Adding multi digit numbers with place value

What we're going to do in this video is get some practice adding multiple digit numbers. But the point of it isn't just to get the answer, but to understand why the method we use actually works.

So we're going to add 40,762 to 30,473, and you can pause the video and try to solve it on your own. But I encourage you to watch this one because it's really about understanding how things happen.

So what I'm going to do is first think about this in terms of place value. Let me write out my place values. Let's see, this goes all the way up to the ten-thousands place. So let's see, ten-thousands, and then we have thousands, thousands, then we have hundreds. I could write these words out, but it's a little bit faster to write it this way. And then we could have a tens place, and then we could have a ones place. I want to do that in a different color.

So then we have a ones place. Let me make a table here. I'm going to express both of these numbers in terms of ten thousands, thousands, hundreds, tens, and ones, and then I'm gonna at the same time use what's sometimes known as the standard method or the standard algorithm. Algorithm's a fancy word for a system, a way of doing something.

But let's first represent these numbers. So here I have four ten thousands—one, two, three, four. Here I have three ten thousands—one, two, three. I'm gonna add these two together eventually. In both of these numbers, I have zero thousands, so I have nothing in this column right now. Here I have seven hundreds—one, two, three, four, five, six, seven. Here I have four hundreds—one, two, three, four.

Then I go to the tens place. Here I have six tens—one, two, three, four, five, six. Here I have seven tens—one, two, three, four, five, six, seven. And then last but not least, here I have two ones—one, two. And here I have three ones—one, two, three.

Now let's just rewrite this number up here. This is just—this is four ten thousands, so this is forty thousand. I have zero thousands right over there, and then I have seven hundreds, seven hundreds. I have six tens, six tens, and I have two ones. I'm just rewriting the number I want. Let me write the tens in that blue color—so, six tens, and then two ones. Having trouble switching colors—two ones.

So there you have it. This and this are just different ways of representing the same number. And then this down here, I have three ten thousands, and then I have zero thousands. I have four hundreds here, and then I have—was this seven tens? Seven tens, and then I have three ones.

And so now let's add up everything. So I can add it here, and then I can also add up things right over here. So in the standard method, we would start at the lowest place, and we'd say, "Okay, two ones plus three ones is equal to five ones." And similarly, two ones plus three ones would be one, two, three, four, five ones—fair enough, nothing fancy there.

Now let's go to the tens place. Well, in the tens place, we have six tens plus seven tens, and in the standard method, what you say is that's thirteen tens. But thirteen tens is the same thing as three tens and one hundred.

So what you do is you would regroup. You would say, "Hey, look, this is three tens and one hundred." Sometimes people say, "Oh, you're carrying the 1. 6 plus 7 is 13, carry the 1," and it seems somewhat magical. But all you're doing is you're taking 10 of the tens and you're regrouping it as a hundred.

It'll be a little bit clearer here. So we have six tens, and then we have seven tens. You add them all together: you get one, two, three, four, five, six, seven, eight, nine, ten, eleven, twelve, thirteen tens. And all we're doing right over here is we're saying, "Look, this is equal to a hundred."

So let's just convert that into a hundred right over here—let's just convert that. And so what we do is we just write the 3 in the tens place and then we add an extra 1 in the hundreds place.

And so what are we going to have in the hundreds place now? And actually, let me do it here because it's a little bit more interesting. So I have one, two, three, four, five, six, seven, eight, nine, ten, eleven, twelve hundreds, so I could write them down here. One, two, three, four, five, six, seven, eight, nine, ten, eleven, and twelve.

But with the same thing, we don't have a digit for the number 12 in our traditional number system. And so what I could do is I could take 10 of these and I can convert it to a thousand. So I'm going to take 10 of those and give myself a thousand. And we're going to do the exact same thing over here. One plus seven plus four is 12.

So you'd write that's 2 hundreds because this is 12 hundreds—that's two hundreds plus a thousand. So we just regrouped again. Now in the thousands place, one plus zero plus zero is one thousand, and you see that right over here—you have one thousand.

And then finally, in the ten thousands place, four ten thousands plus three ten thousands is seven ten thousands. Four ten thousands plus three ten thousands is one, two, three, four, five, six, seven ten thousands.

And so this number is seventy-one thousand two hundred and thirty-five—71,235. So hopefully, it all makes sense how these two things will fit together; that what's going on over here—you're not just magically carrying numbers or magically regrouping, you're just representing the same number in different ways.