Next, we want to do subtraction with whole numbers. On the board, I've written here that the difference of a and b is a - b,
or a subtract b. So, when I say the difference of a and b in words, I mean this in symbols: a - b; and this is the same as subtracting b from a.
So, subtract b from a looks like this; or find the difference of a and b looks like this.
So, the word "difference" is always associated with subtraction. And when I say the difference of a and b, I mean in that order: a - b.
Now, as far as subtraction goes, it's sort of the inverse operation of addition. So, I say that 8 - 5 = 3 because 3 + 5 = 8.
Every subtraction problem like this has an associated addition problem like this: 4 - 1 = 3 because 3 plus 1 gives us 4; 3 + 1 = 4.
9 - 6 = 3 because 3 + 6 = 9. So, because of these addition statements right here, we have these subtraction statements.
So, subtracting is really the inverse operation from addition. Let's go to the board now and work a couple of problems.
First, I want to subtract 24 from 56. So, what I'm going to do is take my 56, and from it, subtract 24.
So, "subtract 24 from 56" means 56 - 24, horizontally; or vertically, it would look like this.
Now, I subtract in columns: 6 - 4 = 2; and then, 5 - 2 = 3.
So, 32 is the answer to 56 - 24. So, if I say subtract 24 from 56, the answer's going to be 32. Here's my next problem.
Find the difference of 29 and 19. Well, from my original definition that I had on the paper that we began the section with,
"the difference of 29 and 19" is 29 - 19â that's how I'd write it horizontally. Vertically, I would write it like this:
9 - 9 = 0, and 2 - 1 = 1. So, 29 - 19 = 10; and that's the same as the difference of 29 and 19.
Let's look at some other subtraction problems. Here again, I have subtract 9,876 - 8,765. Again, I'll subtract in columns.
I'll take 5, and I'll subtract it from 6, that gives me 1; 6 from 7 is 1; 7 from 8 is 1; and 8 from 9 is 1. So, I get 1,111 for the answer to that problem.
And, again, I just simply subtracted in columns: 6 - 5 = 1, 7 - 6 = 1, so on, and so forth.
Let's go on now and look at our next problem. We have 52 - 37. And right away, I'm going to have difficulty because I have 2,
but I can't subtract 7 from it because 2 is less than 7. If I try to say 7 from 2, I don't know how to do that because the 2 is smaller.
So, I'm going to unravel this a little bit and write it in terms of the place value of the digits: I have 5 tens plus 2 ones.
And, for 37, I know that means 3 tens and 7 ones. Now, here's where my problem is: 7 ones from 2 ones, I can't do that. So, what I do instead is,
I go over to my 5 tens, and I'll take one of those tens and rewrite it over here as 10 ones, plus the 2 ones I already have gives me 12 ones.
So, what I do is I'd borrow one of the tens from this 5 ten right here that I have, and write it over here with the ones.
So, I borrow a ten; I write it as 10 ones, plus the 2 ones I already have, that's 12 ones.
Now, 7 from 12 is a problem that I can do. So, when I do that subtraction, I end up with 5 ones; and then over here, 4 - 3 = 1 ten.
So, that answer is 15. So, let me go back over to my original problem and see if I can do this in a shorthand form: 2 subtract 7 won't work;
I go over to the tens column; I borrow a ten. So, instead of 5 tens, I have 4 tens; add that to the 2 ones I already have, that's 12 ones.
12 subtract 7 is 5; and now, 4 subtract 3 is 1. So, this is a shorthand version for all of this work over here.
This is the explanation for why we do borrowing the way we do. Let's try another problem. Problem 5, I have 329 - 234.
Let's write this in column form over here: 329 - 234. Now, subtract in columns, starting in the ones column.
9 - 4 = 5. 2 - 3, I can't do because 2 is less than 3. I go to the next column over, the hundreds column. I have 3 hundreds right here.
I borrow 1 of them, so that leaves me with 2. The one that I borrow, I write as 10 tens, plus the 2 tens that I already have gives me 12 tens.
Now, 12 - 3 = 9; and then, 2 - 2 = 0. So, my answer is simply 95. So, this time I did my borrowing in the tens column, right here.
To work in the tens column, I borrowed from the hundreds column, but the idea is the same as the idea I had before
when I did it in the problem above. Here's another problem: 79,040 - 32,957. I go over to the ones column.
0 - 7, I can't do because 0 is less than 7. I go to the 4, I borrow 1 from it, and that gives me 10 ones in this column. Now, 10 - 7 = 3.
Next, I have 3 - 5, which I can't do because 3 is less than 5. So, I go to the next column; I have 0. So, I go to the next column over here,
and I'll borrow 1 of these thousands. So, now, I have 8 thousands left. That gives me 10 hundreds over here,
that's the same as 1 thousand, 10 hundreds. Now, I can borrow one of those, that gives me 9 hundreds.
I take the 1 hundred that I borrowed, and write it as 10 tens over here, plus the 3 tens I already have gives me 13 tens.
So, it looks a little messy right here, but that is 13 tens in this column; subtract that, from that 5, and I end up with 8.
Now, 9 - 9 = 0; 8 - 2 = 6; and 7 - 3 = 4. So, my answer ends up to be 46,083,
and I had to do some pretty fancy borrowing right here. But in any case, if I borrow 1 thousand over here, that gives me 1 less than I had:
instead of 9, now I have 8; and the thousand that I borrow, I write as 10 hundreds. So, I have an extra 10 hundreds here;
and then, I can borrow from that if I want. Let's go on and look at one last problem that involves estimating.
We want to estimate by rounding each to the nearest hundred.
So, first I'm going to take each number in this problem and round to the nearest hundred.
So, this rounds to 3,500 when I round to the nearest hundred; and this rounds to simply 500. Now, to do my subtraction:
3500, or 3,500 - 500 = 3000. So, we can also use this idea of rounding here to do estimates for subtraction.
So, I've just rounded real conveniently, right here: this, to 3500; this, to 500. I subtract real quickly,
and I realize that this answer's going to be somewhere around 3,000. So, that's a look at subtraction of whole numbers.