Re: i dont like irrationals

From: The Ghost In The Machine (ewill_at_sirius.athghost7038suus.net)
Date: 10/09/04 

Date: Sat, 09 Oct 2004 08:00:06 GMT

In sci.math, Keckman
<keckman@welho.com>
 wrote
on Sat, 09 Oct 2004 03:22:35 +0300
<opsfktzxve3uk9lu@cs81133.pp.htv.fi>:
> Sorry, this is stupid, i agree. I just could not get sleep
> and wondered how to get rid of endless decimal numbers,
> which i don't like

The simplest method is arguably to assign a symbol thereto.
Instead of dragging around 3.14159265...everywhere, we
call it pi; instead of writing 2.718281828459... every
time we need a certain concept, we call it Euler's Number,
or just e,

> ---
> We say that there are irrational numbers like sqrt(2)
> because the ratio between square's diagonal and side is
> sqrt(2), but the diagonal's length itself could
> be some n in N, if we decide to make a standard
> measure unit so.

More or less correct, although if one makes the side 1, one
gets a hypotenuse of sqrt(2), and if one makes the hypotenuse
of 1, one gets a side of 1/sqrt(2). Pick your poison.

>
> We have no method to tell about some measure unit
> if it is irrational or not.

Correct; one can only determine that two lengths are
incommesurable with each other. Many people will
use standard lengths (1 meter, 1 inch, 1 mile, etc.)
for one of them; it depends on the problem.

> And the choose of
> that measure is not in math, nature or god but
> it is from human.
>
> So. I accept only Natural numbers. Even i had
> to deny rational numbers. Then there is not that
> ratio between square and diagonal but only
> natural numbers and some function out of them
> which is not number but result.
>

So, if one had a cake recipe that served 8 and took
a pound of butter to make, but only invited 4 people
over, what does one do? :-) Most people would use
a 1/2 pound of butter, although in a pinch one might
say 1 lb = 16 oz and use 8 oz of butter, but that
method only goes so far (suppose the cake served 9,
instead?).

If one had an automobile which got 30 miles to the gallon,
gasoline cost $2.40 per gallon, and the corner grocer was
4 miles away, how much does a round trip thereto cost?
Even if one counts pennies ($2.40 = 240 pennies), one
encounters some rather insidious fractions. The simplest
method of computing the answer, of course, is to:

[1] compute the total mileage: 8 since it's round-trip.
[2] compute the number of gallons -- 8/30
[3] multiply that by the cost: 8/30 * 2.40 = $0.64 .

If one wants to avoid fractions entirely, one can compute
the cost of 30 trips (which would take 8 gallons or
$19.20) then divide by 30, but what's the difference,
really? I could just as easily pick 29 mpg and a gas price
of $2.23...

On a more theoretical basis, one can get to the complex plane
by starting with the natural numbers, and appropriate extensions.
The steps might be as follows.

[1] Peano's Axioms, defining the natural numbers.
[2] Addition.
[3] Subtraction of a - b where a > b.
[4] "Discovery" of 0 = a - a, and identifying its properties.
[5] Subtraction of a - b where a < b -- negative numbers/arithmetic inverses.
[6] Multiplication a * b and division a / b, if b evenly divides a.
[7] "Discovery" of rational numbers, and extension of division.
[8] Endlessly repeating decimals.
[9] Dedekind cuts and power sets -- Cantor's two proofs.
[10] Square roots of positive numbers.
[11] sqrt(-1) and the complex plane.

At this point I'll have to look up the Fundamental Theorem of Algebra,
since it's been a long time since I've had to work with it, but it
turns out the complex plane is "enough"; one needn't extend the
number system much farther, although one can create specialized subsets
(e.g., transcendental numbers, algebraic units).

Do not fear rational numbers, for such fear is decidedly irrational,
and will give you a complex. :-) Transcend your difficulties,
and you'll see the whole. (Besides, all numbers are imaginary anyway. :-) ) 

-- 
#191, ewill3@earthlink.net
It's still legal to go .sigless.
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