Non integer square roots and pi = irrational?

[Hurkyl]

"In mathematics, any Quantity can be divided in half no matter how small" ...
Where quantity means something from a structure like the reals or rationals, but not the integers.

Only if you maintain that the quantity must keep its identity

I only maintain that "x/2" be defined for one to be able to say that the quantity x can be divided in half.

1/2 is not defined, when we're working over the integers.

Let me reiterate. When working over the integers, it's not that 1/2 is something that isn't an integer: it's that 1/2 has no meaning at all.


Or, if you prefer that "divided in half" means that there exists a thing y, such that y + y = x, then 1 still cannot be "divided in half", because there does not exist a thing y such that y + y = 1. (the domain of + is the integers -- anything that is not an integer cannot be plugged into +)

One cannot find the exact sum of an infinite series of numbers, only the finite number that it may approach.

Of course one can -- it's a standard exercise in calculus classes, and is often done even earlier with things like geometric series.

The sum of an infinite series is exactly the limit of the sequence of partial sums. Nothing more, nothing less.

This is the reasoning behind the paradox; since one can always find a number smaller than another, an infinite amount of finite distances must be traveled to travel at all.

The problem with the reasoning is that it's always presented as a non sequitor -- there is nothing that even resembles a deductive chain of reasoning from "there are an infinite amount of finite distances" to "you cannot travel".

 

[HallsofIvy]

Actually, your previous post made this relate specifically to this thread. One cannot find the exact sum of an infinite series of numbers, only the finite number that it may approach. This is the reasoning behind the paradox; since one can always find a number smaller than another, an infinite amount of finite distances must be traveled to travel at all.

If that is the reasoning behind the paradox, it's no wonder you get a paradox! One certainly can find the exact sum of an infinite series of numbers. For example the sum 1+ 1/3+ 1/9+ 1/27+ ... is exactly 1.5. That is not a "finite number that it may approach". I'm not certain what you mean by "it" here. If you mean the sum, it is not approaching anything, it is 1.5. If you mean the sequence of partial sums (which is what many people mean when they talk about something like this), there is no "may" that sequence is approaching that number and so sum is, by definition, 1.5.

 

[matt grime]

Actually, your previous post made this relate specifically to this thread. One cannot find the exact sum of an infinite series of numbers, only the finite number that it may approach. This is the reasoning behind the paradox; since one can always find a number smaller than another, an infinite amount of finite distances must be traveled to travel at all.

I am not able rightly to apprehend the kind of confusion of ideas that could provoke such a question.

Charles Babbage.

 

[Leonardo Sidis]

I am not able rightly to apprehend the kind of confusion of ideas that could provoke such a question.

Charles Babbage.

Anyone who conducts an argument by appealing to authority is not using his intelligence; he is just using his memory.

-Leonardo da Vinci

 

[HallsofIvy]

Anyone who conducts an argument by appealing to authority is not using his intelligence; he is just using his memory.

-Leonardo da Vinci

Would you mind pointing out where, in this thread, there was an "appeal to authority"?

(as opposed to stating standard definitions.)