What's your favorite transcendental number?

[Janderson]

Damn, I don't remember learning this in Kindergarten. What is this anyway, super-advanced mathametics or some form of physics? And I ask this with respect (because maths teachers tend to explode at me) How does could these numbers be used in problems/life?

[ducon]

Janderson said:

How does could these numbers be used in problems/life?

Why? They should?

[Janderson]

Well no, but why are you learning them?

[Grazza]

In some cases, the very reason these numbers are of interest is because they arise from physical problems, or the functions to which they are related are involved in the solution of physical problems (e.g. in engineering).

pi has obvious physical importance.

The exponential function (e^x) is the basis for a huge variety of functions and methods.

Yeah, it's not kindergarten stuff - more like early undergraduate level, and the distinction between transcendental and non-transcendental is more important in Pure Mathematics than Applied - though even in Applied Mathematics it is important that there is reasonably firm basis for what you're doing. If you want to call upon some theory, then you'll want to be sure it does actually apply to the numbers you're dealing with. Having said that, I'm not aware of any instances of a building falling down because of an issue of that type. ;)

[AndrewB]

[Volumetric capacity of a cube] / [Volumetric capacity of the largest sphere that could fit inside the cube]

Could someone tell me what the resulting constant number is? I've always wondered. I'm guessing it's something around 1.7.

[AndrewB]

How about this. Take a random alteration between two symbols:

OOXXOXOXXOXXXOXOOXXXX

There is 1 group of four (being the four X's at the end), 1 group of three, 4 groups of two, and 6 groups of one (heh).

The average group size in this particular case is 1.75. However, the average group size for an arbitrarily long, random set approaches a constant number. What is that number?

[Fredrik]

AndrewB said:

[Volumetric capacity of a cube] / [Volumetric capacity of the largest sphere that could fit inside the cube]

Could someone tell me what the resulting constant number is? I've always wondered. I'm guessing it's something around 1.7.

6/π = 1.909859... (result of simple algebraic manipulation of the expressions for the respective volumes of a cube and a sphere, with radius set to an arbitrary constant).

[AndrewB]

Pff. I don't know anything about algebraic manipulations, nor do I care. I just want to know how many spheres I can cram into a cube.

[ducon]

Janderson said:

Well no, but why are you learning them?

Because it’s my work, and because I’m a geek.

[Fletcher`]

AndrewB said:

Pff. I don't know anything about algebraic manipulations, nor do I care. I just want to know how many spheres I can cram into a cube.

Stuff a bunch of balls in a crate and tell me.

[exp(x)]

ravage said:

* rf` riles a geek's nerves: .9999~=1
<Carnevil_> WHAT THE FUCK
<Carnevil_> NO IT ISN'T
<Carnevil_> GOD THAT'S LIKE SAYING e = 2.5

Heh, too bad the sum of 9/10^n from n=1 to infinity does equal 1.

[spank]

Fredrik: I had thought about that, but then I realized how functions are sets too. (Probably not in the context of Lambda calculus, you tell me)

[Fredrik]

AndrewB said:

Pff. I don't know anything about algebraic manipulations, nor do I care. I just want to know how many spheres I can cram into a cube.

If you want more than one sphere, the problem is more difficult.

[Doom-Child]

Jesus. You people are just freaking weird.

DC

[Fletcher`]

exp(x) said:

Heh, too bad the sum of 9/10^n from n=1 to infinity does equal 1.

Heh, I'm like everyone else who isnt a jenius.

[Quasar]

e will always be my favorite. It tries really hard to be rational, and then goes totally nuts. Plus it has a cool fractional approximation: 271801/99990 ^_^

[Fredrik]

Not to mention its fabulous continued fraction representation.

[DOOM Anomaly]

Yikes.

I like math and all, but this blows me away. :P

It's all far too complicated for my short-attention span. :D

Remind me to come to you fellas when I need help with math. :D Today I learned what a transcendental number is, I'll be the coolest hobbit on the block with this knowledge under my belt. :D (Okay I admit I'm still a bit shady on it.)

I will feel like a big man when I ask my math teacher "WHAT IS E?" and they answer "something you should never do." I can't stump them with Pi, I think they already know that one.

I'm compelled to learn, but, it's just so much easier to spawn camp Baal's minions. :D

[Doom-Child]

DOOM Anomaly said:

I will feel like a big man when I ask my math teacher "WHAT IS E?" and they answer "something you should never do."

Congratulations, you win. That is awesome.

DC

[spank]

It's just lim (n+(1/n))^n

[ducon]

spank said:

It's just lim (n+(1/n))^n

No, lim (1+1/n)^n.

[exp(x)]

or lim(x->0) of (1+x)^(1/x). Whichever floats your boat.

[Fredrik]

(-1)^(-i/π) floats mine.

[ducon]

Fredrik said:

(-1)^(-i/π) floats mine.

There is just one problem: your definition is not univoque.

(−1)^(−i/π)=e^(ln(−1)×−i/π)
=e^(i(π+2kπ)×−i/π) (k∈ℤ)
=e^(1+2k) (k∈ℤ)

No?

[spank]

ducon: oops, you're right.

[Fredrik]

ducon said:

There is just one problem: your definition is not univoque.

(−1)^(−i/π)=e^(ln(−1)×−i/π)
=e^(i(π+2kπ)×−i/π) (k∈ℤ)
=e^(1+2k) (k∈ℤ)

No?

You're saying e^n = e for odd integer n, which is clearly wrong. The third expression doesn't follow from the second.

[ducon]

Fredrik said:

You're saying e^n = e for odd integer n, which is clearly wrong. The third expression doesn't follow from the second.

I say that ln(−1) has not only one value, that’s the meaning of the third expression.
(−1)^(−i/π)=e^(ln(−1)×−i/π)
=e^(i(π+2kπ)×−i/π) (k∈ℤ)
because ln(−1)=i(π+2kπ) and so,
(−1)^(−i/π)=e^(1+2k) (k∈ℤ)
because i(π+2kπ)×−i/π=1+2k.

[Kristian Ronge]

Quasar said:
e will always be my favorite. [...] Plus it has a cool fractional approximation: 271801/99990

Well, if we're going to discuss coolest fractional approximation I have to say I prefer 355/113, which approximates π to 6 correct decimals...

[Fredrik]

ducon said:

Right. So allow me to rephrase that as the smallest value of (-1)^(-i/π) that is larger than 1.

[Quasar]

Kristian Ronge said:

Well, if we're going to discuss coolest fractional approximation I have to say I prefer 355/113, which approximates π to 6 correct decimals...

Yes but 271801/99990 is correct for e to NINE decimal places, probably enough for any non-microscopic/non-astronomical engineering application ;) It's easy to derive through the formula used to convert a repeating decimal to a fraction, by using the approximation 2.718281828... for e.