Pi, it is said, when expressed in base 10, is an irrational and transcendental number. I wonder is anyone can prove this, because this is quite an assertion. Of course if pi can not be know with rational accuracy, it is rather hard to prove that the value of pi is really fuzzy. Its fuzzy nature maybe due to our lack of accurate methods for determining its actual value.
Pi was proved to be trancendental in the 19th century by Lindemann (I think I spelled his name right). Base has nothing to do with it.
>> Pi was proved to be trancendental Ah ha, is this an absolute algebraic proof or is it based upon the calculus of limits or approximations.... ?
It's a very simple algebraic proof, but depends directly on the proof that e is transcendental, which was established by Hermite a few years before. Not sure how that one works, but it had been accepted by just about everyone with any knowledge of number theory. I see no reason to think that pi might be algebraic. Nor do I understand what you mean by "accurate methods for determining its actual value."
>> "accurate methods for determining its actual value." well you can not directly measure the length of a curve accurately... you can not determine the true length of a curve by adding infinitesimal segments via a geometric method. We know the length of a straight line length, we define it as a length. But to equate the circle curve length to the diameter (straight length) must rely on some inaccurate mathematical logic.... therefore any result is only an approximation unlike 1 = 1 or a = b Indeed we make up a ratio value and we call it 'pi' such that pi*Diameter = length of circle arc drawn on diameter. so the ratio pi = arc length / diameter length But length must be an absolute value... and finite. Logic would then say that finite/finite is rational. Therefore diameter and arc length are absolute values, but pi (the ratio) can not accurately be determined by any method that I am aware of, because the actual arc length can not be accurately determined. IMO
URI: It is transcendental regardless of which base it is expressed in. Do you believe otherwise? If so, please give reasons. The value of pi is not fuzzy. It can be calculated exactly, to any degree of accuracy. Do you believe otherwise? If so, please give reasons.
You seem confused about the difference between a mathematical curve and a physical curve you would measure with a ruler. Do you appreciate the difference? You seem to be confusing the concepts "finite" and "irrational". Do you understand that a number can be both finite and irrational?
Sure you can. It's called a line integral. Although I suppose that won't satisfy you if you have some kind of problem with limits, but suffice to say... No, integer/integer would rational. Finite/finite doesn't tell you much at all. How accurate do you want? Pi has been calculate to over a trillion decimal places...
>> a mathematical curve and a physical curve ... what is the difference, other than accuracy ? >> a number can be both finite and irrational? in mathematics yes, but in reality if a length is of finite length then it can be defined as having a (definite) beginning and end.... and an absolute length in between. If I am using the wrong 'word' please correct me. >> It's called a line integral. Although I suppose that won't satisfy you if you have some kind of problem with limits, >>> yes that is the problem. accuracy that is not there, so to call pi transcendental is pushing past the limits of the mathematical method, IMO. >> Finite/finite doesn't tell you much at all.>> when applied to length, it defines the fraction as rational.... it is all a matter of scale... but length must be a whole number. >> Pi has been calculate to over a trillion decimal places... yes indeed, but is that the real "pi" ? that is the question.
In other words, pi has been calculated to the point that it is far more accurate than any measurements we can ever hope to make, not off by even a micron over the circumference of the known universe. The number of microns in that distance has less than 100 digits. 7 digits in the number of seconds in a year. 14 digits in the number of microns travelled in a second. 10 digits in 20 billion light years, as this is the American billion. Add another digit for the value of pi. Pi might as well be considered to be a whole number.
Something that is irrational and transcendental can also have an absolute value. Ermm... Do you understand that "transcendental" has a definite and technical meaning, here? What makes you think there is a scale such that only whole numbers is involved? Consider sqrt(2). That's simply the diagonal of a square's ratio to it's side length. But once we've proven the irrationality of sqrt(2), we know for certain that we can never measure both the side length and the diagonal to be a whole number. There's simply is no such square, regardless of scale and ruler length.
Which has no bearing on the issue of rationality. The only word that applies to such a length is that it will be a real number. This includes all rationals, irrationals and, yes, even transcendental numbers. I'm sure I have no idea what you mean by that. Why would the length from a to b have to be an integer? There's simply no basis for that... Well, it's the first trillion decimal places of it. What is the question again?
>> sqrt(2), we know for certain that we can never measure both the side length and the diagonal to be a whole number. >> Interesting 1^2 + 1^2 = 2 therefore length of square side for area =2 ----> 2^0.5 but as you can see this is a mathematical power manipulation. But I can see your statement is correct, the ratio can not be rational in base 10 Wondering if it helps though. It is not a direct equivalence such as a*x = y or pi*D = arc I will ponder this. Thanks >> Something that is irrational and transcendental can also have an absolute value.>> Maybe I am getting confused with terms I have long ago put down. Can you expand this a wee bit ? If it has an absolute value then it can be expressed as a whole number ??
Pi base ten? are you talking about the logarithm of pi? cuz if you are, the answer to *log(base10)pi=0.497149872* Arc the god
"...not be rational in base 10" makes no sense, a number is rational (or irrational) irregardless of the base you use to represent it.
What's wrong with power functions? The example shows that one must allow for irrational lengths, at least if one wants to talk about right triangles. It also shows that sqrt(2), while irrational, is algebraic (as opposed to transcendental), since it is the solution of the integer-coefficient polynomial equation x^2 - 2 = 0. Perhaps it should be mentioned that the definition of a transcendental number is that it is *not* the solutions to any integer-coefficient polynomial equation of finite order. No, absolute value and whole numbers have little to do with eachother. The absolute value of a real number x is simply sqrt(x^2). This works for any number, rational or irrational, algebraic or transcendental. Read this page: http://mathworld.wolfram.com/TranscendentalNumber.html
What else would it be? How could we find the "real pi" if the pi we are using is not real? You're rambling.
Thanks, 'absolute' in math is positive.... >> How could we find the "real pi" if the pi we are using is not real? best Q so far... Anyone know why the volume of a sphere is (4/3)*pi*r^3 ?
Yes, see the UniKEF's Balls thread and the Jello thread. It comes up in the integrals all the time. Basically, if you integrate 1 over a sphere you get that formula. I don't know why you think this is such a good question, it is usually a simple homework problem in high-school calculus. Offhand I can think of 3 ways to solve it (I have used all 3 methods) and I am sure there are more (that I haven't used). -Dale
