The unusual notation ...999 = -1

[Martin]

Exactly. "If it walks like a duck, smells like a duck, ... " then it might just be a duck.

No doubt someone will be along presently to insist that it's really a turkey.

 

[Trollbane]

We have a terminating sequence of 9s, but the length is infinity.

A terminating sequence of 9s that has infinite lenght is equally logical to a square circle. If it terminates at any point it is not infinite. If the number is infinite it is pointless to say it has any value and any calculation made with that number can have any value you want it to have.

If you quantisize infinity it no longer is infinity, since infinity is the largest possible number and for a quantisized number there always is a n+1.

/edited a bit for spelling

 

[bjornart]

I disagree here.

...999 wouldnt be infinity since by definition infinity cannot end and thats what that string does . If it was infinity a better way to state it would be:

999...

What's better with that? Both of them are infinity. The good thing about the first one is that numbers are best 'read' from the desimal point outwards, so the nines there can be read as 9, 90, and 900. While the nines in 999... are infinity, infinity and infinity.

 

[patoco12]

A terminating sequence of 9s that has infinite lenght is equally logical to a square circle. If it terminates at any point it is not infinite.

When I said it terminates, I meant it terminats at the decimal point but has NO starting point. I could have easily said that it starts at the decimal point and has no termination point. Sorry about my confusing terminology.

If the number is infinite it is pointless to say it has any value and any calculation made with that number can have any value you want it to have.

If you quantisize infinity it no longer is infinity, since infinity is the largest possible number and for a quantisized number there always is a n+1.

You are confusing the value of the number with the length of the number. This caused 400+ posts in the .9~=1 forum.

I REALLY don't want to start another argument over infinties and such; that would be missing the point of this forum topic.

 

[Trollbane]

You cant use a logical system, like math, to evaluate things from outside of its scope. By the way this is going we might as well start discussing what came first the chicken or the egg.

Alternatively the same proof could be used to prove that:

x=..1111
10x=..1110
x-10x=1
...1111=-1/9

etc..

But it is a limitation of math, but rather a false dilemma, since you cant take the start nor the end of an infinity, since it has neither. This is going a bit on the philosophy side though.

But the key point is that if any chain ends or starts at any point it cannot be infinite.

/edited to clarify a bit and to add

Now the bloody Matrix tagline- Everything that has a beginning has and end- is in my head... I really need a drink.

 

[bjornart]

Why do you say that? The congruences modulo to the number base are the same.

Maybe because I can hardly remember what 'congruences' mean.

Or maybe because, like Trollbane pointed out, you can use that same 'logic' to prove that:

x = ...777

10x = ...7770

x-10x = 7

-9x = 7

x = -7/9

Which made sense in the .999... debate, but which, at least to me, doesn't make sense when it comes to ...999.

Exactly. By not stopping at the obvious notion that ...999 is not a real number and has no finite value in the traditiona sense, we can still ask questions about the nature of those kinds of "numbers". The results may seem counter-intuitive, which is typical when dealing with infinity, but p-adics are a legitimate and consistent number system, just not what we're used to.

Just out of curiousity, and to avoid reading a lot about p-adics. How does ...777 = -7/9 fit into that system?

 

[LW]

You cant use a logical system, like math, to evaluate things from outside of its scope.

But, as xouper wrote, you can extend the system to handle things that you can't handle in the original.

Let's consider the set of natural numbers for while, that is, the numbers 0, 1, 2, ...

When you consider natural numbers, the expression 2 - 3 doesn't make any sense. If I have two apples I can't give you three of them.

But the expression starts to make sense when we extend the natural numbers to integers and we get 2 - 3 = -1.

This p-adic system is a way to extend rational numbers. It is not the conventional way and they are not real numbers that are used in the mathematical or everyday sense. So you don't have to expect them to behave like real numbers do.

But it is a limitation of math, but rather a false dilemma, since you cant take the start nor the end of an infinity, since it has neither. This is going a bit on the philosophy side though.

What we have here is a sequence of digits that is of infinite lenght. Such a sequence can have a start point. For example, the ordered sequence of all integers is infinite, but it has start, namely zero.

As an example of an infinite sequence that has an end but not beginning, consider the set of negative integers: ..., -3, -2, -1. This sequence ends in -1 but doesn't have a beginning.

 

[slimshady2357]

Posted by Trollbane...since you cant take the start nor the end of an infinity, since it has neither. This is going a bit on the philosophy side though.

I hope you can see how wrong that is now.

Think of a rope of infinite length, but you have one end in your hand.

Or with the Reals, the interval from 0 to 1 has a start AND an end point and yet contains an infinity of numbers.

I'm interested in what xouper thinks of Bjorn's ...777 = -7/9 result. Maybe I should just go read the link xouper provided!

Adam

 

[xouper]

Trollbane: You cant use a logical system, like math, to evaluate things from outside of its scope.

Sometimes it's amusing when someone who doesn't seem to have a firm grasp of mathematics tries to tell mathematicians what they can or cannot do. And sometimes it's simply boring.

 

[Trollbane]

Sometimes it's amusing when someone who doesn't seem to have a firm grasp of mathematics tries to tell mathematicians what they can or cannot do. And sometimes it's simply boring.

Thank you for telling me how firm of a grasp I have at maths.. I just suck at the stupid theoretics/mental masturbation part. Come to think of it I might just use this proof on the pipeline project.. Come to think of it in thermodynamics calculations ...999=-1 is almost an acceptable margin of error

 

[xouper]

Trollbane: Thank you for telling me how firm of a grasp I have at maths.. I just suck at the stupid theoretics/mental masturbation part.

You're welcome. And thank you for admitting you have no business criticizing those who are good at the stupid theoretics/mental masturbation part.

Edited to add - If it wasn't for the people who are good at the stupid theoretics/mental masturbation part, mathematics would be nothing more than an alchemy of numbers.