It’s Professor Dave, let’s look at big numbers.
When we introduced the concepts of addition and subtraction, we were looking at very small
numbers.
Two apples plus three apples is clearly five apples, which can be verified by simply counting
the five apples.
But sometimes we need to do arithmetic with very large numbers, where counting becomes
impractical.
When this is the case, we must rely on computation to give us answers.
Let’s say one market sold one thousand two hundred ninety three apples today, while another
market sold two thousand six hundred fourteen.
How many were sold in total?
We definitely don’t want to count all those apples, or we will be here all night.
Instead we can calculate the sum.
So how can we do this exactly?
Well if we are using a pen and paper, the convention we have come up with is to place
one number over the other, with a plus symbol and a line below.
This signifies that we are adding the two numbers together, and the sum will be listed
below this line.
We must make sure that the numbers are lined up in such a way that the place values are
in perfect correspondence.
The units place goes with the units place, the tens with the tens, and so forth.
Now, the math is very easy, because we just add two digits together at a time, starting
with the units place.
Three plus four is seven, so we put a seven here.
With the tens place, it’s a little trickier, because nine plus one is ten, and we can’t
put ten here, as we want a single digit.
So instead, we must realize that we can put a zero if we then increase the hundreds place
by one, signified by adding a one on top.
This is called “carrying the one”, and it works because ten tens are the same as
one hundred.
We do this any time we go over the capacity of a certain place to report value.
So in the sum we have a zero for the tens place, and we move on the hundreds place.
The one that we carried, plus two, plus six, gives us nine for the hundredths place, and
then one plus two is three.
Three thousand nine hundred and seven apples were sold today between the two markets.
This works the same way for subtraction.
Let’s say one market vendor started with four hundred seventy three apples and sold
two hundred eighty six.
How many does that vendor have left?
Let’s arrange these numbers just as we did for addition, making sure to put the larger
number on top.
Now, starting with the units place we have three minus six.
That’s not going to work, so what if we borrow one of the tens from the tens place?
If we cross out this seven and put a six instead, we’ve borrowed ten units that we can give
to the units place.
That means we can now do thirteen minus six, which is seven.
Now in the tens place, we have six minus eight.
Again this won’t work, but if we borrow one of the hundreds from the hundreds place,
four hundred becomes three hundred, and we’ve borrowed ten tens that we can use for the
tens place.
This means it’s like we have sixteen tens minus eight tens.
That gives us eight for the tens place.
Lastly, three minus two is one.
The vendor has one hundred eighty-seven apples left to sell tomorrow.
So now we have the basic algorithm for adding and subtracting larger numbers.
We can use this method to add long lists of numbers beyond just two at a time.
Say we want to add all of these numbers together, seventeen, twenty-two, eleven, thirty-four,
and forty-six.
We could add two together, and then a third, and so forth, but that’s time consuming,
and becomes increasingly impractical the longer the list gets.
Instead, let’s use the method we just learned to add them all together at once.
In the units place, seven plus two plus one plus four plus six is twenty.
So let’s put a zero in the units place and carry a two to the top of the tens column.
Now the two we carried plus one plus two plus one plus three plus four gives us thirteen,
and there’s no hundreds place to carry anything, so we just write thirteen, which gives us
one hundred thirty as the sum.
And now we have a basic algorithm for adding and subtracting large numbers.
Let’s check comprehension.