Thread: Mathematical constants

Is there any mathematical reason why mathematical constants all need to be occuring between 0 and say 5? (i.e. why they are all relatively small).

The largest known constant seems to be the Feigenbaum constant at about 4.66 (happens to be the last one to be discovered as well).

Is this related to the fact that many of these constants can be produced by convergent series (i.e. starting with 1 and plus/minus an infinite number of diminishing terms)?

Originally Posted by accountabled

Is there any mathematical reason why mathematical constants all need to be occuring between 0 and say 5? (i.e. why they are all relatively small).

The largest known constant seems to be the Feigenbaum constant at about 4.66 (happens to be the last one to be discovered as well).

Is this related to the fact that many of these constants can be produced by convergent series (i.e. starting with 1 and plus/minus an infinite number of diminishing terms)?

10 is a constant.

So is

Both are larger than 5.

If X is the largest constant, then what is X+1 ?

I think what he is referring to is this:
http://en.wikipedia.org/wiki/Mathema...ical_constants

One attempt at an explanation: you generally start with unity, and for e.g. geometrical constants, you'd need rather high aspect ratio's. If the numbers used to get a constant are all in the same order of magnitude, you get a number between 0.1 and 10.

Another one is that we might prefer a smaller number. Instead of , we could define another constant equal to , which wouldn't be unreasonable, but you'd get a higher number, and maybe people don't like higher numbers.

it makes the math easier

One fundamental problem with the concept of 'mathematical constants' is already described on the Wiki site:

"What it means for a constant to arise "naturally", and what makes a constant "interesting", is ultimately a matter of taste, and some mathematical constants are notable more for historical reasons than for their intrinsic mathematical interest."

"Matter of taste" doesn't of course easily translate into the language of mathematics. Although to a certain extend 'beauty' and 'complexity' of theorems can actually re-inforce or discourage the mathematician's intuition about whether he/she is on the right track or not.

My initial interpretation of these constants is that they act as a kind of numerical 'atoms' from which more complex structures can be created. They can't be split into other atoms. And it actually doesn't matter from which field of mathematics they have been derived.
Therefore or are imo not mathemical constants. And the same for is true for . But then why should or be one?

The challenge is that any number that can't be factored into smaller components should then be classified as a mathematical constant. The primes come to mind, since they serve as the basic building blocks for all the composites. However, all primes can be factored into sums of integers and I therefore could find some logic to rule them out.

But then what about all irrational numbers? There are an infinite number of them and most can't be factored into smaller irrational numbers.

I give up. Too many definition problems to give any meaning to my hypothesis that all mathematical constants tend to grow like weed between the numbers 0 to 5...

Originally Posted by Bender

I think what he is referring to is this:
http://en.wikipedia.org/wiki/Mathema...ical_constants

One attempt at an explanation: you generally start with unity, and for e.g. geometrical constants, you'd need rather high aspect ratio's. If the numbers used to get a constant are all in the same order of magnitude, you get a number between 0.1 and 10.

Another one is that we might prefer a smaller number. Instead of , we could define another constant equal to , which wouldn't be unreasonable, but you'd get a higher number, and maybe people don't like higher numbers.

My point is really that mathematics has essentially no use for constants, at least in the sense that physics does.

In physics there are some fundamental constants with deep meaning -- Planck's constant, the fine structure constant, etc.

In mathematics there is no emphasis on "constants". and are well known and important transcendental numbers, but are not treated as constants in the sense of physics.

In quite a few years studying mathematics in graduate school, I can tell you that not once were "constants" discussed as a topic. Of course everyone knew that the circumference of a circle is but the fact that the ration of the circumference of a circle to the radius is a constant is not very difficult to prove and what you get is a number that has some interest but not a "fundamental constant" that affects the structure of mathematics.

I would be willing to guess that if you took a pole of professioinal mathematicians, most would not have heard of more than one or two of the constants in that list, and would not care either.

Originally Posted by DrRocket

(...)
I would be willing to guess that if you took a pole of professioinal mathematicians, most would not have heard of more than one or two of the constants in that list, and would not care either.

I understand your logic, however I also do recall from reading James Gleick's book 'Chaos', that the first thing Feigenbaum did after he discovered the new constants, was to try to relate them back to and . I guess that's the same thing Riemann tried to do when he found the first non-trivial zero at . Also when Euler managed to prove the closed form of he probably immediately started to look for series that would produce some other "mathematical constants".

I believe these constants could therefore well reside at the forefront of the minds of many mathematicians (and most likely with those that are into number theory).

Originally Posted by accountabled

Originally Posted by DrRocket

(...)
I would be willing to guess that if you took a pole of professioinal mathematicians, most would not have heard of more than one or two of the constants in that list, and would not care either.

I understand your logic, however I also do recall from reading James Gleick's book 'Chaos', that the first thing Feigenbaum did after he discovered the new constants, was to try to relate them back to and . I guess that's the same thing Riemann tried to do when he found the first non-trivial zero at . Also when Euler managed to prove the closed form of he probably immediately started to look for series that would produce some other "mathematical constants".

I believe these constants could therefore well reside at the forefront of the minds of many mathematicians (and most likely with those that are into number theory).

If they reside there I have seen no expression of that fact from any of the many mathematician that I know, including me.

There are several proofs that , including one that uses Fourier series that I find interesting. But I still don't think of as a fundamental constant.

Every convergent series produces a constant. And given any number at all it can be realized as an infinite series. It is only when a useful quantity is related to a series in some natural way that anything interesting arises.

BTW I must admit that I have assiduously avoided reading Glieck's book. That is for two reasons. First, I generally only read popularizations written by people who have actually been involved in the research being described. I do not normally read popularizations by professional popularizers. Second, I have a strong bias against may things that go under the name of "chaos". I respect mathematicians who work in topological dynamics, ergodic theory, dynamical systems, etc. Poincare, who initiated many of the ideas with his work in celestial mechanics and stability was clearly a first-rate mathematician. But, I have yet to meet anyone who claimed to work in "chaos theory" who could prove anything, or who really understood mathematics. If you would like to read a real mathematics book that actually covers a bit of chaos in a fully rigorous way I can recommend Bob Devaney's book An Introduction to Chaotic Dynamical Systems.

Originally Posted by DrRocket

Originally Posted by accountabled

Originally Posted by DrRocket

(...)
I would be willing to guess that if you took a pole of professioinal mathematicians, most would not have heard of more than one or two of the constants in that list, and would not care either.

I understand your logic, however I also do recall from reading James Gleick's book 'Chaos', that the first thing Feigenbaum did after he discovered the new constants, was to try to relate them back to and . I guess that's the same thing Riemann tried to do when he found the first non-trivial zero at . Also when Euler managed to prove the closed form of he probably immediately started to look for series that would produce some other "mathematical constants".

I believe these constants could therefore well reside at the forefront of the minds of many mathematicians (and most likely with those that are into number theory).

If they reside there I have seen no expression of that fact from any of the many mathematician that I know, including me.

There are several proofs that , including one that uses Fourier series that I find interesting. But I still don't think of as a fundamental constant.

Every convergent series produces a constant. And given any number at all it can be realized as an infinite series. It is only when a useful quantity is related to a series in some natural way that anything interesting arises.

BTW I must admit that I have assiduously avoided reading Glieck's book. That is for two reasons. First, I generally only read popularizations written by people who have actually been involved in the research being described. I do not normally read popularizations by professional popularizers. Second, I have a strong bias against may things that go under the name of "chaos". I respect mathematicians who work in topological dynamics, ergodic theory, dynamical systems, etc. Poincare, who initiated many of the ideas with his work in celestial mechanics and stability was clearly a first-rate mathematician. But, I have yet to meet anyone who claimed to work in "chaos theory" who could prove anything, or who really understood mathematics. If you would like to read a real mathematics book that actually covers a bit of chaos in a fully rigorous way I can recommend Bob Devaney's book An Introduction to Chaotic Dynamical Systems.

Fair enough, but I'd like to defend Gleick's book as a good read for non-mathematicians, and a good way to start thinking about dynamical systems.

Secondly, I suspect that pi and e, being explored in terms of their properties, during the 1700s and 1800s, were much more in the forefront of mathemticians' minds at the time, and, when investigating other mathematical formulae, mathematicians would naturally strive to see if there was a connection between them because that would actually simplify their analyses.

Thirdly, pi and e are implicitly present in a number of modern mathematical situations - the sine wave for instance, a key component of modern electronics, needs pi (or a similar irrational quantity) to function. And e is, of course, the base of the natural logarithm, and therefore implicit in the original prime number formulae of Gauss etc. As you point out, however, this doesn't make them 'constants' in the way physics has constants, but it does make them numbers that crop up quite frequently in modern theoretical, and applied, mathematics (as it happens I have read quite a few mathematicians who suggest that pi is not an intrinsically interesting number except to laymen!)

Finally, to accountabled's OP, if these so-called constant's were in binary, you wouldn't notice their size as much, so the base alone can make something of a difference...

Mod note: Please try to avoid quoting posts in their entirety, and most especially try to avoid "nesting" quotes.

Originally Posted by sunshinewarrior

Thirdly, pi and e are implicitly present in a number of modern mathematical situations - the sine wave for instance, a key component of modern electronics, needs pi (or a similar irrational quantity) to function. And e is, of course, the base of the natural logarithm, and therefore implicit in the original prime number formulae of Gauss etc. As you point out, however, this doesn't make them 'constants' in the way physics has constants, but it does make them numbers that crop up quite frequently in modern theoretical, and applied, mathematics (as it happens I have read quite a few mathematicians who suggest that pi is not an intrinsically interesting number except to laymen!)

What is far more important than the number is the exponential function associated with it and the extension to complex numbers which yields the Euler formula . The exponential function has enormous implications, is used extensively in physics and electrical engineering and is the foundation for complex analysis.

is less important and arises in modern mathematics as just 1/2 of the period of the function .

I don't understand your statement that the sine needs or a similar irrational quantity to function. Certainly .being the period, is important, but the sine and cosine really find their origin in the exponential function and Euler's formula. Euler's formula actually serves as a rigorous definition for the sine and cosine, one that can be made without the necessity to draw pictures, as is usually done in an elementary trigonometry class. (Drawing the pictures is still important for understanding and intuition, but it is essential to have a rigorous definition that can be traced directly back the Peano Axioms.)

Now that we're on the subject of and , I think the best way to settle the debate about which one is more important is to test whether one could be derived from or expressed in the other.

Therefore checked the web:

The most obvious one that comes to mind is of course , but there's not a really nice way to isolate or . Although is probably correct, it doesn't feel good enough.

Then tried a connection via the primes with , and there is indeed also a way to derive from primes by:

with being the sequence of primes.

But the strongest connection I found is from Gosper:



Although in the latter appears to be "born" from 's only, I'd still opt for an ex equo since in neither connection there is a clear solution for the "chicken and egg" problem. :-D

Originally Posted by accountabled

Is there any mathematical reason why mathematical constants all need to be occuring between 0 and say 5? (i.e. why they are all relatively small).

That must be because scientists often express physical quanitities in the form where is an integer and The fact that may be what gives you the illusion that the number is “small”.

In hind sight I still would like to declare as the winner since I managed to express fully in and all positive integers.




Who can do the same for in terms of and positive integers?

But the thing is, both and arise out of different natural situations. Neither actually comes from the other. They just happen to have various relationships that are significant to mathematics.

Originally Posted by Chemboy

But the thing is, both and arise out of different natural situations. Neither actually comes from the other. They just happen to have various relationships that are significant to mathematics.

Chemboy,

I understand. But how do you know for sure that there isn't another underlying "constant" that unifies both of them? I mean, is there any evidence that we have hit the 'atom' state of and or are they half-fabricates of something deeper?

Originally Posted by accountabled

I understand. But how do you know for sure that there isn't another underlying "constant" that unifies both of them? I mean, is there any evidence that we have hit the 'atom' state of and or are they half-fabricates of something deeper?

I would say hints a a such relationship.


One thing that confuses me: in math, a constant is a non-variable. Therefore, 1,000,000,000,000,000,000,000,000,000,000,000,000, 000,000 is a constant, and is much much bigger then 5.

Check out Gelfond's constant, found at http://en.wikipedia.org/wiki/Gelfond%27s_constant

Quote from the page:
"In mathematics, Gelfond's constant, named after Aleksandr Gelfond, is , that is, e to the power of π..."

The value of Gelfond's constant is Ironically, subtract 20 and the first three digits are 3.14