How long of an answer do you want?
No, rationality isn't dependent on base.
But it's still irrational.
So
.
Yes, but it has a nicer description in other representations.
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My Issue With Pi
- Thread starter Skwim
- Start date
How long of an answer do you want?
No, rationality isn't dependent on base.
Yes, but it has a nicer description in other representations.
Skwim
Veteran Member
Skwim
Veteran Member
Actually there is a base where pi is rational. Use base pi. In that base pi equals a single unit “1” of that base pi. Of course rational numbers from our base 10 system become irrational using that base, but it would make pi completely rational and solve the “neatness” thing for OP.
Subduction Zone
Veteran Member
Or as I like to write it ln(-1) = i*pi.
I used to feel unconfortable with the existence of irrational numbers.
This disconfort subsided somewhat once I realized that quantities generally lack any reason to confortably coincide with such strong restrictions as those of rational numbers.
But it took quite a while.
This disconfort subsided somewhat once I realized that quantities generally lack any reason to confortably coincide with such strong restrictions as those of rational numbers.
But it took quite a while.
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Subduction Zone
Veteran Member
@Shaul already brought that up. I don't know how one would do an irrational base, but it seems that would be the case. In base Pi Pi would be 10. I am trying to figure out how one would do that. zero is always zero regardless of base and 1 should also always be 1. Since n^0 is equal to (n^x)/(n^x) = n^(x - x) = n^0 = 1. I have a feeling that an irrational base may not be possible for a number system.
Altfish
Veteran Member
Yes I know the equation. But when I said I didn’t know how to type it, I meant that I wasn’t sure how to get ‘powers’ on this message board.
exchemist
Veteran Member
Ah yes indeed, so I see.
I can use superscripts for powers for numbers and Roman alphabet letters but not for Greek so it doesn't help when it comes to π, which is why I resort to exp( ) . But some people use ^ I notice, which has the advantage of being usable for other things than exponents.
It isn't *rational* in that base. It merely has a terminating expression in that base. Different things.
You have to be careful here, as the complex logarithm is multiple valued.
It is possible, but computations of things like addition and multiplication are horridly difficult in irrational bases. And yes, ordinary integers would have a non-terminating expression in an irrational base.
Here's a treatment of Archimedes' method:
Archimedes' Approximation of Pi
And a bit of video for the same method:
There are other, much more efficient methods to calculate pi now, mostly based on some trigonomeric formula and some sort of series expansion.
exchemist
Veteran Member
I confess I've never worked out how to use LaTex. Is it a feature that comes with websites or is it some extra gizmo I need to install on my PC?It is also the standard for LaTeX, the type-setting program for math.
It is a typesetting program that is freeware and needs to be installed (along with fonts, etc). For a Windows machine, I'd recommend MikTeX. For Linux, there is usually a package for LaTeX. I don't know what the typical version is for Macs.
There is a learning curve for the use of LaTeX and it might not be useful unless you plan to publish mathematics (or physics...sometimes chemistry). Most math journals require LaTeX submissions (Windows Word not accepted).
exchemist
Veteran Member
Thanks, that's what I suspected. I don't think I'll bother, as I only need to use equations of any complexity very rarely. I can usually muddle through with the Mac symbols, most of the time.
Skwim
Veteran Member
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That's only because you're using base 10. If you use base pi, then pi is expressed with a single digit.I recognize this is purely personal, but it's one of those disturbing things in the universe that just doesn't seem right.
Please bear with me.
Here we have the simplest shape in the universe, a circle.
A shape in which a straight line drawn through its center from any point on the circle to another, its diameter, (d) is always the same length. Neat and tidy, right? Now, regardless of the size of the circle its circumference will always have the same relative length to its diameter. And as it turns out, the length of the circumference is always 3.1415926535897932384626433. . . . times as long as the diameter. This ratio of 3.1415926535897932384626433. . . . to 1 is called pi, π.
You will note the ellipses following the numbers. These indicate there are more numbers to follow. The ratio of the circumference of a circle to its diameter is not just 3.1415926535897932384626433. . . . to 1 but something far more precise. In fact, in November of 2016 pi was calculated to be at least 22.4 trillion digits long, and with no end in sight. Such numbers, with no repeating pattern of digits, are called irrational.
Now here's my vexation. WHY? Why does the simplest shape in the universe have such an unsettled, Inexact, Irrational, Stupid nature? One would think (I do anyway) that in nature the very simple wold be at least relatively simple throughout. And if anything in nature should be neat and tidy one would expect it to be the circle. It's just a ....simple! ....round! ....shape!....for crying out loud.
digits long. ..............................It's almost enough to make one believe in god, and not one of those a nice gods either.
Okay, rant over.
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