In this section, we want to start our work with addition with positive and negative numbers.
Over here on the board, I've written some of the vocabulary and properties associated with addition.
First of all, we say if a, b, and c are any three numbers; then, the sum of a and b is written a plus b.
So, we see that the word sum and the operation addition always go together.
So if we want the sum of two numbers a and b, then what we are talking about is what we get when we add them.
So, a plus b is the sum of a and b. The addition property of zero says that 0 is a special number with addition.
If we add 0 to any number with addition, we always end up with that number back, no matter whether we add it on afterwards
or we add it on before. So, a + 0 = a; likewise, 0 + a = a.
The commutative property of addition says that a + b = b + a.
So, the order that we add two numbers in with addition is not going to affect the result: a + b and b + a are exactly the same number.
The associative property for addition says that the quantity (a + b) + c is always equal to a + (b + c).
So, if we're going to add three numbers, we can add any two of them first, and the result will be the same.
So, if I add a + b, and then add c to what I get, it will be the exact same thing as adding b + c and then adding a to what I get.
So, commutative property, associative property, both important properties for addition; 0 is an important number with addition;
and the word sum is always associated with a + b. So, let's go to the board now, and take a look at our first problem.
For problem number 1, I've written, "Find each sum," and here I have the quantity (6 + 3) + 7.
So, I'll do inside the parentheses: (6 + 3 = 9), + 7. I write my equal sign; what I get when I add 6 + 3, and then I add the 7 on: 9 + 7 = 16.
So, this sum of (6 + 3) + 7, adding 6 + 3 first and then the 7, comes out to be 16. Here is our next example:
the quantity (2 + 1) + (6 + 3). I'll begin by adding these two numbers together to get 3;
here's my plus sign; and then, 6 + 3 = 9. My next line, then, is what I get when I add 3 and 9; and 3 + 9 = 12.
So, 12 is the result of simplifying this expression right here. Let's look at our next example. Let's rewrite 6 + 4 using the commutative property:
well, the commutative property says that we can change the order of two numbers in a sum without changing the result.
So, 6 + 4 will be equal to 4 + 6, using the commutative property.
Now, notice it doesn't say to simplify 6 + 4; that is, I don't want to add 6 + 4; I simply want to rewrite this using that commutative property.
So, we're trying to get used to these properties of real numbers by taking expressions that we may know the answer to,
but that's not significant in this case, and rewriting them using those properties. Here's our next problem.
Problem 4: we want to rewrite using the associative property; and I've written the quantity (1 + 2) + 3.
Now, the associative property says that we can change the grouping of three numbers of a sum without changing the result.
So here, 1 and 2 are grouped together, and then I add on 3; I'm going to rewrite that as 1 + (2 + 3) grouped together.
Now, again, we're not trying to simplify this expression; we're simply trying to apply this associative property to this expression.
When I do that, I rewrite the expression as 1 plus the quantity 2 + 3.
So, here the 2 + 3 are grouped together; here the 1 + 2 are grouped together.
The property that allows us to do that is the associative property.
Let's look at another example. I've written here, "Find the solution," and I have problems 5 and 6, they're both equations.
Here's the variable n; and what I know is that n + 8 = 13.
So, I want to know what number I add to 8 to end up with 13; and the answer is 5. n = 5 because 5 + 8 turns out to be 13.
So, what we're doing here when we solve this equation by writing n = 5 is, we're solving the equation by inspection.
We're simply looking at it, and knowing what we know about addition with single-digit numbers, or with small numbers,
we know that 5 is the number that we add to 8 to end up with 13. Here's our next one:
17 = n + 9. So, what number do I add to 9 to end up with 17?
See, in a way it's just a puzzle. 17 = n + 9 is the same as saying, what number would I add to 9 to end up with 17
and the answer would be n = 8. So, 8 + 9 is going to be 17; and that's why the solution to this equation is n = 8.
Let's go on now and look at some other problems. We want to do some addition problems here, and we have 9 + 1 + 8 + 2.
Now, the commutative and associative properties tell us that we can rearrange the order of these numbers,
and we can group them together anyway we want, as long as it's all addition.
So, if I want to, I can begin by adding 9 + 1 = 10; to that, I will add the sum of 8 + 2, which is also 10; and then, 10 + 10 = 20.
So, I can work this problem this way by grouping the numbers together two at a time.
If I wanted to, I also could have added the 9 and the 2 together, and to what I get there, add the 1 and the 8.
Or I could add 9 and 1, then to that add 8, and then to that add 2.
The commutative and associative properties tell us that we can add and regroup in any order we want, as long as it's all addition.
Here's our next problem: let's add 4,281 with 3,016. I'll draw a line there. I'm just going to add in columns.
I'll add the numbers in the ones column: 6 + 1 = 7; in the tens column, 8 + 1 = 9; in the hundreds column, 2 + 0 = 2; put in my comma;
and then, in the thousands column, 4 + 3 = 7. So, when the numbers are a little larger like this, we can write one under the other,
and then just add in columns. Add the numbers in the ones column, tens column, hundreds column, and so on. Here's our next problem: 49 + 16.
Now, if I do the same thing I did in the previous problem, I'm going to add the two numbers in the ones column, 9 + 6 = 15.
I don't want to write 15 right here; I'm going to have to do what is called carrying.
So, before I do that, and see why I do that, you probably already have done problems like this beforeâ
let's write these numbers out in terms of the columns, that the numbers that are digits in the columns right here,
and see if we can get a handle on what this carrying is all about.
So, I'm going to rewrite 49 as 4 tens + 9 ones; and I'm going to rewrite 16 as 1 ten + 6 ones.
Now, I can add the ones together, and I have 15 ones; and to that, I'm going to add 4 + 1 = 5 of these tens.
Now, I know that 15 ones is the same as 1 ten and 5 ones. And so, to that I'm going to add this 5 tens right here.
So, you see what I have is my 5 tens from here; and then, my 15 ones I've rewritten as 1 ten + 5 ones.
Now, this 5 tens + 1 ten is going to give me 6 tens + 5 onesâ and that is why my answer comes out to be 65.
Now, let's see how that looks over here with the original problem, how I'm going to do this problem without writing all of this out.
9 + 6 = 15; I'll write down the 5, and I'll carry the 1 over here, and that represents this 1 ten. Now, 1 + 4 = 5, + 1 = 6, and there's my 65.
So, you can compare these two answers; see that they are the same,
we're not going to do this each time, but this is the reason why carrying works the way it does.
Let's go on now, and look at our next problem. A little more complicated, the numbers are a little larger; but again, we'll just add in columns.
If we come out with a number bigger than 9, we'll simply carry. So, I'll start here, 5 + 2 = 7, + 1 = 8, + 7 = 15; I write the 5, and carry the 1.
Now, I add 1 + 4 = 5, + 7 = 12, + 4 = 16, + 2 = 18; and I'll write the 8, and carry the 1.
1 + 2 = 3, + 6 = 9, + 3 = 12; write the 2; carry the 1. 9 + 1 = 10, + 8 = 18. So, that result is 18,285.
And I found it by simply adding the numbers in the same column, and then anytime I had a number, anytime I had a sum that is more than 9,
I wrote down the first number and then carried the second number like this. Our next problem is a word problem:
"In one term at school, Indra spent $126 for books, $258 for tuition, $489 for rent, and $108 for food.
How much did she spend for tuition and books?" Well, if I look up here, I see she spent $126 for books, so I'm going to write down 126, here,
and those are dollars. And then, I also want to know what she spent on tuition; so tuition was 258, so, $258 for tuition.
I'll add those two numbers up, and I end up with 8 + 6 = 14; 5 + 2 = 7, + 1 = 8; and I don't have anything to carry here, so 2 + 1 = 3.
So, $384 is what Indra spent in one term for tuition and books.
So, something I want to point out here with this problem, like a lot of the problems that you're going to see:
there are extra numbers in the problem. We didn't have to use all the numbers in the problem to solve the problem.
We want to get used to that in this book because I have written the problems that way.
So, you can't just tell by looking at the problem, the numbers in it, what to do to solve the problem.
Some numbers you'll use; other times, there will be numbers that you won't use.
Let's look at a problem that involves a little geometry. We want to find the perimeter, and I've drawn this little figure right here.
The perimeter is the distance all the way around the outside. So, I want to add the lengths of all these sides.
Well, you see, I don't have the length of this side, and I don't have the length of this side.
But isn't it true that if I put this and this and moved it up right here,
that this total distance here would have to be the same as this distance right there?
Likewise, if I took this line segment here and moved it out to here, wouldn't this distance have to be the same as this distance?
So, because of that, the perimeter's going to be 18 + 20;
plus this distance right here, which is this plus this, which is going to be the same as this, 18;
plus this distance right here, plus this, which is the same as this distance, or 20.
So, if I add these numbers up, I'm going to have 18 + 20, = 38; + 18 + 20, which is 38.
I'll add 8 + 8 = 16, carry the 1; 3 + 3 = 6, + 1 = 7.
So, 76 inches is the perimeter of this figure right here, and I found it by doing a little reasoning here,
saying that this, the perimeter of this figure, is essentially the same as the perimeter of this figure right here.
That is, if I move this up, it will be the same as this; and if I take this and move it over, it will be the same as this line segment;
and this and this have the same length, as do this and this. So, there's a look at some problems that involve addition with whole numbers.