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Visualization of Pi being irrational.
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Previously:
https://www.homemadetools.net/forum/...698#post209675
https://www.homemadetools.net/forum/...444#post231337
https://www.homemadetools.net/forum/...730#post225945
https://www.homemadetools.net/forum/...075#post224569
Another way to appreciate the infinite nature of the expansion of pi is to examine a trick used by programmers too lazy to memorize pi out to the precision their computer could handle.
Most programming languages have a built-in arctangent function. Now, it's true that:
arctan (1) = π / 4 (π/4 radians = 45 degrees)
so:
π = 4∙arctan (1)
The inifinite series for arctan (1) looks like:
arctan (1) = 1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11 + ...
Since the series continues like that out to infinity, so does the decimal expansion of π.
It is interesting to think that our universe could not exist if Pi had a different value. The same is true for other mathematical constants. One theory is that there are an infinite number of universes, one for each value of these constants. It blows my mind!
Rick
Perhaps, it Is not that our universe could not exist unless Pi had this value. Rather it is that the universe does exist, as it does, because it does have this value.
We do not know what would exist if it had some other value.
And that is what challenges my mind!Quote:
Originally Posted by hemmjo
What is so special about the value of Pi? It helps us explain many things in our lives. Did the physical world come first and then Pi set to explain it or did Pi come first and the physical world confirmed to it?
I look forward to seeing some brilliant experimental physicist proving it all.
Rick
Newton characterized gravity as an inverse square force. It's good he did because, if that exponent differed from two even a tiny bit, stable orbits could not exist and our world could not exist. It's another example of "things are the way they are because it would be impossible for them to be any different".
Now, if you like to soothe your insomnia with mind-bending questions, here's another one for you...
Science has had amazing success using mathematics to understand our world and even predict things unknown that after being predicted were discovered. Now, is mathematics simply the handiwork of human genius or is it a fundamental of the universe that we are gradually discovering and putting to use?
I think the behavior is there and we discover the math that describes it. Change the math and the universe would laugh.
Rick
I agree. But, there is a large part of mathematics which do not relate to phenomena. You might argue, for example, that prime numbers are found in Nature, in the Fibonacci series. But a glance at a copy of Dictionary of Curious & Interesting Numbers reveals something called Graham number, which cannot be expressed at all using conventional notation, or powers of powers. There is an interesting chapter in “Pi in the Sky” by John Barrow, entitled transhuman mathematics. In it, the author asserts that there are non-computable functions. It was published in 1992Quote:
Originally Posted by rgsparber
Even the philosophy of mathematics is a wide subject and who can say they have mastered it?