Last week, in Number Sense 020, we applied the area model, and the distributive property, to do multiplication of large numbers. The numbers weren't very large, they were only two digit numbers, but they were larger than any numbers found in a multiplication table. We couldn't just look up the answer, we had to figure it out.
Methods for figuring out answers that we can't look up are called algorithms, and there are a lot of them. Last week I showed you an algorithm for multiplication called multiplication with partial products. This week I thought I'd take a closer look at place value, an idea that underpins many of the arithmetic algorithms we use today.
Place value has to do with the notion that the value of a digit depends on where it is written.
this number, for example, has the digit “2” in it twice. But each digit represents a different value.
There is a big difference between the number of red squares represented by
and the number of green squares represented by
Each time we move to the left one position, the value of the digit increases 10 times. So we could look at this number as
or
which is the expanded notation we saw last week.
The reason this place value idea underlies many arithmetic computations is that, if we have a way to account for the zeros, the only real computations we need to do are all simple ones: single digit by single digit.
So, for adding 243 with 351 we don't need an addition look up table that goes into the hundreds. All we need to solve this is the sums for 2+3, 4+5 and 3+1.
If we wanted to multiply those two numbers, we would do more calculations, but they would still be simple:
1x3, 1x4, 1x2, 5x3, 5x4, 5x2, 3x3, 3x4 and 3x2
the tricky part is accounting for all those zeros in expanded notation, which is just another way of saying we have to account for the position of those digits in our place value notation.
With expanded notation, the addition problem becomes
(200 + 40 + 3) + (300 + 50 + 1)
We get our three simple sums (2+3, 4+5 and 3+1) by recalling that addition is commutative, so we can add those numbers in any convenient order. Here is the order we choose:
(200 + 300) + (40 + 50) + (3 + 1) = 500 + 90 + 4
In our standard addition algorithm, we place addition problems in columns rather than rows
which we can also do with expanded notation. Using place value saves us from writing all the extra zeros, if we impose strict columns on our numbers, and remember to always write the digits of our numbers in the correct column.
It wouldn't do to get sloppy and try to add these numbers written like this
Because we would get the wrong answer.
Have fun in the comments.