So another way, it is not... it is not even. So now that we know what an odd or an even number is, let's think about what happens when we operate on odd or even numbers. So let's think, in particular, actually let me move over to the side right over here. Let me move over.... right over here. Let's think, whoops. Fell off the screen. Let's think about what happens if I take an even number, an even number, plus another even number. Is this number, the sum, is it going to be even or odd? Well you could take some examples. If I say two plus six, that is going to be equal to eight. Well eight is an even number. If I say 14 plus four, well that equals 18. Once again, eight in the ones place. This is an even number. If I say 150... 156 plus... plus 100... 100 and... why don't we do a simpler number, plus four, that is equal to 160. Zero in the ones place, this also is an even number. So it looks like all the examples that I've done so far, when I add an even to an even, I get an even number. And I encourage you to keep trying this out. I encourage you to keep trying, pick some even number, then another even number, add them together and you'll see that you keep getting even numbers. And it makes sense. Because if you had one number that's a multiple of two, and you add it to another number that's a multiple of two, it makes sense that the sum is going to be a multiple of two. Now what happens if you add an odd and an odd? What happens if you add an odd number plus another odd number? Well let's try. What is one plus three? Well that's equal to four. You actually get an even number. You add two odd numbers, you get an even number. Well maybe that was just that special case. What if I add 15 plus seven? These are both odd numbers, when I add them together I get 22. I get another even number. This is interesting. If I take, let me say 19 plus 3. Actually that gets us 22 again. We have an even number. This is interesting. What about, what about, let's see... 23 plus 5, well that gets us to 28. Once again, get an even number. So the pattern that seems to be forming here is that if I have an even plus an even I get an even, but also if I have an odd plus an odd, then I also get an even number. And once again, I encourage you to try out as many numbers as you can to see if this pattern holds. And you will see that it is true. An odd plus an odd number is an even. Now let's think about one last combination. What about an even, what about an even plus an odd? An even plus an odd. So let's say I have two plus one. Two plus one, well what is that going to be? Well that's going to be equal to three. That's going to be an odd number. Well what about, what about four plus three? Four plus three, well that's equal to seven. That's equal to an odd number. And so it looks like if I have an even plus an odd, and I've only tried out two cases here, but I encourage you to try out many, many, many more to make sure that you feel good about this, but an even and an odd is always going to give you an odd. So this is always going to give you an odd. So let's just remind ourselves, even numbers, they're multiples of two, including zero. Odd numbers are just numbers that aren't multiples of two, that aren't even. And an even plus an even is going to be equal to an even. An odd plus an odd is also going to be equal to an even. We saw that, we saw that multiple times. But an even plus an odd? That is odd.