Last week, in Fundamental Understanding of Mathematics XXXV, we discussed multi-digit multiplication, the general case of multiplying two numbers. We discovered that the multiplication algorithm used the same underlying principle as the addition algorithm: do simple steps a number of times, and, to make calculation a bit quicker, memorize the simple steps. And way back, in Fundamental Understanding of Mathematics XX we took a look at the division algorithm, and some practical ideas about what it means to divide. This week, I'll talk a bit about combining the ideas of multiplication and division.
First off, I'd like to point out there are four ways of writing down a division problem.
The first is fractional notation.
means the number a divided by the number b is the same as the number c. Some concrete examples:
The second is using the division sign
We sometimes use a forward slash
And then there's that long division bracket
As I pointed out in FUM XX, long division is simply a method for converting fractional notation into decimal notation, because for many purposes, decimal notation is easier to work with. It's much simpler to decide which pen to buy if one costs $0.85 and another costs $0.95 than if one costs $68/80 and the other costs $57/60.
We will use fractional notation and the division sign (which, if you look closely, mimics fractional notation with its fraction bar and a dot above and a dot below the bar.)
Earlier, we gave some practical definitions of division, the measurement definition (we know the size of each group, we find the number of groups) and the partitive definition (we know the number of groups, we find the size of each group.) Now, we will give the mathematical definition: division is an alternative way of writing multiplication.
Let's look at a diagram:
This rectangle has a short side 3 units long, a long side 5 units long, and an area of 15 square units. We can verify this by counting. Mathematically, there are three values associated with this rectangle, 3, 5 and 15. These values are in a fixed mathematical relationship, but we can look at that relationship starting on the outside looking in, or starting on the inside looking out.
If we start on the outside, we begin with the length of the two sides. If we know the length of both sides then the area can be calculated. If we start on the inside, we begin with the area, and if we know the area and the length of one of the sides, we can calculate the length of the other side. These calculations use the same numbers to describe the same rectangle, so they are equivalent. Both calculations tell us the same thing: the length of the sides has a fixed mathematical relationship to the area. Here is how we write these two calculations:
These two statements tell us exactly the same thing about the rectangle, one is written using a multiplication sign, the other with a division sign. They are equivalent.
The order, though, is important. We cannot say
In the division sign statement, the area value is divided by the length of one of the sides to get the length of the other side. We cannot divide a side by a side and get an area. To find the area using values for the sides, we must multiply. To understand why, we return to our definition of multiplication: repeated addition.
To write this definition using variables, we have
Another way to look at it is this: in a multiplication, we have two factors and a product. If we know the factors, we can find the product, because these three numbers have a mathematical relationship. Another way to express the same relationship is to start with the product and one of the factors, and use division to find the other factor.
On a meta note, I am planning, starting next week, to move Fundamental Understanding of Mathematics to Sunday morning rather than it's current slot on Saturday morning. It's a matter of vegetables: the best Farmers Market near me is on Saturday morning, and I haven't made it there for a while. It's also a matter of time: my weekly workload is piling up, and the switch will let me devote some quality time to these essays on Saturday, if I move the publishing day to Sunday. Hope this doesn't inconvenience any of my regular readers and commenters.
Speaking of commenters: have fun in the comments. :)