At 12:25 PM -0800 1/15/98, Bill Studenmund wrote:
>On Thu, 15 Jan 1998, Dave Huang wrote:
>
>> On Wed, 14 Jan 1998, Colin Wood wrote:
>> Algebraic functions are functions that can be made by performing a finite
>> number of algebraic operations (i.e. +, -, *, /, reciprocal, power, root)
>> on the argument of the function and rational constants. So like sqrt(x^5 -
>> 1/(sqrt(2/3 + x^9))) is freaky, but still an algebraic function.
>
>I agree with the jist of what you're saying, but I wouldn't put root and
>power to a non-integral value in there. Power to an integer exponent is
>just shorthand for multiplying and maybe reciprication.
>
>I'd say the big difference is in the rules about combining things. Like is
>the operation associative and commutative (sp?). +, -, and * are.
>Reciprication doesn't fit nicely. Nor does x**y.

I agree with the gist (correct sp) of what you're saying, but I'm pretty
sure / and reciprocal are included by definition.  The following indicates
that roots are also included.

Merriam Webster's 10th:
TRANSCENDENTAL  2 a: incapable of being the root of an algebraic equation
with rational coefficients <Pi is a ~ number>  b: being, involving, or
representing a function (as sin x, log x, e^x) that cannot be expressed by
a finite number of algebraic operations <~ curves>

ALGEBRAIC  2: involving only a finite number of repetitions of addition,
subtraction, multiplication, division, extraction of roots, and raising to
powers  <~ equation> --- compare TRANSCENDENTAL
>
>> Some (maybe all?) transcendental functions can be expressed as an infinite
>> sum of algebraic functions though. Taylor series and stuff like that...
>> Like sin x = x/1! - x^3/3! + x^5/5! - x^7/7! + x^9/9! - ... forever
>
>I think they all do, if you look in a good-enough reference book. :-)
>
Anything can be represented with an infinite series if you try.  It's
finding the simpler forms that takes the work.

>x**y is often written as exp(y*ln(x)), which is why I'd say it's a
>transcendental function too. And there's a Taylor series for sqrt too.
>sqrt(1+x) = 1 - 1/2 x + 3/8 x^2 ... (I'm pretty sure it's 3/8).

My interpretation of the definitions I found is that anything represented
as a finite number of exponentiations to a rational exponent qualifies
[e.g. a**(b**(c/d))].  Not what I thought.

Computationally, the sqrt is just a minor variation on division.  I believe
(though I can't quite see it this instant) that raising a number to a
rational power can be done with less general (possibly faster or more
accurate) routines by using the binary representation of the exponent and
"factoring" the computation into sequences of squaring and/or
square-root'ing.  Certainly any good FORTRAN compiler checks for a number
of special case exponents like 3/2 and such and calls special routines.

Second answer:

CRC Math Handbook says:

Real numbers [as distinguished from integer, rational, or complex] are
often divided into two subsets.  One subset, the algebraic numbers, are
real numbers which solve a polynomial equation in one variable with integer
coefficients.  For example; 1/sqrt(2) is an algebraic number because it
solves the polynomila equation 2 x^2 - 1 = 0, and rational numbers are
algebraic.  Real numbers that are not algebraic numbers are called
transcendental numbers.  Examples of transcendental numbers include PI and
E.

(Note that saying integer coefficients is the same as saying rational
coefficients since you can always multiply the equation by all the
denominators without changing the roots.)

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