Last week in Fundamental Understanding of Mathematics XX we took a look at the operation of division, in terms of both the notation we use to write division problems, and the underlying meaning of division in real world problems.
This week we are going to take a look at what it means for division to be the inverse of multiplication, and talk a bit about multiplying fractions.
NOTE: The diaries won't accept tex formatting, so I can't use regular top and bottom fraction notation, and use slash notation instead. It's a bit harder to read, and of course I refer to the numbers as top and bottom. So, I'd suggest getting pencil and paper and where ever you see fractions written in slash notation, jot them down in top and bottom notation
More over the fold.
Division undoes multiplication. If we multiply two numbers, say, positive integers, or counting numbers, we will get some larger counting number: 2 times 3 is 6, 3 times 5 is 15. Think of floor tiles: a patch that's 3 tiles wide and 5 tiles long has 15 tiles in it.
Division is the inverse of multiplication. With division, we'd start with the number of floor tiles, 15, and ask, if a rectangle made of these tiles is 3 wide, then how long is it? We'd get 5, the other number we started with when we multiplied to get 15.
This is the geometric analogy: if we know two sides, multiplication will give us the area. If we know the area and one side, division will give us the other side.
a / b = c if c x b = a
This is the definition of division. If we take a pile of a things, and divide it up into b groups, each group will have c things in it. Or, if we take a square tiles, and form a rectangle b tiles wide, the rectangle will be c tiles long. Or, if we take a string a inches long, and cut it into pieces each b inches long, we'll get c pieces of string.
You'll notice that none of these formulations make sense if b is zero. Take a pile and divide it into no groups? Make a rectangle zero tiles wide? Cut nothing off the end of the string? Whatever you might do here, it's not division. Division by zero is undefined.
The multiplicative identity, the number you can multiply by to get the same number, is 1.
Let's substitute 1 for b in the above definition of division
a / 1 = c if c x 1 = a
We know that c times 1 is c, so in the second part of the definition, a = c. We then take that fact and rewrite the first part of the definition:
c / 1 = c if c x 1 = c
We know the second part is always true: c times one is always c, whatever number c is. So if c is an integer, we have a simple way to convert an integer to fraction notation. c, an integer, is c over 1, a fraction.
Now let's take a look at this:
1 x d = d
this turns into
d / d = 1
if we follow our division definition, and substitute 1 for c, d for a and d for b. A number, divided by itself, is equal to 1.
Ok, that makes sense. If division is repeated subtraction, how many times can we subtract a number from itself? One time. If we have 15 things and divide them into 15 groups, how many things will be in each group? One. If we have a string, 15 inches long, and we cut 15 inch pieces, how many pieces will we have when we're done? One.
This simple fact turns out to be very useful when dealing with fractions. Before we can demonstrate how useful, though, we need to talk about multiplying fractions.
Up until now, all the numbers we have multiplied have been larger than one. But what happens when we multiply with fractions? In an earlier diary's comments, algebrateacher lamented the lack of text books that proclaim the simple facts that one multiplies by the top number, and divides by the bottom number in a fraction.
Multiplying a fraction by an integer can be seen as a variation of repeated addition. Let's say we want to multipy
1/3 x 7
in repeated addition, this looks like
1/3+1/3+1/3+1/3+1/3+1/3+1/3
and we obviously get seven thirds. We multiply the top number, 1, by 7, and the bottom number does not change. Since it's repeated addition, and we are adding thirds, no matter how many of them we add, they'll still be thirds.
If we multiply
2/3 x 7 is the same as 2/3 + 2/3 + 2/3 + 2/3 + 2/3 + 2/3 + 2/3 = 14/3
the top number is multiplied, the bottom number stays the same.
But what if the second number is also a fraction? What do we do with this?
1/3 x 1/2
There are a couple of ways to look at this, here is one: suppose we multiply by 1. Then
1/3 x 1 = 1/3
But we can write one as one half plus one half, like this:
1/3 x (1/2 + 1/2 ) = 1/3
It's still equal to one third. Then we apply the distributive property of multiplication over addition
(1/3 x 1/2) + (1/3 x 1/2) = 1/3
and we get two of what we are trying to compute, one third times one half, but both of them together are still equal to only one third. Obviously, each one is worth one sixth.
1/3 x 1/2 = 1/6
The bottom numbers are multiplied together to get the 6, and, while it isn't obvious in this example, the top two numbers are also multiplied together to get 1. Try working through the example, but start with two thirds instead of one third, to convince yourself that the top two numbers are also multiplied.
Well, I think that's enough to chew on for one diary, so I'll continue with more fractional ideas next week.
Have fun in the comments.