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Adding and Subtracting Positive and Negative Numbers

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Today we're going to talk about adding and subtracting positive and negative numbers. We're going to do some examples using both a number line and real-world examples to help you really understand the concept behind it. And so at first, before we do our examples, I want to go over how we would use this number line and a couple of interesting things about it.

So you might be used to seeing a number line horizontal instead of vertical. But I'm going to use this vertical one, because it makes it a little bit easier to think about adding and subtracting. So on my number line, I have 0 in the middle. And then I have positive numbers and negative numbers.

Anything that is above 0 is going to be a positive number. And anything that is below or less than 0 is going to be a negative number. I think the best way to think about a negative number is owing someone money or being in debt. If you have less than $0 in your bank account, you have a negative amount of money, or you owe money to somebody. So that's a good way to think about a negative number.

So a couple other things about our number line. When we have positive and negative numbers, any number that has the same value-- or I'm sorry, the same number, like 2 and negative 2, but have opposite signs, one positive and one negative, we call those opposites. So a positive 2 and negative 2 are called opposites. Similarly, 4 and negative 4 are also called opposites.

So what you'll notice is that any two numbers that are opposite of each other are going to be the same distance from 0 on the number line. So 3. Positive 3 is 3 units away from 0. And negative 3 is also 3 units away from 0.

Another thing to notice is that when you add two numbers that are opposites of each other, their sum is going to be 0. So let's look at an example. If I have negative 3 and I add its opposite, positive 3, it's going to equal 0.

On a number line, we can see that, because negative 3-- if I start at negative 3 and I add 3 to it, 1, 2, 3, it's going to give me 0. So negative 3 plus positive 3 equals 0.

So let's do some examples. I think the best way, like I said before, of thinking about positive and negative numbers is in terms of money. So getting money would be a positive number. Owing someone money or giving somebody money would be a negative number.

So let's think about if I had $4 and my brother gave me another $1. So 4 plus 1, we already know that this is going to be equal to 5.

If we looked at that on our number line, if I started at $4 and I added $1 to that, that's going to give me at $5. This is nothing new. But something that's important to think about is that adding a positive number is going to increase the amount of money that you have, or increase the value. So adding a positive will increase the value. We're going up.

So let's think about it in another way. Let's say that I had my $5 and then I realized or my brother realized that I owed him $2 from last week. So since I'm owing him money, I'm going to think about that as a negative number. I'm owing him $2.

So what happens in that scenario? I'm starting at $5. And I owe him $2. So since I'm owing him $2, that's going to be negative. It's going to decrease my value.

I start at 5 and I'm going to go down 2, which would give me 3. So 5 plus a negative 2 will give me 3. And so the important thing to see there is that adding a negative number is going to decrease the value.

You can also think about adding a negative number as subtracting. And I'll give you an example. Let's say we go to the store and we want to buy a jug of milk for $4. And let's say I also have a $1 off coupon.

So my problem becomes $4 plus my $1 off coupon. Since it's $1 off, you can think about that as a negative number. So I have my $4. But I am adding to it a negative $1.

So as we looked at it before, if I start at 4 on my number line and I'm adding a negative, I know I'm decreasing my value. So 4 plus negative 1 will give me 3.

Now you might notice something here. You might be saying, but I also know that 4 minus 1 is 3. And you're exactly right. So this is what I want you to notice, that adding a negative number is the exact same as subtracting the opposite of it, or subtracting a positive number.

So 4 plus negative 1 is the same as 4 minus 1, which we know is 3. Again, adding a negative number is the same as subtracting its opposite. Negative 1 and positive 1 are opposites, so adding negative 1 is the same as subtracting a positive 1. In both cases, we get 3.

So I'm going to do our last example to show how adding a negative number is the same as subtracting. We're going to look at it in both ways and see how we get the same answer. Let's say you're feeling lucky and you buy some new stock called ABC stock. It started at $6.00 per share when it opened on Monday.

However, right away on Monday it fell 8. On Tuesday it rose 3. On Wednesday it rose 4. On Thursday it fell 2. And on Friday it fell 1 more.

So let's try and figure out what the closing value at the end of Friday would be. If you want to write down this problem, go ahead and pause and write it down. But we're going to convert it to a numerical expression, so you can also just write that down.

So first let's think about what our positive and negative numbers are. Actually, first let's look at our number line and see what our answer's going to be. All right. So we started at $6.00 per share.

And on Monday, it fell 8. So 3, 4, 5, 6, 7, 8. So we went down to negative 2 on Monday.

On Tuesday, we went up 3. 1, 2, 3. On Wednesday, we rose 4. 1, 2, 3, 4. On Thursday, it fell 2. 1, 2. And then on Friday, it fell 1.

So we already know that our answer is going to be at 2. So let's go back and see how we can write those numerical expressions and get that same answer.

All right. So we know that if our value is rising, then that's going to be a positive number. And if our value is falling, that's going to be a negative number.

So if I start at 6 and it fell 8, I'm going to be adding a negative 8. Tuesday it rose 3, so I'm adding a positive 3. Wednesday it rose 4, adding positive 4. Thursday it fell 2, so I'm going to be adding a negative 2. And on Friday, it fell 1, so adding negative 1.

So I now have 6 plus negative 8 plus 3 plus 4 plus negative 2 and plus a negative 1. So we know that our answer is going to be 2. We already did that using our number line.

So here's the other way to think about it, using subtracting instead of adding your negative numbers. So if I know that I'm adding a negative 8, I can also think of that as subtracting 8. So 6 minus 8, add my 3, add my 4, plus negative 2 will turn into minus 2, and adding negative 1 will turn into minus 1.

Let's try that same thing using our number line. Again I start at 6. I subtract 8, so I'm coming down 8. That will bring me to negative 2.

I'm adding 3. That will bring me up to 1. I'm adding 4. That will bring me up to 5. But then I have to subtract 2 and subtract 1. So we see that our answer is again 2.

So I hope that the use of this number line and these real-world examples helped you understand a little bit more about adding and subtracting positive and negative numbers. Make sure that you review your notes and the examples that we did with the use of your number line. And pretty soon you probably won't even need to use the number line. You'll be a pro. Thanks for watching.