to infinity, and then some...

[Michael C]

x^nm=x^{n * m}

That would be (xⁿ)^m.

x^nm is usually considered to be x^(nm).

LandR already gave this link, which gives the number of digits in 2¹⁰¹¹⁸, although it doesn't say how many of these digits are zeros. But maybe the number of digits was what Ambrosia wanted anyway.

 

[LandR]

Post removed. Just noticed someone noted this above.

 

[JoeTheJuggler]

That would be (xⁿ)^m.

x^nm is usually considered to be x^(nm).

LandR already gave this link, which gives the number of digits in 2¹⁰¹¹⁸, although it doesn't say how many of these digits are zeros. But maybe the number of digits was what Ambrosia wanted anyway.

Thanks for the correction.

 

[LandR]

Unfortunately WolframAlphas DigitCount[n,b] doesnt work on such large numbers it seems.

DigitCount[2^(10^118),10,0]

Returns no result.

 

[Ambrosia]

But maybe the number of digits was what Ambrosia wanted anyway.

Yes. I just want to try to visualise what that number looks like in standard notation, my feeble math-fu didn't realise it was in base 2 (DOH!)

Stuff like considering infinity makes me think that life as we know it isn't "real" and we are all running on a virtual reality like simulation on some computer somewhere.

If there are an infinite number of universes, then it has to be true that some other life form evolved and attained sufficient technology to run such a simulation, we are but one of many such simulations, which means there is a "God" if you define "God" a simply a higher power (technologically gained or 'supernatural' - not that we'd be able to tell the difference if the tech was sufficiently advanced)

That in turn implies that stuff like the "face on Mars" or the layout of apparent structures on the Cydonia plain, things like Venus having a runaway greenhouse effect and our own planet exhibiting the beginnings of a greenhouse effect, questions about why Mars seems to be missing half of it's crust and used to feature running surface water and an atmosphere, whether ancient monuments are really the result of 6000odd years ago civilisations or perhaps could be remnants of a much earlier age of civilisation, all these things could just be puzzles set for us to figure out an answer to as some kind of giant experiment...

Sometimes maybe I think too much...

 

[JoeTheJuggler]

Yes. I just want to try to visualise what that number looks like in standard notation, my feeble math-fu didn't realise it was in base 2 (DOH!)

It'd be easy enough to answer the question in base two.

In binary, it would be a 1 followed by 10^{118[/sub] zeroes.}

 

[Fredrik]

When you see a number like 2¹⁰¹¹⁸, you can use that 2¹⁰=1024>1000=10³. So...

2¹⁰¹¹⁸>10³¹¹⁸=10^3*118=10³⁵⁶
This is of course a pretty crude way to (under)estimate the number, but it's sufficient for many purposes.

More fun with infinites...

There are as many numbers in the interval (0,1) as in (1,∞). Proof: Consider the function that takes x to 1/x and restrict it to one of these intervals.

There are as many integers there are rational numbers. Proof: Every rational number can be expressed as n/m, and can therefore be represented as the point (n,m) in a plane. We can clearly construct a curve that starts at (0,0) and goes through each point with integer coordinates exactly once. Define q(0)=0, and q for each n=1,2,3... to be the nth "new" rational number that the curve reaches. (By "new" I mean e.g. that we skip points like (2,4) because 2/4=1/2, and when we reach (2,4) we have already passed through (1,2)).

There are more real numbers in the interval (0,1) than there are integers. Proof: Suppose that that each number in the interval (0,1) can be written as x_n, with n a positive integer, and that x_n≠x_m when n≠m. (We assume that the statement we want to prove is false, and show that this leads to a contradiction). Any x in (0,1) can be expressed in decimal form, x=0.a₁a₂..., with each a_i an integer between 0 and 9. So we can write

x₁=0.a₁₁a₁₂a₁₃...
x₂=0.a₂₁a₂₂a₂₃...
x₃=0.a₃₁a₃₂a₃₃...
x₄=...
.
.
.
But there certainly exists a number y=0.b₁b₂b₃... such that b₁≠a₁₁, b₂≠a₂₂, b₃≠a₃₃,... Any such number is in the interval (0,1), but isn't equal to x_n for any n.

 

[Ziggurat]

x^nm=x^{n * m}

No, this obviously wrong. x^a=x^b only if a = b, but n^m does not in general equal n*m

I think you've mistaken this for the relationship (xⁿ)^m = x^{n * m}. But the order of operations is different in this case, and the order of operations matters a lot.

If you want to know how many zeroes 2¹⁰¹¹⁸ has, use the relationship x^y = 10^log(x)*y. That means 2¹⁰¹¹⁸ = 10^log(2)*10118=10^3*10117. That's not 1180 zeros, that's 3*10¹¹⁷ zeroes. The number describing the number of zeroes has 117 zeroes.

Edit: I see this has been partly addressed already. Note also that by "zeroes" I mean how many digits after the first one. Most of the digits won't actually be zero, and I've rounded off with the numbers I used.

 

[sol invictus]

What gets annoying is when you take something finite, like a ball, and divide it up into infinitely many pieces. Then you spin the pieces, put the pieces back together, and you have two balls the same size as the original.

You can do that by dividing the ball into a finite number of pieces.

 

[Badly Shaved Monkey]

Look, I know this has already been answered, but I still wonder about the business of

Infinity - infinity not equal to 0

I understood the explanation given in the programme with the hotel rooms again, and I understood Dave Rogers' equation on page 1 of this thread, but it feels like the hotel room example isn't quite the same as 'minus' when you do it in ordinary arithmetic (accepting that number theory is founded in set theory and I really don't understand that).

It just feels like when I say 6-6=0 I am dealing with one thing and when I say "I have a hotel with infinite rooms and one guest stays behind etc etc" I am dealing with a different kind of thing. So, whereas I find the other arguments about the set of all numbers and the set of even numbers being the same size of infinity subjectively acceptable, I don't see the argument about subtraction as being acceptable to me in the same way. Can someone tell it to me a different way?

 

[blutoski]

Contrary to the wide-spread belief, all those monkeys would never type all the works of Shakespeare.

[Last Exit To Springfield]
Burns:

It was the best of times, it was the.. blurst of times!! You stupid monkey!

 

[Dilb]

You can do that by dividing the ball into a finite number of pieces.

Right, I should have said an infinitely dividable ball can be split into 5 pieces, have two of the pieces rotated, and the pieces can be reassembled into 2 balls the same size as the original.

I don't think a non-continuous collection of points is a very good 'piece' of something, but it's really not an area where I should try and use my intuition about volume.

 

[RussDill]

Look, I know this has already been answered, but I still wonder about the business of

Infinity - infinity not equal to 0

I understood the explanation given in the programme with the hotel rooms again, and I understood Dave Rogers' equation on page 1 of this thread, but it feels like the hotel room example isn't quite the same as 'minus' when you do it in ordinary arithmetic (accepting that number theory is founded in set theory and I really don't understand that).

It just feels like when I say 6-6=0 I am dealing with one thing and when I say "I have a hotel with infinite rooms and one guest stays behind etc etc" I am dealing with a different kind of thing. So, whereas I find the other arguments about the set of all numbers and the set of even numbers being the same size of infinity subjectively acceptable, I don't see the argument about subtraction as being acceptable to me in the same way. Can someone tell it to me a different way?

Infinity isn't a number from what I understand, so you can't do math with it like you are suggesting. It is a description of the cardinality of a set. So you don't really want subtraction, you want set operators.

BTW, I love watching people work through the logic of the cardinality of infinite sets, it certainly isn't common knowledge or even common sense, and it's really easy to glaze over someone saying "Cantor's diagonal argument" without really thinking about it.

 

[Ziggurat]

Look, I know this has already been answered, but I still wonder about the business of

Infinity - infinity not equal to 0

The short answer is that you can't meaningfully define arithmetic operations on infinity.

Now, you CAN do arithmetic operations on variables and look at limits of those variables as you approach infinity. And if you do that, then the answers will generally be well defined.

 

[Third Eye Open]

actually while I've woken up the maths people here.

10¹¹⁸ is a 1 with 118 zeros after it.

if you wrote out 2¹⁰¹¹⁸ longhand how many zeros are in that number?

You couldn't write it. you would have to write more digits than there are atoms in the visible universe.

 

[nathan]

You can do that by dividing the ball into a finite number of pieces.

IIUC, 5 pieces (the pieces are disjoint)

 

[nathan]

I always thought that:
infinity-infinity=0
infinity/infinity=1

No. This is where treating infinity as-if it were a specific very large number breaks down.

It depends on precisely which infinity you are measuring. For instance, the number of non-negative integers and the number of positive integers are both infinite. However, their difference is 1 (because one set contains zero and the other does not). Now, in order to do that calculation, I've had to extend the subtraction operator into being a difference operator between sets, and then show a 1:1 correspondence between (all but one) members of the set of non-negative integers and members of the set of positive integers.

 

[nathan]

Oh and the monkeys, well the chances of a monkey typing the complete works of shakespeare at random is roughly equivalent to winning the UK lottery (14millionish to 1) every week, week in, week out for 29000 years.

This presupposes that monkeys type at random. They do not.

From http://en.wikipedia.org/wiki/Infinite_monkey_theorem:

They left a computer keyboard in the enclosure of six Celebes Crested Macaques in Paignton Zoo in Devon in England for a month ...

Not only did the monkeys produce nothing but five pages[24] consisting largely of the letter S, the lead male began by bashing the keyboard with a stone, and the monkeys continued by urinating and defecating on it. The zoo's scientific officer remarked that the experiment had "little scientific value, except to show that the 'infinite monkey' theory is flawed". ... He concluded that monkeys "are not random generators. They're more complex than that. … They were quite interested in the screen, and they saw that when they typed a letter, something happened. There was a level of intention there.

 

[RussDill]

No. This is where treating infinity as-if it were a specific very large number breaks down.

It depends on precisely which infinity you are measuring. For instance, the number of non-negative integers and the number of positive integers are both infinite. However, their difference is 1 (because one set contains zero and the other does not). Now, in order to do that calculation, I've had to extend the subtraction operator into being a difference operator between sets, and then show a 1:1 correspondence between (all but one) members of the set of non-negative integers and members of the set of positive integers.

I could provide you with a 1:1 mapping between the two sets that would disagree.

 

[dlorde]

A straight line in which direction? Or are they saying we're surrounded by sphere of countless trillions of duplicate earths located exactly (2^10¹¹⁸)*10²⁶m away from us?

(There probably aren't even enough earth-like planets in the universe that you wouldn't have to be insane to expect to find one with flora and fauna that exactly match that of earth, never mind the absurdity of finding a perfect replica.)

This sounds like complete bullpoop to me, unless they're basing the claim on an (unproven) theory that the universe is curved, and you'll end up back where you started.

No, what they were saying was that, in an infinite universe - i.e. not curved, just continuing out infinitely far - if you go far enough, the chances of coming across a planet identical to this one are 1 (in fact, there would be an infinite number of them), and, given the number of particles that make up the Earth, you can calculate how far the distance would be to be certain that you encountered a duplicate. I presume they meant within a sphere of that radius, but they weren't very specific, and I'm not convinced by the calculation, although the concept seems sound enough (given that infinities are involved).

What people tend to forget, what with the monkeys typing Shakespeare and the duplicate planets all over the place, is that there would be infinite number of in-betweens - from monkeys that typed out just a few words of Shakespeare among a mountain of gibberish, to monkeys that got any number of variations a few words short of correct. Likewise with the planets - there would be an almost infinite number of nearly Earths before you found an identical duplicate. This possibly gives a better sense of the numbers involved.

This also troubled me about the traditional idea of the multiverse, where the universe xfurcates at every event according the probabilities of the various outcomes, which means that everything, however unlikely, that can happen does happen (somewhere), and that universes where every outcome of every event is almost infinitely unlikely, must exist... but seeing the whole thing as a single wavefunction containing all possibilities at their respective amplitudes makes it more bearable.