2012-01-27

Very basic math

I was playing around with the idea of presenting fractions in the same way as negative numbers. Instead of 1/x, you'd write /x. Just like instead of 0-x, you write -x. And since multiplication with single-letter symbols is often annotated with putting the symbols next to each other, marking the inverse with /x looks quite natural: A x /B = A/B, 9 x /7 = 9/7.

It also makes you think of the inverse in less magical terms. Consider the addition rule for fractions:


A   C   AD   BC   AD + BC
- + - = -- + -- = -------
B   D   BD   BD     BD

There's some crazy magic happening right there. The literal meaning is (A x D x 1/B x 1/D) + (C x B x 1/D x 1/B), but you wouldn't know from looking at that formula. And it gets even more confusing when you start multiplying and dividing with fractions. Think about the following for a moment:

A   C   AD
- / - = --
B   D   BC


Right?


In linear notation with /B and /D and suchlike, this all actually sort of makes sense in a non-magical way. Here's the first of the above two examples (with intermediate phases written out):


(A x /B) + (C x /D)
= [1 x (A x /B)] + [1 x (C x /D)]
= [(D x /D) x (A x /B)] + [(B x /B) x (C x /D)]
= [(A x D) x (/B x /D)] + [(B x C) x (/B x /D)]
= (/B x /D) x [(A x D) + (B x C)]

 [here's where you go: "oh right, /7 x /4 = /28", analogous to 7 x 4 = 28]

And the second one:

A x /B x /(C x /D)
= A x /B x /C x D
= (A x D) x (/B x /C)


Note the similarity with addition:


A + -B + -(C + -D)
= A + -B + -C + D
= (A + D) + (-B + -C)


Now, you might notice that there is a bit of magic there. How does /(C x /D) magically turn into (/C x D)? Or -(C + -D) to (-C + D) for that matter. Let's find out! Here's how it works:


/(C x /D)
= 1 x /(C x /D)
= [(/C x D) x /(/C x D)] x /(C x /D)
= (/C x D) x /(/C x C x D x /D)
= (/C x D) x /(1 x 1)
= (/C x D) x /1 -- Remember the axioms 1 x N = N and N x /N = 1. Since 1 x /1 = 1 we get /1 = 1.
= (/C x D) x 1 = (/C x D)


For the -(C + -D) case, replace / with -, x with + and 1 with 0.


And there you have it, my small thought experiment. And derivations for some basic arithmetic rules. I kinda like how breaking the magic bits down into the basic field axioms makes things clearer.


[edit]

Why is /A x /B = /(A x B)?

/(A x B) x (A x B) = 1 


1 x (/A x /B) = (/A x /B)
/(A x B) x (A x B) x (/A x /B) = (/A x /B)
/(A x B) x (A x /A) x (B x /B) = (/A x /B)
/(A x B) x 1 x 1 = (/A x /B)
/(A x B) = (/A x /B)
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