Reply to: Maths query

[OwlHoot]

The reason they are the same number is nothing to do with limits; rather the definition of real numbers.

In classical maths, which covers most requirements, the real numbers can be rigorously defined in terms of equivalence classes of Cauchy sequences, and that involves limits.

The main reason I emphasised limits, distinguishing them from sequences, was that confusion over the two is the main reason these "does 0.999.. equal 1?" debates crop up over and over again on unmoderated maths & tech forums.

Your "axiomatic" approach might satisfy a professional; but it doesn't help dispel obstinately held misconceptions, about what is really no more than a matter of notation once the relevant concepts are clearly grasped.

[jamesbrown]

Which part of "the real number system" did you not understand?

Exactly. It has a goddamn Wikipedia page FFS. They also have it in the FAQ over at sci.math and in countless other forums. Mathematics is all about internal consistency, based on precise rules. If you don't know the rules, you have no hope of discussing their implications.

[expat]

Which part of "the real number system" did you not understand?

There is no number (in the real number system, with standard analysis) between 0.99999.... and 1. Therefore (in the real number system, with standard analysis) 0.99999... and 1 are simply different representations of the same number.

The reason they are the same number is nothing to do with limits; rather the definition of real numbers.

You can tell the few people here who actually have an inkling of how hard the real number system is.

[NotAllThere]

That's feeble

There are no integer values between 1 and 2; doesn't make them the same thing.

Which part of "the real number system" did you not understand?

There is no number (in the real number system, with standard analysis) between 0.99999.... and 1. Therefore (in the real number system, with standard analysis) 0.99999... and 1 are simply different representations of the same number.

The reason they are the same number is nothing to do with limits; rather the definition of real numbers.

[vwdan]

I love that we have people arguing until they're blue in the face over widely accepted mathematical constructs. Yes, all those people who have devoted their entire lives to maths must be wrong and incompetent, because CUK knows better.

[OwlHoot]

If it's an infinite sequence, isn't 0.999... the infiniteth element of the sequence?

There's no such thing as "the" infinitieth element. For any element you choose, there's always a larger one.

The sequence represents a potential infinity, as opposed to an actual or completed infinity.

[Rhino 888]

If it's an infinite sequence, isn't 0.999... the infiniteth element of the sequence?

But there's more of the sequence after that....

[Bunk]

That's absolutely true, the sequence and its limit are not the same thing.

But 0.999.. is defined as, or rather denotes, the limit of the sequence.

It doesn't represent the sequence, or (in this case) any term of the sequence, and it therefore doesn't share the sequence terms' property of being less than 1.

You're almost there, but you still somehow end up identifiying 0.999.. with an element of the sequence (presumably because their notation looks much the same - but that is slightly misleading).

If it's an infinite sequence, isn't 0.999... the infiniteth element of the sequence?

[OwlHoot]

This is the value beyond which the sequence can not progress, it is NOT the value that the sequence reaches. They are not the same thing.

That's absolutely true, the sequence and its limit are not the same thing.

But 0.999.. is defined as, or rather denotes, the limit of the sequence.

It doesn't represent the sequence, or (in this case) any term of the sequence, and it therefore doesn't share the sequence terms' property of being less than 1.

You're almost there, but you still somehow end up identifiying 0.999.. with an element of the sequence (presumably because their notation looks much the same - but that is slightly misleading).

[MyUserName]

Surely 0.9 recurring is not equivalent to 1.

Even if, functionally, it might as well be. It will not because there will be an infinitesimally small difference as the limit tends towards infinity.

The difference might become so small that we have trouble expressing it but it will still be there.

[Bacchus]

That rather negates the principle of limits, summation of converging infinite series, etc.

No it doesn't, quite the opposite, as above.

[Bacchus]

How many more fecking times?

The infinite sequence {0.9, 0.99, 0.999, ...} "gets increasingly close to 1 but never reaches it" (to use your words).

But "by its very definition" 0.999 .. is not that sequence but defined as the _limit_ of that sequence, which is 1.

The persistently muddled thinking (and it isn't just you) around these sequences and limits is astounding!

The key word here being "limit"

This is the value beyond which the sequence can not progress, it is NOT the value that the sequence reaches. They are not the same thing.

[OwlHoot]

By its very definition 0.999... gets increasingly close to 1 but never reaches it, mathematically they simply are not the same thing in any shape, size, or form. Arithmetically they may as well be.

How many more fecking times?

The infinite sequence {0.9, 0.99, 0.999, ...} "gets increasingly close to 1 but never reaches it" (to use your words).

But "by its very definition" 0.999 .. is not that sequence but defined as the _limit_ of that sequence, which is 1.

The persistently muddled thinking (and it isn't just you) around these sequences and limits is astounding!

[d000hg]

That's feeble

There are no integer values between 1 and 2; doesn't make them the same thing.

By its very definition 0.999... gets increasingly close to 1 but never reaches it, mathematically they simply are not the same thing in any shape, size, or form. Arithmetically they may as well be.

That rather negates the principle of limits, summation of converging infinite series, etc.

[Bacchus]

There is no number (in the real number system, with standard analysis) between 0.99999.... and 1. Therefore 0.99999... and 1 are simply different representations of the same number.

Anyone who says different either doesn't know what they're talking about or is using non-standard analysis. I suspect strongly the former.

That's feeble

There are no integer values between 1 and 2; doesn't make them the same thing.

By its very definition 0.999... gets increasingly close to 1 but never reaches it, mathematically they simply are not the same thing in any shape, size, or form. Arithmetically they may as well be.