Math Is Fun Forum

I came up with this idea last night while brushing my teeth!

Let K denote the field of rationals.
Let U := { u | u = a[sub]1[/sub][sup]r1[/sup]a[sub]2[/sub][sup]r2[/sup] … a[sub]n[/sub][sup]rn[/sup]} where {a[sub]1[/sub] … a[sub]n[/sub]} are prime, and  {r[sub]1[/sub] … r[sub]n[/sub]} are in K
For all u,v in U and k in K:
let u+v := uv,
let ku := u[sup]k[/sup],
let 0 in U := 1
Then we have U is a vector space over K.

Proof:

Associativity:
(u+v)+w = (uv)w = u(vw) = u+(v+w)

Zero vector existence and properties:
0+u = (1)u = u = u(1) = u+0

Negative vector existence and properties:
For every vector u = a[sub]1[/sub][sup]r1[/sup]a[sub]2[/sub][sup]r2[/sup] … a[sub]n[/sub][sup]rn[/sup],  there exists (-u) = a[sub]1[/sub][sup]-r1[/sup]a[sub]2[/sub][sup]-r2[/sup] … a[sub]n[/sub][sup]-rn[/sup]  such that:
u+(-u) = (-u)+u = (u)(u[sup]-1[/sup]) = 1 = 0

Commutativity:
u+v = uv = vu = v+u

Distributivity of scalers:
k(u+v) = k(uv) = (uv)[sup]k[/sup] = u[sup]k[/sup]v[sup]k[/sup]= (u[sup]k[/sup])+(v[sup]k[/sup]) = ku + kv

Distributivity of vectors:
u(k1+k2)= u[sup](k1+k2)[/sup]=u[sup]k1[/sup]u[sup]k2[/sup]= u[sup]k1[/sup]+u[sup]k2[/sup]=k1u + k2u

Scaler Associativity:
(ab)u = u[sup](ab)[/sup]=u[sup](ba)[/sup]= (u[sup]b[/sup])[sup]a[/sup]=a(u[sup]b[/sup])=a(bu)

Unit Scaler:
(1)u = u[sup]1[/sup]=u

Here is an amusing method for finding the sine of an angle that makes intuitive sense:

suppose we  want to find the sine of 1 radian, about 57 degrees.

First we choose a power of 2 that is large relative to the angle. I'll choose 32.

1/32 is fairly small and so sin(1/32)  ≈ 1/32, and cos(1/32)  ≈ 1.

Now that we know sin(1/32) and cos(1/32) we can use 5 applications of the double angle identities: sin(2x) = 2sin(x)cos(x) and cos(2x)=cos^2(x) - sin^2(x)  to get the final sine and cosine!
I worte a simple program to do it for me. Have a look:

sin(1/32): 0.03125 cos: 1
sin(2/32): 0.0625 cos: 0.999023
sin(4/32): 0.124878 cos: 0.994142
sin(8/32): 0.248293 cos: 0.972723
sin(16/32): 0.48304 cos: 0.884541
final (32/32): sin: 0.854537 cos: 0.549085

results by windows calculator:
sin: 0.8414709848078965066525023216303, cos: 0.54030230586813971740093660744298

Hey! Thats pretty close! Of course! you can make it more accurate by choosing a larger power of 2. Choosing 2^10 for instance yields: sin: 0.841881 cos: 0.540566

Its hardly the fastest way to do it, but its easy to understand why it works and I think that makes it fun!
Originally I envisioned this as a series of line reflections. But I eventually realized I was essentially using the double angle formulas. Maybe I'll post a diagram later.

I started avoiding these forums because one member was making things a bit unpleasant for me but it appears that member isn't posting so much anymore.

Anyways! I got my associates degree in computer science and i should have one in math but I was one humanties class shy! but I changed schools and I'm continuing my Computeer Science education though i'm considering switching my major to math. I am unhappy with how little mathematics I've learned since I entered college, relative to before when I was self taught,  and I'm trying to find time for more self study. These forums are also a great place to learn and practice math, by asking and answring questions, so I thought I should frequent them more!

anyways, good to be back everyone!

that i made up myself!

Whats a mathematicians favorite soda?

square root beer!

Whats a mathematicians favorite beer?

Natural Lagger!

Why did einstein cross the road?

well he insists the road crossed him!

Why did the mathematician cross the road?

he never crossed it, he approached it asymptotically

what is a helicopter bagel?

its like a plain bagel only it takes off vertically!

:]