Last week, in Number Sense 022, we invented a number system with only four digits, 0, 1, 2 and 3. We were not, of course, the first to have invented this number system, it's well known in the mathematics community as base 4. We figured out how to count in base 4, then we drew a base 4 number line, and finally developed a base four addition facts table and did an addition problem.
This week, I'd like to play a bit more with our toy number system, and use it to multiply.
First things first. In order to multiply, we need to establish our multiplication facts. So we need to fill in this table.
We can fill in more than half the table by recalling that the multiplication property of identity says that multiplying by 1 doesn't change the other factor. Consider multiplication as repeated addition: some thing added once is simply that thing. For example:
doesn't even have to be numbers. Three goats times 1 is three goats. We repeatedly add three goats to the pen but we do it only one time. Result?
Three goats in the pen.
So, 1 x 1 = 1, 1 x 2 = 2 and 1 x 3 = 3
Well, that's not quite half the table, but if we also recall that multiplication is commutative, we know that 2 x 1 = 2 and 3 x 1 = 3
and there are five out of the nine entries, or more than half done, just by recalling two properties of multiplication.
Now we have to actually get into our toy number system to fill in the rest of the table. We can use our base four number line, and add 2 twice, to get 2 x 2 (the top red and blue bars), then add 2 three times, to get 2 x 3 (the bottom red and blue and red bars.)
And the same method for repeatedly adding three
Now, lets do some multiplication! How about 31 x 23?
As with addition, we are going to use expanded notation to help us keep track of the zeros.
So this multiplication problem becomes
When ever we have a sum multiplied by a number, we can use the distributive property to calculate the product. But here we have a sum multiplied by another sum. We are going to perform a little conceptual trick. We are going to (temporarily) replace one of the sums with a goat.
Applying the distributive property to this new expression gives us
It should be pretty clear that, when dealing with goats, 31 goats is the same as (30 +1) goats and is also equal to 30 goats + 1 goat.
Now we can let the goat go about his business, and replace him with the sum he stood in for.
This gives us two separate expressions we can apply the distributive property to:
Taking them one at a time, we get
This leaves us with four multiplication problems we can look up in our multiplication table.
If we add all these numbers up, we'll get the product of 31 x 23 in base four.
Ok, zero plus zero plus zero plus three is three, one plus two is 3, two plus two is ten (remember, base four. If you forgot your base four addition facts, recall that 2 + 2 is the same as 2 x 2, both mean two goats go in the pen, two times.) So, two plus two is ten, write the zero and carry the ten to the next column, and one plus one is two.
If we wrote this using the standard multiplication form, instead of using expanded notation, it would look something like this:
Same answer, more compact form. There are a lot of shortcuts embedded in this form. We have the distributive property applied three times, four separate multiplications, dealing with zeros by leaving them out in favor of place notation (which is a rule about where we write a number on the paper) and that 'carrying' trick which is really breaking a two digit sum into expanded notation, then using the commutative property to move it to a column were it can be added as if it were a single digit number. Whew. No wonder learning how to multiply is complicated.
One thing is interesting to notice. Even though we are calculating in a different number base, a number system we aren't used to dealing with on a day to day basis, the mathematical properties are the same. The distributive property is still the distributive property, it works exactly the same as it does in base ten. Likewise the identity property, the commutative property, equality, and so on. We use the same logic in figuring out how to count, how to add, how to multiply...
In fact, the only thing different about base four compared to base ten is the labels on the tick marks on the number line.
Have fun in the comments.