So, have we established that
ab x cd = (a x c) / (b x d) ?When we multiply fractions, the product of the top numbers is divided by the product of the bottom numbers. I think we left the last part of that demonstration as homework, in Fundamental Understanding of Mathematics XXI. Let's do an example over the fold:
Let a = 1 , c = 1, b = 2 and d = 3, then our formula becomesPeople who have followed this series from the beginning may want to chide me for using two equal signs in a row, since I cautioned against the practice in the comments of an earlier diary. I do think it is a bad practice when solving algebraic problems, since my method is to do as much copying and as little thinking as possible, and stringing problems out in a line makes that more difficult.
12 x 13 = (1 x 1) / (2 x 3) = 16
Moving right along, then. I mentioned earlier that the identity property for multiplication (1 times another value equals that other value) would lead to some interesting mathematical tricks. Let's take a look at some of them.
If you divide a value by itself, you get 1.
44 = 1. 1717 = 1. Hatbox / Hatbox = 1
How does this work? Well, look at division as repeated subtraction. If you have a John Deere tractor, how many times can you subtract that same John Deere tractor? Once. Whatever you start with, you can subtract it from itself one time, and one time only.
If you divide a value by one, you get the value back.
Hatbox / 1 = HatboxSo, 1 is not only the identity number for multiplication, it's also the identity number for division. We could use a string example for this: If we have a piece of string, and we cut it into one piece (divide it by one), how long is the piece? Why, the same length as the original string. Or we are dealing a deck of cards, but to only one player. How many cards does our single player get? All of them, the original number of cards in the deck.
Now we've taken a look at Hatbox / 1, and Hatbox / Hatbox, and you'll notice in both cases, Hatbox is the top value of the fraction. When we divide one by Hatbox, we have what's called a reciprocal:
1 / Hatbox.Any number, multiplied by its reciprocal, is one. Let's look at an example.
4 x 14 = 1. This makes sense. If we start with 1 cookie, and cut it into 4 pieces, each piece is 1/4 and we have four of them. So the reciprocal is a single fractional piece. One quarter of the cookie. Multiplying by four reassembles the original cookie.
We can prove this with Hatboxes.
We start with
Hatbox x (1 / Hatbox)We know that Hatbox / 1 = Hatbox, so we substitute Hatbox / 1 for the first Hatbox, and get
(Hatbox / 1) x (1 / Hatbox)We multiply the numbers on top, and the numbers on the bottom, and get
(Hatbox x 1) / (1 x Hatbox)Which is the same as
Hatbox / HatboxSince multiplying by 1 doesn't change the value.
Anything divided by itself is one, so
(Hatbox x 1) / Hatbox = 1Have fun in the comments.