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View Full Version : Insane Ramblings #45 (I think I calculated 0)



Fiendish_Dire_Moose
2010-02-23, 01:34 PM
This is a hypothetical.

This ramble will use the term googolplex quite often. For those of you who do not know what 1 googolplex is, you can find the information here (http://en.wikipedia.org/wiki/Googolplex). It will be represented by g.


As we all know math is a fickle mistress. The first sign of an improperly calculated slope and she'll leave you for your brother. Much like dating an absent minded schizophrenic, she leaves us with an enigma to solve.
0 =/= nothing. In fact, 0 is technically a whole number represented by an incalculable decimal. Thus 0/5, is undefined, because we simply don't know what it is. However, with all the enigmas based on 0, it is still applicable.
So, I came up with a possible calculation for 0. Bare with me, for I'm nuts.

By now you know what g is. g is a number we cannot actually show, and we cannot show 0 either. You can say you have 0 dollars, but all you can show is that you have no dollars. Right? Fine, I may be wrong, but this is a hypothetical anyway.
So, if g is incalculable, and 0 is incalculable, then perhaps this insane number can help us define 0.
Atoms are so small that they are calculated in sizes of pictometres (http://en.wikipedia.org/wiki/Picometre), however, they are viewable, we can show them, and we most certainly know they exist. Which means that the scale used to measure atoms is still too large to calculate 0. Which brings us again, to g.
Now to get to where I am going.

Could 0 = 1/g? There are plenty of good arguments that would say no. The most notable is using scientific notation to show g. However scientific notation can show ideas that cannot be shown physically, so thus we use g.
For this purpose, I am in fact going to state that 0=0=1/g. This will mean that 0 = 0.g zeros with a 1 at the end. Or it could be -1/g, proper notation escapes me some times.

So this means that, though we can still no longer show 0, we now know what it is equal to, and can find a new application or two for it. Of course, this use of 0 could cause planes to fall out of the sky. I'm not in physics yet, so I do not know.

What do you think? Could 0=1/g?

Arti3
2010-02-23, 02:03 PM
No. 0=0 and 1/g=1/g, but no matter how close you get to zero, if you don't get to zero, it's not zero, even if it's 0.000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000 001.

Dogmantra
2010-02-23, 02:04 PM
One over infinity might be nought though.

Arti3
2010-02-23, 02:16 PM
One over infinity might be nought though.

True; however 1/g!=1/infinity.

Astrella
2010-02-23, 02:20 PM
Not really, one over infinity, is just as dividing by zero, a undefined operation.
The limit of 1/x for x going to infinity is zero though.

Dogmantra
2010-02-23, 02:21 PM
Not really, one over infinity, is just as dividing by zero, a undefined operation.
The limit of 1/x for x going to infinity is zero though.
Exactly

I have no idea what I've proved, but... I guess I've proved something?

Zanaril
2010-02-23, 02:23 PM
One over infinity might be nought though.

I thought 1/infinity was 0.0000...001, or 1 - 0.9r

:smallconfused:


Then again, 0.9r = 1, so maybe you're onto something.

Dogmantra
2010-02-23, 02:26 PM
I thought 1/infinity was 0.0000...001, or 1 - 0.9r

They're both nought. If you have an infinite number of noughts, then that 1 on the end is meaningless. Also 0.999... = 1 as is provable:
x = 0.999...
10x = 9.999...
10x-x = 9 = 9x
therefore x = 1

You got there first
So yeah, if you're accepting the OP's reasoning, then 1/infinity is 1 - 0.999... :smallwink:

PirateMonk
2010-02-23, 02:28 PM
Thus 0/5, is undefined, because we simply don't know what it is.

0/5=0. 5/0 is undefined.


By now you know what g is. g is a number we cannot actually show

We can't show it because we don't have enough time or space to make or fit all those zeroes at the end. It's not a mathematical limitation.

WalkingTarget
2010-02-23, 02:29 PM
I thought 1/infinity was 0.0000...001, or 1 - 0.9r

:smallconfused:


Then again, 0.9r = 1, so maybe you're onto something.

Well, the form 0.0000...[something other than 0] isn't really proper notation.

Dallas-Dakota
2010-02-23, 02:30 PM
I thought this was some sort of weird variation or wrongly done new thread of RB.


But nooooooooooooooo, you just had to go and hurt my head with maths.

Zanaril
2010-02-23, 02:30 PM
x = 0.999...
10x = 9.999...
10x-x = 9 = 9x
therefore x = 1


Ah, so that's how that's proved. Thanks for posting that.

Dogmantra
2010-02-23, 02:32 PM
Ah, so that's how that's proved. Thanks for posting that.

That's the simplest one, or at least the easiest to remember.
There's a whole page about it here (http://qntm.org/?pointnine) and I recommend the other stuff on there too because it's rather interesting.

Symmys
2010-02-23, 02:34 PM
A while ago I figured out that 1/0 is both the smallest and the largest positive number in existence.

On one hand, it's one of zero. In other words, zero. On the other hand, one is, multiplicatively speaking, infinitely larger than zero*. So 1/0 is infinitely larger than 0/0.

Of course, if zero really is what you just defined it as, 1/0 only equals 1 g. I think.
Zero is a very mysterious number indeed.


*=I am not a mathematician. I cannot be held accountable for any inaccuracies in this statement.

Edit: Stupid ninjas.

Mando Knight
2010-02-23, 02:38 PM
What do you think? Could 0=1/g?

Nope. Because g/G (http://en.wikipedia.org/wiki/Graham%27s_number) < 1. :smalltongue:

Graham's Number: Consider, if you would, a number that goes on forward as an irrational number goes back. A number so large that we only know the last thousand or so digits. A number that is to googolplex as googol is to the square root of two. Ridiculously over-sized.

Fiendish_Dire_Moose
2010-02-23, 02:45 PM
Crud.

So then let's play with numbers. 3/3 =/= 1, by decimal standards. The square root of 1, is 1. 0 fictoral(sp?) is 1. So, using these utterly ridiculous rules, is 0/0 technically 1? No? Well let's pretend it is, I now need someone to make an insane math problem using this.

PirateMonk
2010-02-23, 02:50 PM
Crud.

So then let's play with numbers. 3/3 =/= 1, by decimal standards. The square root of 1, is 1. 0 fictoral(sp?) is 1. So, using these utterly ridiculous rules, is 0/0 technically 1? No? Well let's pretend it is, I now need someone to make an insane math problem using this.

0=0*2
0/0=2
1=2

Under those principals but otherwise normal mathematics, all numbers are equal.

Ashery
2010-02-23, 03:44 PM
Ya'll can divide by 0 after you give me some number x that works here:

0 * x = 1

Give me that x and you can divide by 0 in peace ;p

Others have already addressed 1/g not being 0.

Dogmantra
2010-02-23, 03:52 PM
Ya'll can divide by 0 after you give me some number x that works here:

0 * x = 1

Give me that x and you can divide by 0 in peace ;p

Others have already addressed 1/g not being 0.

Well clearly x = 1/0
*world explodes*

Ashery
2010-02-23, 04:02 PM
Well clearly x = 1/0
*world explodes*

But you're already dividing by 0 there. Nice try with the circular logic, though ;p

Fiendish_Dire_Moose
2010-02-23, 04:36 PM
Well clearly x = 1/0
*world explodes*

HA! You almost got away with it Goldfinger!

Maximum Zersk
2010-02-23, 05:33 PM
Well clearly x = 1/0
*world explodes*

Oh Sna-

SPACE FILLER

Thufir
2010-02-23, 05:44 PM
So then let's play with numbers. 3/3 =/= 1, by decimal standards.

Where the hell did you get that idea?

x/x=1, for all x in the set of all complex numbers (I need a way to type mathematical symbols). If someone comes up with a set of numbers bigger than the complexes, it probably holds in that too.

curtis
2010-02-23, 05:47 PM
I may be about to sap the fun, thought and whimsy out of all this, so feel free to skip...

Humans invented maths (or math, if you prefer). The rules that we put in place are meant to help define this concept we use to understand the universe. Anything we can do with numerical rules ( like 0.999r=1) is sheer folly, as you're not finding loopholes in the laws of the universe, you're finding loopholes in the laws that we've created.

Sorry.

Thufir
2010-02-23, 06:07 PM
I may be about to sap the fun, thought and whimsy out of all this, so feel free to skip...

Humans invented maths (or math, if you prefer). The rules that we put in place are meant to help define this concept we use to understand the universe. Anything we can do with numerical rules ( like 0.999r=1) is sheer folly, as you're not finding loopholes in the laws of the universe, you're finding loopholes in the laws that we've created.

Sorry.

1) 0.999...=1 is not a loophole, it's a fact, a consequence of how numbers and infinity work.

2) Maths describes the laws of the universe. In some cases it describes abstract concepts, but they are universally true. Humans derived them and gave them names, but they were true anyway. To reference Shakespeare, that which we call 1+1=2 by any other name would still be a fundamental mathematical identity. Rename the numbers and operators, use different symbols for them, but the maths will always be true, by the nature of the universe. We didn't create it, we just figured some of it out.

lobablob
2010-02-23, 06:39 PM
If I'm remembering correctly, didn't the original poster here make a thread a while ago trying to argue that you could divide by zero? And if I remember correctly, that ended with just about everyone else proving that was nonsense and they OP deciding that they could believe it if they wanted or something. Cause that went pretty badly and if my recollections are correct, this one seems liable to end badly as well.

Fiendish_Dire_Moose
2010-02-24, 02:36 PM
If I'm remembering correctly, didn't the original poster here make a thread a while ago trying to argue that you could divide by zero? And if I remember correctly, that ended with just about everyone else proving that was nonsense and they OP deciding that they could believe it if they wanted or something. Cause that went pretty badly and if my recollections are correct, this one seems liable to end badly as well.

It was a hypothetical. A very poorly written hypothetical which caused me to rethink my habits of not checking my statements before posting them. Severe posting habit overhaul over the past couple months.

This, is another hypothetical. One a little more (or less) thought out, but still just as weird. The last one was taking knowledge of 0 and throwing it away in favor of 0=absolute nothingness. This one is a fun poking at trying to calculate zero.

As for the other questions: 3/3 does not equal 1. Many teachers will tell you that. However, we accept that it is roughly equal to 1 because the difference can't be shown, or noticed. But 1/3 = .33, .33*3 =/= 100. The difference between the two is just no negligible that we call it 1, just because it's easier. Fractions are division problems after all.

Anyway, as for math. Math is an observation of natural laws. We have a name for a specific phenomenon, and we call it 2. We add two of them together to make the phenomenon we call 4. Phenomenon 2 added to phenomenon 2, is always phenomenon 4, unless 2 =/= 2. But, in most cases it does.
Math is not a natural law, it is a man made one, but it is pretty much an undeniable truth that we have learned to bend like crazy.


So, since we have the best math minds on the board (see what I did there?), who wants to pool minds to calculate 0 and win this website a Nobel Prize for Sheer Awesomesause?

WalkingTarget
2010-02-24, 03:31 PM
As for the other questions: 3/3 does not equal 1. Many teachers will tell you that. However, we accept that it is roughly equal to 1 because the difference can't be shown, or noticed. But 1/3 = .33, .33*3 =/= 100. The difference between the two is just no negligible that we call it 1, just because it's easier. Fractions are division problems after all.


Sure we can show the difference between 3*0.333r and 1.

It's 0.000r :smalltongue:

Mando Knight
2010-02-24, 03:31 PM
Crud.

So then let's play with numbers. 3/3 =/= 1, by decimal standards. The square root of 1, is 1. 0 fictoral(sp?) is 1. So, using these utterly ridiculous rules, is 0/0 technically 1? No? Well let's pretend it is, I now need someone to make an insane math problem using this.

Zero factorial is defined as one.

As for the other questions: 3/3 does not equal 1. Many teachers will tell you that. However, we accept that it is roughly equal to 1 because the difference can't be shown, or noticed. But 1/3 = .33, .33*3 =/= 100. The difference between the two is just no negligible that we call it 1, just because it's easier. Fractions are division problems after all.
Nope. See the proof by Dogmantra above, and applied here: 1/3 is not 0.33, it's 0.33333... ad infinitum. 3*0.3333... is 0.9999..., which when multiplied by 10 becomes 9.9999..., and if we call 0.9999... x, the equation becomes 10x=9.9999...
Now subtract 1x from 10x, so you have 9x:
10*0.9999...-0.9999...=9.9999...-0.9999...=9
And, if 10x-x=9x=9, then x must be equal to one.

Furthermore, declaring that 3/3 is not equal to one and then stating that fractions are division problems in the same paragraph is invalid. Every math teacher will tell you that dividing any non-zero number by itself will yield an answer of exactly one, since the ratio of the two is in fact one to one.

Ashery
2010-02-24, 03:36 PM
As for the other questions: 3/3 does not equal 1. Many teachers will tell you that. However, we accept that it is roughly equal to 1 because the difference can't be shown, or noticed. But 1/3 = .33, .33*3 =/= 100. The difference between the two is just no negligible that we call it 1, just because it's easier. Fractions are division problems after all.

3/3 very much equals 1. So long as x is non-zero, x/x = 1.

Regardless of that, someone already showed why 0.99r = 1.


Anyway, as for math. Math is an observation of natural laws. We have a name for a specific phenomenon, and we call it 2. We add two of them together to make the phenomenon we call 4. Phenomenon 2 added to phenomenon 2, is always phenomenon 4, unless 2 =/= 2. But, in most cases it does.
Math is not a natural law, it is a man made one, but it is pretty much an undeniable truth that we have learned to bend like crazy.

So, since we have the best math minds on the board (see what I did there?), who wants to pool minds to calculate 0 and win this website a Nobel Prize for Sheer Awesomesause?

...what?

(Looks like Mando beat me to this.)

Player_Zero
2010-02-24, 03:49 PM
If you don't understand limits then stop talking about them. It annoys us maths-studying people.

Fiendish_Dire_Moose
2010-02-24, 06:11 PM
If you don't understand limits then stop talking about them. It annoys us maths-studying people.

Has very little to do with limits, or understanding. If you have a problem with the subject matter you're welcome to find a different thread.

As for 3/3=1: Double crud.
This is probably one of the reasons I'm taking math at the college anyhow.

But, since absolutely NOBODY seems to want to have any fun and just wants to state rules of math, whatever. Fine, we can be done here.
This are my sad face.

Mando Knight
2010-02-24, 11:51 PM
But, since absolutely NOBODY seems to want to have any fun and just wants to state rules of math, whatever. Fine, we can be done here.
The rules of reality are the rules of reality. If math didn't work the way it does, then Apollo 11 really was faked, and the Internet is a lie. Sometimes, trying to break a system that provides far more potential awesome when followed isn't as awesome-sounding an idea.

Ashery
2010-02-25, 04:16 AM
Has very little to do with limits, or understanding. If you have a problem with the subject matter you're welcome to find a different thread.


I don't have a problem with the subject matter. I have a problem with it being a based around a bastardized form of mathematics.

You claimed that 1/g was 0. What about 1/(g+1)?

So now you're stuck with a non-negative number that's less than zero.


But, since absolutely NOBODY seems to want to have any fun and just wants to state rules of math, whatever. Fine, we can be done here.
This are my sad face.

These rules that you're complaining about form the very foundation of mathematics. And there's a *reason* they've been established as rules. Without them, the system collapses into contradictions.

If you want "fun" mathematics, go study up on topology. Your intuition will just get you killed in that subject, heh.

Reinboom
2010-02-25, 04:39 AM
This has quite a bit to do with limits, really...
The limit of a sequence of numbers xn as it approaches a number y (but never reaches) is y, but this does not mean that xn = y
The reason why this is only useful is due to not easily seen oddities in math... that exists no matter which base you are using.

Let's observe:

x = 0.9
10 * x = 9.9 << (in decimal) 1 (NOT 9.9 - I'll give an example why in a bit)
10 * x - x = 9.09 (I do not know the notation for this, it's with 9 on the end that was shifted out of range)
therefore... x = 0.9

Now, as a different way to view this problem, let's take the problem as written by Dogmantra:


x = 0.999...
10x = 9.999...
10x-x = 9 = 9x
therefore x = 1

And let's... replace something here.

x = 0.222...
10x = 2.222...
10x-x = 2 = 2x
therefore x = 1

Did you see what I did there?/did u c wut i did thar?

0.2 does not approach 1. 0.9 does for the purpose of common math, but is not 1.
This is why limits are used.


Here's a different example, imagine if the number system worked in base 30 (oh... how I wish it did), which would mean that numbers range from 0 - T ( {0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F,G,H,I,J,K,L,M,N,O ,P,Q,R,S,T} - it's what is most commonly used for this sort of thing )

So that 1T + 1 = 20, and that 19 + 1 = 1A.

0.9 approaches... well, closest approaching number by sticking to a low number of digits is 0.A, not 1
0.S approaches... low digit again... 0.T, yet, base 30: 0.S > base 10: 0.9

So... yeah (http://tvtropes.org/pmwiki/pmwiki.php/Main/SoYeah)

[edit]
Ah incidentally, I must apologize to the fellow math nerds for misusing "limits" at the end of that. It's just... difficult to laymen.

Ashery
2010-02-25, 05:52 AM
x = 0.222...
10x = 2.222...
10x-x = 2 = 2x
therefore x = 1

Did you see what I did there?/did u c wut i did thar?


Not even dealing with the first part, there's one very clear mistake here.

10x - x = 9x, not 2x ;p

It should be:
x = 0.222r
10x = 2.22r
10x - x = 9x = 2
x = 2/9, aka 0.222r


Here's a different example, imagine if the number system worked in base 30 (oh... how I wish it did), which would mean that numbers range from 0 - T ( {0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F,G,H,I,J,K,L,M,N,O ,P,Q,R,S,T} - it's what is most commonly used for this sort of thing )

So that 1T + 1 = 20, and that 19 + 1 = 1A.

0.9 approaches... well, closest approaching number by sticking to a low number of digits is 0.A, not 1
0.S approaches... low digit again... 0.T, yet, base 30: 0.S > base 10: 0.9


0.99r doesn't approach 0.A if we're talking about base 30 (Why did you choose base 30, anyways? Base 12 would've been more than enough to illustrate the point you tried to make).

To illustrate this more, in base 10, add *any* positive number to 0.99r and you will always get something greater than one. The same does not hold true in base 30. 0.99r + 0.001 = 0.99A9r, etc.

Not even sure what relevance the 0.SSr > 0.99r comment has.

(Edited to add positive a couple lines back)

Reinboom
2010-02-25, 06:32 AM
Ashery: I like to emphasize points in sarcastic, intentionally false, extremes. Which is mostly what you quoted me as messing up on.


Not even dealing with the first part, there's one very clear mistake here.

10x - x = 9x, not 2x ;p

It should be:
x = 0.222r
10x = 2.22r
10x - x = 9x = 2
x = 2/9, aka 0.222r
...though this was an honest mistake. I will not change it, however, due to the point of me going through and replacing all the 9s with 2s was only intended to emphasize the absurdity of what dogmantra had said.


0.99r doesn't approach 0.A if we're talking about base 30 (Why did you choose base 30, anyways? Base 12 would've been more than enough to illustrate the point you tried to make).

To illustrate this more, in base 10, add *any* positive number to 0.99r and you will always get something greater than one. The same does not hold true in base 30. 0.99r + 0.001 = 0.99A9r, etc.

Not even sure what relevance the 0.SSr > 0.99r comment has.

(Edited to add positive a couple lines back)

Oh yes, I knew my misuse of a term was going to bite me there.
And yes, I know that one needs 0._n where N is the highest accountable single digit in a base in order to approach.

However, the ultimate goal of that - and the reason I used base 30 - was still illustrated I believe most significantly by the 0.S > 0.9 comment.

Since T would be required to technically approach 1 (and thus would designate the base 30 equivalent to trying to say 0.9 = 1 as 0.T = 1), yet 0.S does not approach 1 for the same reason that 0.8 does not approach 1. Yet, provably, 0.S > 0.9

Base 12: 0.A < Base 10: 0.9
Which kind of fails to emphasize that. I believe the earliest this could have been shown is base 21?
I just made the base nice and high to make it not seem so randomly chosen.

Thufir
2010-02-25, 06:48 AM
Ashery: I like to emphasize points in sarcastic, intentionally false, extremes. Which is mostly what you quoted me as messing up on.

You realise that something intentionally false can't prove anything mathematically except in a prrof by contradiction, right?


...though this was an honest mistake. I will not change it, however, due to the point of me going through and replacing all the 9s with 2s was only intended to emphasize the absurdity of what dogmantra had said.

Absurdity? It's mathematical fact.


However, the ultimate goal of that - and the reason I used base 30 - was still illustrated I believe most significantly by the 0.S > 0.9 comment.

0.SSS... > 0.999... in base 30. 0.SSS... in base 30 < 0.999... in base 10.


Since T would be required to technically approach 1 (and thus would designate the base 30 equivalent to trying to say 0.9 = 1 as 0.T = 1), yet 0.S does not approach 1 for the same reason that 0.8 does not approach 1.

I assume you meant 0.T there. And yes, 0.TTT... = 1.


Yet, provably, 0.S > 0.9

Then prove it.

Dogmantra
2010-02-25, 07:17 AM
Absurdity? It's mathematical fact.

The blobfish is absurd, but it still exists. :smallwink:

Ashery
2010-02-25, 07:18 AM
Ashery: I like to emphasize points in sarcastic, intentionally false, extremes. Which is mostly what you quoted me as messing up on.

Ah, always fun to do. Although, I'd definitely argue that it's best to avoid those "sarcastic, intentionally false, extremes" when other people are saying similar things but believe them to be true.


Oh yes, I knew my misuse of a term was going to bite me there.
And yes, I know that one needs 0._n where N is the highest accountable single digit in a base in order to approach.

However, the ultimate goal of that - and the reason I used base 30 - was still illustrated I believe most significantly by the 0.S > 0.9 comment.

Since T would be required to technically approach 1 (and thus would designate the base 30 equivalent to trying to say 0.9 = 1 as 0.T = 1), yet 0.S does not approach 1 for the same reason that 0.8 does not approach 1. Yet, provably, 0.S > 0.9

Base 12: 0.A < Base 10: 0.9
Which kind of fails to emphasize that. I believe the earliest this could have been shown is base 21?
I just made the base nice and high to make it not seem so randomly chosen.

First off:

0.S (base 30) > 0.9 (base 10), but 0.SSr (base 30) < 0.99r (base 10).

Why?

1 - 0.S = 0.2 (base 30) or 0.066r (base 10)
1 - 0.9 = 0.1 (base 10), which is larger than 0.066r

1 - 0.SSr = 0.11r (base 30) or 1/27 (base 10)
1 - 0.99r = 0 (base 10)

I'd also be willing to bet that it's technically impossible to write something like 0.00r1, 0.000....01, 0.0_1, or whatever your preferred notation is due to the fact that you're writing an infinitely long series that terminates.

Read that again, an infinitely long series that terminates.


I assume you meant 0.T there. And yes, 0.TTT... = 1.

I do believe that, what you responded to, was simply a misuse of a variant of the squeeze theorem rather than saying T instead of S. This was all based upon 0.SSr being larger than 0.99r. Using that, you'd have that 1 = 0.99r < 0.SSr < 0.TTr = 1. That is, we would have a contradiction that would prove that 0.99r is not 1 (or 0.TTr for that matter). This, however, breaks down, as 0.SS is NOT greater than 0.99r.

Reinboom
2010-02-25, 07:50 AM
Indeed. In my attempts to spell out the proof, I seem to have made an error in judgment.
I apologize. (Apologizing for me being wrong, that is)

Assuming:
base-n, with x being the greatest single digit of base-n
base-m, with y being the greatest single digit of base-m
∑ (∞,k=1) { n-k*x } = ∑ (∞,k=1) { m-k*y }

Am I correct in this, at least?


I'd also be willing to bet that it's technically impossible to write something like 0.00r1, 0.000....01, 0.0_1, or whatever your preferred notation is due to the fact that you're writing an infinitely long series that terminates.

Read that again, an infinitely long series that terminates.

*throws a p-adic number system smoke bomb and hides*

Ashery
2010-02-25, 08:11 AM
Indeed. In my attempts to spell out the proof, I seem to have made an error in judgment.
I apologize.

Assuming:
base-n, with x being the greatest single digit of base-n
base-m, with y being the greatest single digit of base-m
∑ (∞,k=1) { n-k*x } = ∑ (∞,k=1) { m-k*y }

Am I correct in this, at least?

I do believe so.


*throws a p-adic number system smoke bomb and hides*


In this notation, 10-adic expansions can be extended indefinitely to the left, in contrast to decimal expansions, which can be extended indefinitely to the right.

Hurray for wikipedia!

Winterwind
2010-02-25, 08:12 AM
I don't quite understand why you are trying to calculate 0 via decimals. That's a completely backwards approach, when it comes to how numbers are actually defined. Mathematics is built from base axioms, and everything it consists of is just a conclusion of these base axioms (this is pretty much the first thing you do in university-level math courses):

0 and 1 are introduced via axioms (specifically, the Peano axioms). They are not calculated, they are stated to be. The same axioms also introduce a + relation, which expands 0 and 1 into the entire set of natural numbers (so, yes, even as simple a thing as the number, say, '4', is not something that exists right from the start; all you have are 0 and 1, and these two define what 4 actually means).

Then, you move on to the concept of a group and the + operation, which expands the set of natural numbers into the group of integers. Which, then, with the * operation, can in turn be expanded into the ring of rational numbers.

And only once you have that, you define real numbers (i.e., the ones you have been working with all along) by means of Cauchy sequences of rational numbers.

Trying to define 0 via decimals is absurd. 0 is necessary to define what a 'decimal' is in the first place.

Ashery
2010-02-25, 08:31 AM
I don't quite understand why you are trying to calculate 0 via decimals. That's a completely backwards approach, when it comes to how numbers are actually defined. Mathematics is built from base axioms, and everything it consists of is just a conclusion of these base axioms (this is pretty much the first thing you do in university-level math courses):

0 and 1 are introduced via axioms (specifically, the Peano axioms). They are not calculated, they are stated to be. The same axioms also introduce a + relation, which expands 0 and 1 into the entire set of natural numbers (so, yes, even as simple a thing as the number, say, '4', is not something that exists right from the start; all you have are 0 and 1, and these two define what 4 actually means).

Then, you move on to the concept of a group and the + operation, which expands the set of natural numbers into the group of integers. Which, then, with the * operation, can in turn be expanded into the ring of rational numbers.

And only once you have that, you define real numbers (i.e., the ones you have been working with all along) by means of Cauchy sequences of rational numbers.

Trying to define 0 via decimals is absurd. 0 is necessary to define what a 'decimal' is in the first place.

Z also happens to be a ring and Q a field under the standard addition and multiplication, heh.

Good points, though. People generally don't see modern/abstract/whatever algebra (Groups, rings, and fields) until the last year or two of their bachelors here in the states. Meaning that, unless you're majoring in math, you won't see it at all.

Winterwind
2010-02-25, 08:35 AM
Z also happens to be a ring and Q a field under the standard addition and multiplication, heh.Erm, yeah. To my excuse, it's been seven years or so since I did that at uni. :smallredface:


Good points, though. People generally don't see modern/abstract/whatever algebra (Groups, rings, and fields) until the last year or two of their bachelors here in the states. Meaning that, unless you're majoring in math, you won't see it at all.Really? That was the very, very first thing we did in math at uni. Seems odd to me to leave the basis on which everything else is founded for later. :smallconfused:

Ashery
2010-02-25, 08:58 AM
Erm, yeah. To my excuse, it's been seven years or so since I did that at uni. :smallredface:

And to further your defense, I double checked myself on wikipedia (Faster than digging through a closet full of old notebooks) to make sure my gut feeling was right ;p


Really? That was the very, very first thing we did in math at uni. Seems odd to me to leave the basis on which everything else is founded for later. :smallconfused:

I should clarify my second sentence a bit (The last one is accurate, though): On my campus it's required that we complete all but one calculus course, a linear algebra course, and an introduction to proofs course before we are allowed to take our undergrad modern algebra course.

My campus could very well have an atypical math program, but I'm pretty sure it's fairly standard.

It probably doesn't help a student's timely progress in their major when they're swamped with so many GE courses, either.

Heliomance
2010-02-25, 09:18 AM
I do believe that 0 is defined either as the cardinality of the empty set (if you don't have any apples, how many apples do you have? 0) or as the additive identity (ie that number which when added to any number, produces the same number - x+0=x for all x)

Thufir
2010-02-25, 09:25 AM
Really? That was the very, very first thing we did in math at uni. Seems odd to me to leave the basis on which everything else is founded for later. :smallconfused:

Well, you don't actually need to know that stuff to do everything else.
I imagine it varies from uni to uni. In my case, in 2nd year there was a module on algebra, covering rings and fields, and possibly some other things I can't remember, then a module on group theory in 3rd year. Both optional.

lobablob
2010-02-25, 05:51 PM
But, since absolutely NOBODY seems to want to have any fun and just wants to state rules of math, whatever. Fine, we can be done here.
This are my sad face.

Yeah. This is what I was talking about. You make a claim regarding maths, get proven wrong with maths and then whine at people for using maths.