art with code

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)
Post a Comment

About Me

My Photo

Built art installations, web sites, graphics libraries, web browsers, mobile apps, desktop apps, media player themes, many nutty prototypes, much bad code, much bad art.

Have freelanced for Verizon, Google, Mozilla, Warner Bros, Sony Pictures, Yahoo!, Microsoft, Valve Software, TDK Electronics.

Ex-Chrome Developer Relations.