Re: Chipmunk Basic arrays (Attn: Ron Nicholson)

On Thu, 27 Sep 2007 03:31:16 -0400, Tom Lake wrote:

"David Williams" <david.williams@xxxxxxxxxx> wrote in message
news:1190858989.877.1190835036@xxxxxxxxxxxxx
-> How can a transcendental number have "an exact value"?

They do have exact values. They just can't be written down in numerical
form.

That's like saying, "This car uses no hydrocarbons. It just runs on gas."

Not quite. It's an admission that it is impossible to express some
things exactly in a way that we would prefer.

A number doesn't have to be transcendental before it "can't be written
down in numerical form."

The square root of two is (merely) an irrational number -- it is not
transcendental. If we try to write that number down in decimal form,
it is a non-terminating, non-repeating string of digits.

There is no "last" digit in the square root of 2. If there is no last
digit, then there can never come a time when anyone might truthfully
say, "I have finished writing down the exact decimal representation
for the square root of 2." If that time can never come, then it
"can't be written down in numerical form."

Nor is there a "last" digit in one-third. We can write it in a kind
of shorthand with a trailing ellipsis:

0.333...

Which means we would continue writing 3s forever if we could and they
were important to us -- but we can't and they aren't.


In math we consider PI *not* to have an exact value.

But it does.

Consider the following function:

y = 4 * arctan(x)

It produces "one and only one" exact y-value for each exact x-value
over a certain range of x-values.

The inverse tangent of 1 is exactly pi/4.

The value of pi is given exactly by the following formula:

pi = 4 * arctan(1)


Transcendentals are numbers that have no algebraic representation.
e and PI are two of those.

1/3 is not transcendental since it is algebraic (can be represented by
a polynomial)

http://en.wikipedia.org/wiki/Algebraic_number
an algebraic number is a root of a non-zero polynomial
with rational (or equivalently, integer) coefficients.


At any rate, transcendental numbers are not a special case when it
comes to storing values which cannot be represented exactly in
floating point format.

A 20-digit truncation of pi is not transcendental, irrational, or
non-terminating. It has a manageable number of digits if one wants to
write it down. But it can't be stored exactly in a double-precision
variable either.

Adam

--
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.

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