The .999... = 1 "debate"

[phyti]

All knowledge is by definition, which would include math.

The root of the problem/disagreement is the concept of infinity.

The integers/(natural numbers) and rationals, have a constant value, and they can be formed in a finite number of steps.

The irrational numbers are formed by an infinite process which can never be completed.

The integers and rationals can be constructed and correspond to real world objects because they are discrete models.

The irrationals cannot be constructed because they are continuous.

The representation '.9...' can be defined as '=1' but that doe not make it '=1'.

It has to be defined as '1' to make it work, because the process cannot do it.

It really is a variable by definition, it approaches 1 in the limit. Note it only equals 1 when it reaches 1.

Irrationals are beneficial in the analysis of complex problems.

Their value as an analytic method is not in question.

The continuous process issue is not confined to irrationals but all concepts that depend on it.

Pi is a good example. No one knows what the millionth decimal is until it is constructed.

All applications use an approximation because no one can wait for the theoretical/ideal value.

For a wagon wheel 3.1 is sufficient. For a ball bearing 3.1416 is sufficient.

The precision depends on the purpose, but in each case a different value is used.

The ideal value of pi is never used because it can never be known.

For some the leap from the finite to the infinite does not need justification.

The mind does not experience anything infinite, so it imagines what it could be.

In simplistic fashion it extrapolates the properties and processes of the finite to the infinite.

Eg. we can list a finite set of objects, but can we list an infinite set?

Eg. we can count a finite set of objects, but can we count an infinite set?

The objection is the assertion without proof that the irrationals, transcendentals, etc. exist, even as mental constructs.

The result, as the limit of an infinite process, exists only if the process is completed.

This is the failure point and not a proof.

The mind has no experience of the infinite therefore it has no basis for a concept of an object or a process.These objects can only exist as approximations and not absolute values or constants (a misnomer).

The step from finite to infinite is as great as the step from 0 to 1.

[silkworm]

I don't know if this has been mentioned, but I certainly hope so.

 

That is not 0.999 = 1. It is 1 = 0.999(with an infinity of 9s) = 1.

 

There is no application or anything else whatsoever where you'd have significant figures numbering beyond infinity, so anywhere you choose to end, the number will end up being 1. So 0.999(with an infinity of 9s) is 1, unless you need precision beyond an infinity of decimal places, where in which case, you're screwed.

[the tree]

phyti, 0.9... is not an irrational number, an irrational can be identifed as non-terminating and non-recuring in a decimal representation like [math]\pi[/math], [math]\phi[/math] or [math]e[/math].

There is no question of when, mathematics happens the moment you lay down the definitions.

Precision also doesn't come into it because there is no difference between 0.9... and 1.

[KLB]

Okay here we go again, screwing with Ken's brain.

 

I can't add to this new line of discussion, but I'm finding it very interesting, so please keep it up. I'll replace brain coggs later if needed.

[matt grime]

All knowledge is by definition, which would include math.

 

debatable. If I hit your thumb with a hammer, you know it hurts, you're not going to debate what 'hurt' means before screaming.

 

The root of the problem/disagreement is the concept of infinity.

 

no, the root of the problem is that people do not know the definitions.

 

The integers/(natural numbers) and rationals, have a constant value,

 

what does that even mean?

 

and they can be formed in a finite number of steps.

 

from what? and what does this have to do with the problem?

 

The irrational numbers are formed by an infinite process which can never be completed.

 

define 'process' and 'complete'

 

The integers and rationals can be constructed and correspond to real world objects because they are discrete models.

 

define construct, and what it means for something to 'be' constructed so tthat we can get into context the next line:

 

The irrationals cannot be constructed because they are continuous.

 

how is this true? Perhaps if you offer a definition of construct then we can agree or disagree that a continuous object can be constructed.

 

 

The representation '.9...' can be defined as '=1' but that doe not make it '=1'.

 

This is moot. I agree it would be infinitely more illustrative to use the terminology of 'equivalent', however, utility, and consistency dictates that we use equal just as we do when we say 1/2 = 2/4

 

 

It has to be defined as '1' to make it work, because the process cannot do it.

 

what process?

 

It really is a variable by definition, it approaches 1 in the limit.

 

it ain't approaching anything. it is a number.

 

 

Note it only equals 1 when it reaches 1.

 

again, it is not 'reaching' or tending towards anything, it is a number.

 

 

Irrationals are beneficial in the analysis of complex problems.

Their value as an analytic method is not in question.

The continuous process issue is not confined to irrationals but all concepts that depend on it.

 

what process?

 

Pi is a good example. No one knows what the millionth decimal is until it is constructed.

 

and?

 

 

All applications use an approximation because no one can wait for the theoretical/ideal value.

 

the same surely holds for dividing by 3 (or any number divisible by any prime not equal to 2 or 5)

 

For a wagon wheel 3.1 is sufficient. For a ball bearing 3.1416 is sufficient.

The precision depends on the purpose, but in each case a different value is used.

The ideal value of pi is never used because it can never be known.

 

nonsense. the integral of exp{-x^2} is sqrt(2pi), a perfectly exact representation of the answer. just cos you might want it in decimal notation rather than a symbolic one means nothing.

 

i'll leave the rest (which still uses the word 'process' without defining it properly)