Relate place value to standard algorithm for multi-digit addition

- [Instructor] 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 of this in terms of place value. So 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 a thousands, then we have a hundreds-- I could write these words out but this is a little bit faster to write it this way. And then we could have a tens place and then we can have a ones place. I wanna do that in a different color. So then we have a ones place. Let me make a table here and I'm gonna 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 is a fancy word for a system, a way of doing something. But let's first represent these numbers. 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 four ten thousands, so this is forty thousand, I have zero thousands. Right over there. And then I have seven hundreths. Seven hundreds. I have six tens. Six tens. And I have two ones. I'm just rewriting the number, but 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 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 13 tens," but 13 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 hundreds." Sometimes people say, "Oh you're carrying the one. "Six plus seven is 13, carry the one." And it seems somewhat magical, but all you're doing is you're taking ten of the tens and you're regrouping it as a hundred. It'll be a little clearer here. So we have six tens and then we have seven tens, you add 'em all together, you get one, two, three, four, five, six, seven, eight, nine, ten, eleven, twelve, thirteen tens. All we're doing right over 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 three in the tens place and then we add an extra one in the hundreds place. And so what are we gonna have in the hundreds place now? And actually, let me do it here because it's a little more interesting. So I have one, two, three, four, five, six, seven, eight, nine, ten, eleven, twelve hundreds. So I can write 'em 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 twelve in our traditional number system, so what I could do is I could take ten of these and I can convert is to a thousand. So I'm gonna take ten of those and give myself a thousand. No we're gonna do the exact same thing over here. One plus seven plus four is twelve. So you'd write-- That's two hundreds because this is twelve 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 71,235. 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.