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Unfortunately the author seems uninformed. He claims that 0^0 = 1, not indeterminate which is what Wolfram Alpha claims (and as anyone knows, if you use indeterminates you can prove any number equals any other number, which the author does in uncached screenshots). The problem though is that 0^0 changes its meaning depending on the context. If you have an function f(x,y) = x^y, it's not continuous at (0,0) because the limit of x -> 0 is 0 and the limit y -> 0 is 1, making the value undefined! (Which is still different from 1, 0, or indeterminate.) I used to know of a better blog post on the issue, but here's http://mathforum.org/dr.math/faq/faq.0.to.0.power.html

Oh wait, I found it: http://www.askamathematician.com/?p=4524 "Zero raised to the zero power is one. Why? Because mathematicians said so."

Sort of in the same character of dispute is the fact that 0 and 1 aren't probabilities: http://lesswrong.com/lw/mp/0_and_1_are_not_probabilities/

I'm also unsure what the author is entering since the screenshot isn't cached. If I try http://www.wolframalpha.com/input/?i=summation+x^n%2Fn!+from+n%3D0+to+infinity+where+x+%3D+0 wolfram gives the correct e^0 = 1.

Edit: Ah, he's using http://www.wolframalpha.com/input/?i=summation+0^n%2Fn!+from+n%3D0+to+infinity (substituting 0 in explicitly instead of instructing Wolfram to substitute on x) which is quite…

> There is absolute angular speed.

Not really. If you have two bodies in hydrostatic equilibrium arranged so that you can extend the rotational axis of one such that it overlaps completely with the rotational axis of the other, how do you come up with "absolute rotation"?

You could consider the case where one is exactly spherical and the other is highly oblate. While you're free to choose a system of coordinates which keeps the highly oblate one non-rotating with respect to the coordinates, you have to appeal to fictitious forces to explain the oblateness of the coordinate-stationary body and the exact spherical symmetry of the coordinate-rotating body. Physics typically takes a simpler form if you decide the spherical body should be non-rotating against a set of coordinates that covers both bodies. (On the other hand, if these are large bodies -- planets, for example -- and you can put a lab on the surface of each, you get the Special Relativistic form of physics for practically all possible experiments wholly contained within each lab).

However, a more typical case is that neither body is exactly spherically symmetrical, in which case you might want to appeal to the view of the motions of deep sky objects observable on each body. One might be ultra-Machian and say that "absolute rotation is determined by the movement of fixed stars", but really you probably should be more interested about whether you need to resort to pseudotensors in your write-down of…

Apparently man has long wanted to know
about the universe, God as maker of the
universe, etc.

Well, now we know much more about the
universe, our solar system, earth, and
life on earth than just 100 years ago.

Here is a little of what we know:

At

http://www.askamathematician.com/2017/06/q-where-is-the-middle-of-nowhere/

is

Q: Where is the middle of nowhere?

Posted on June 13, 2017 by The Physicist

with at

http://www.askamathematician.com/wp-content/uploads/2017/06/Nearsc.jpg

a map, amazing, of our local universe,
within about 500 million light years from
earth.

Amazingly there are large voids. Can see
several voids 100+ million light years
across on the map. Off the map there are
voids maybe 1 billion light years across;
considering that the big bang was about
13.7 billion years ago, a void 1 billion
light years across is surprisingly large.
Where there are stars and galaxies, there
are strings, sheets, walls, etc.

Weird stuff.

But we know much more than in such a map.

We must ask:

Q 1. What the heck was the propose of
creating the universe?

Q 2. Then we ask, do we know enough now to
start to guess a purpose?

Maybe the most amazing thing we know about
so far in the universe is human life.
Well, even if human life now is a bit
messy to serve as a guess of the purpose
of the universe, the amazing parts of
human life have increased a lot in just
the last 100 years.

So, we ask,

Q 3. Give human life another short time
to develop, say, another 100…

I don't know about (1), but (2) could be achieved by scanning through the first n digits and building a lookup tree. If we restrict ourselves to Pi then (3) could be achieved by either just keeping all the digits in memory or using the Bailey–Borwein–Plouffe formula [1], which allows the nth digit to be determined without needing to calculate the preceeding digits.

Your're essentially describing a compression mechanism. As you might expect, this isn't a new idea [2] and it generally fails because (as you said) the index quickly gets large as the size of the block of data increases - basically destroying any advantage for non-trivial block sizes. There is some discussion of the space trade-offs in [3], and I wouldn't be surprised if the speed of this approach is poor when compared to conventional techniques.

Another problem is that it is still not known whether number like Pi actually contains every possible finite sequence of digits [3], so not all input blocks can be compressed. In practice, this shouldn't be a problem for small blocks though.

[1] http://en.wikipedia.org/wiki/Bailey-Borwein-Plouffe_formula

[2] https://www.google.co.uk/search?q=pi+compression

[3] http://penduin.blogspot.co.uk/2006/10/pi-compression.html

[4] http://www.askamathematician.com/2009/11/since-pi-is-infinite-can-i-draw-any-random-number-sequence-and-be-certain-that-it-exists-somewhere-in-the-digits-of-pi/

If you're interested in this, you should read up on Bell's theorem. [1] [2]

The explanation of it I like best is this one (I don't remember where I originally read it):

It involves a two player game, the players completely separated from each other. Each is independently shown X or Y. Each player must then give either answer A and B. They win the game if:

- when both are shown X, they choose different answers

- otherwise, they choose the same answer

They can choose a strategy to follow before, but after they're physically separated, each player's decision can only depend on what they're shown (X or Y). Given this restriction, even allowing the players to have (pseudo-)non-deterministic strategies, they'll win with a chance of 75%.

However, if the players' strategy involves each having half of an entangled pair of photons, they can make measurements of its spin at one of two predetermined angles (P1X = 0, P2X = 67.5, P1Y = 45, P2Y = 22.5), based on if they see X or Y. Their answer is based on whether they get a spin-up or spin-down measurement. With this strategy, they win about 85% (cos^2(tau/16)) of the time.

[1] http://en.wikipedia.org/wiki/Bell%27s_theorem

[2] http://www.askamathematician.com/2010/06/q-how-it-is-that-bells-theorem-proves-that-there-are-no-hidden-variables-in-quantum-mechanics-how-do-we-know-that-god-really-does-play-dice-with-the-universe/

http://www.askamathematician.com/2010/12/q-what-does-00-zero-raised-to-the-zeroth-power-equal-why-do-mathematicians-and-high-school-teachers-disagree/

It might be a mistake to try to argue that defining 0^0=1 is the best answer or the most consistent answer. The problem is that there are conflicts. There is no such thing as more consistent, if any rule breaks, they all break. Some math rules lead you to 0^0=0, some to 0^0=1, and some to 0^0=undefined. So math doesn't have an answer for this, and what really happens is we decide that 0^0=1 by convention. There are other examples of this happening in math, and it works reasonably well. But let's not pretend it's "right", let's accept that it's a choice.

We're choosing to say that the rule x^0=1 takes precedence over the rule 0 * x = 0.

Just to play devil's advocate, I like the choice of 0 for intuition consistency, not math consistency. The positive side limit is 0, so if 0^x is defined at x=0, how about we try to make the function more continuous, instead of trying to make it discontinuous?

0^0 is also saying 'start with zero, then multiply it by itself ... Wait, on second thought, don't start'

There's nothing, because you don't multiply zero by anything. The best number we have to represent nothing is zero. Saying that not raising zero to a power is one feels like a random number.

But, I'm wrong. ;)