An infinite question: Why doesn't .999~ = 1?

[Trinopus]

Proof by induction:

0.9 is a number.

If 0.999[to n terms] is a number, then 0.999[to n1 terms] is a number. (Closure property of addition.)

Ergo 0.999… is a number.

[septimus]

@watchwolf49

[0, 0.9]∪[0.9, 0.99]∪[0.99, 0.999]∪… = [0, 1]

No. 1 is not in this “segment”:
[0, 0.9]∪[0.9, 0.99]∪[0.99, 0.999]∪… = [0, 1**)**
Mr. Netz is correct that no finite(*) number of 9’s gets you to 1. The correct response to him is “So what?”, not to post falsities.

• • Yes, Mr. Netz insists that an infinite number of 9’s isn’t enough either. The terms “arbitrarily large” and “infinite” have different meanings; thread denizens are conflating the two terms, some in one direction, some the other.

[The_Great_Unwashed]

@septimus

Mr. Netz insists that an infinite number of 9’s isn’t enough either.

Haven’t you just joined him?

[septimus]

@The_Great_Unwashed

Haven’t you just joined him?

[QUOTE=septimus]
The terms “arbitrarily large” and “infinite” have different meanings;
[/QUOTE]

Mr. Netz has posted much to object to, but so have some of his detractors. A particular controversy seems to be the truth of the mathematical fact
1 ∉ Union {x[sub]j[/sub] | j ∊ ℕ}
where
x[sub]j[/sub] = { r | 0 ≤ r ≤ 1 - 10[sup]-j[/sup] }
Mr. Netz and I agree that this statement of non-membership is true; some of you think it is false. My math fluency isn’t what it used to be (and maybe never was :rolleyes: ), so I hope one of our resident mathematicians will adjudicate.

[TATG]

@septimus

No. 1 is not in this “segment”:
[0, 0.9]∪[0.9, 0.99]∪[0.99, 0.999]∪… = [0, 1**)**
Mr. Netz is correct that no finite(*) number of 9’s gets you to 1. The correct response to him is “So what?”, not to post falsities.

• • Yes, Mr. Netz insists that an infinite number of 9’s isn’t enough either. The terms “arbitrarily large” and “infinite” have different meanings; thread denizens are conflating the two terms, some in one direction, some the other.

The way people are using “…” in this thread sometimes seems to mean “continue this with arbitrarily large n”, and other times means “continue this with arbitrarily large n, and also infinitely many n”. (I’m agreeing with you here, just pointing out that there seems to be further confusion over the use of “…”. But maybe I am being too charitable in re-interpreting “…” for them.)

[TATG]

@Trinopus

Proof by induction:

0.9 is a number.

If 0.999[to n terms] is a number, then 0.999[to n1 terms] is a number. (Closure property of addition.)

Ergo 0.999… is a number.

If you just include finite n, then while you get infinitely many numbers on that list, you don’t get 0.999… .

@The_Great_Unwashed

Haven’t you just joined him?

Septimus hasn’t joined him because:
[0, 0.9]∪[0.9, 0.99]∪[0.99, 0.999]∪… = [0, 1)
only has 0.9n’s, where n is finite (but as above, there are still infinitely many distinct 0.9n’s). (But also as above, I’m not sure everyone is using “…” in the same way.)

[netzweltler]

@septimus

So… Is it your contention that 1/3, or any real number not of the form, for some integers a, b, c,
a ∙ 2[sup]b[/sup] ∙ 5[sup]c[/sup]
has no representation as a decimal fraction? That an equation like 1/7 = .142857 is an abuse of notation?

All I am saying is that .142857 is not a point on the number line. Whereas 1/7, square roots, e, π are points on the number line.

@septimus

Is the following a fair statement of your thesis?:
“While I, netzweltler, do understand that 1/3 and 1/7 are real numbers, they have no legal decimal representation because I, netzweltler, insist that all decimal representations have finite length.”

This seems like a fine (though highly non-standard) position and I won’t argue against it. I do wonder if repeating the nursery rhyme about “moving your pen” was the best way to express this opinion.

Non-terminating decimal representations (except 000…) cannot be located on the number line.

[netzweltler]

@naita

Is [0, 0.9]∪[0.9, 0.99]∪[0.99, 0.999]∪… also equal to [0, 1>?

Yes.

[netzweltler]

@naita

Not with mathematics you haven’t. Whereas the rest of us have shown that any number of threes that hasn’t brought us to 1/3 is not infinite. Just to repeat that:

If 1/3 - (3/10 + 3/10^2 + 3/10^3 + … 3/10^n)=d then n is a natural number.

But of course you’re not saying 0.333… is different from 1/3 as a number, you are saying it isn’t a number. It’s a non-number result of, according to you, mathematical operations.

Can we do math with these non-numbers though? Or to put it another way, can you show that these operations below are in error somehow?

0.333 = [0, 0.3]∪[0.3, 0.33]∪[0.33, 0.333]∪…

30.333 = 3([0, 0.3]∪[0.3, 0.33]∪[0.33, 0.333]∪…)=[0, 0.9]∪[0.9, 0.99]∪[0.99, 0.999]∪…
3*0.333 = 0.999

0.333 / 2 = [0, 0.15]∪[0.15, 0.165]∪[0.165, 0.1665]∪…
0.333/2 = 0.1666

And if you can’t show that these operations are in error (and saying “you can’t do that” isn’t a proof) isn’t it odd that these non-numbers behave exactly like they would do if they were valid representations of fractions and the number 1?

Basically you are asking if we can do math on sequences. Yes we can, even if they are infinite. Because you are doing math with numbers (0.3, 0.03, 0.003, …). But if you multiply 0.333… (a non-number) by 3 you will get another non-number.

[netzweltler]

@watchwolf49

[0, 0.9]∪[0.9, 0.99]∪[0.99, 0.999]∪… = [0, 1]

is your only lame attempt to disprove this fact is “i said so” … And your only logical step is equivalent to “god can’t make rocks he can’t lift”?

Please, upon which pseudo-theorem do you base the statement “it was wrong the first time you said this” on?

As the number of elements in this union approaches infinity, the length of the line segment approaches 1 … Therefore the limit point of the union is 1 … Why do you believe then that 1 is not an element of this union?

[0, 0.9]∪[0.9, 0.99]∪[0.99, 0.999]∪… = [0, 1)

[naita]

@netzweltler

Basically you are asking if we can do math on sequences. Yes we can, even if they are infinite. Because you are doing math with numbers (0.3, 0.03, 0.003, …). But if you multiply 0.333… (a non-number) by 3 you will get another non-number.

If you, for just one second, ignore your perception of their definition and how it differs from a definition of a number, do you recognise that any math done on these infinite sequences work out as if they actual were the fractions the rest of us say they represent?

For instance, if you multiply .142857 by 8, you get 1.142857. If you accepted .142857 to be equal to 1/7, there is no mathematical operations you could do on .142857 that would lead to a different answer than if you did it on 1/7. Agree?

[netzweltler]

@naita

If you, for just one second, ignore your perception of their definition and how it differs from a definition of a number, do you recognise that any math done on these infinite sequences work out as if they actual were the fractions the rest of us say they represent?

For instance, if you multiply .142857 by 8, you get 1.142857. If you accepted .142857 to be equal to 1/7, there is no mathematical operations you could do on .142857 that would lead to a different answer than if you did it on 1/7. Agree?

Of course you can do that. You can do that with any terminating version 0.142857, 0.14285714285714285, 0.14285714285714285714285714285714, and it will also work with the non-terminating version .142857.

[naita]

@naita

If you, for just one second, ignore your perception of their definition and how it differs from a definition of a number, do you recognise that any math done on these infinite sequences work out as if they actual were the fractions the rest of us say they represent?

For instance, if you multiply .142857 by 8, you get 1.142857. If you accepted .142857 to be equal to 1/7, there is no mathematical operations you could do on .142857 that would lead to a different answer than if you did it on 1/7. Agree?

@netzweltler

Of course you can do that. You can do that with any terminating version 0.142857, 0.14285714285714285, 0.14285714285714285714285714285714, and it will also work with the non-terminating version .142857.

That is false. It only works with the non-terminating version.

7 * 1/7 = 1
7 * 0.142857 = 0.999999 != 1
7*.142857 = 0.999 “=” 1 (i.e. = by the rules in my stated hypothetical.)

[netzweltler]

@naita

That is false. It only works with the non-terminating version.

7 * 1/7 = 1
7 * 0.142857 = 0.999999 != 1
7*.142857 = 0.999 “=” 1 (i.e. = by the rules in my stated hypothetical.)

To make the last equation true you need to show that 1/7 = .142857.

[naita]

@netzweltler

To make the last equation true you need to show that 1/7 = .142857.

I’m asking for your response to a hypothetical. Are you really unable to deal with such a concept?

I’ll restate it just for emphasis. If we axiomatically decided non-terminating repeating decimal notations were truly equal to the fractions they derive from, there would be no mathematical operation you could do on the defining sequences that did not yield the same results as if they were done on the fractions themselves.

Can you show this hypothetical to be internally inconsistent?

[netzweltler]

@naita

I’m asking for your response to a hypothetical. Are you really unable to deal with such a concept?

I’ll restate it just for emphasis. If we axiomatically decided non-terminating repeating decimal notations were truly equal to the fractions they derive from, there would be no mathematical operation you could do on the defining sequences that did not yield the same results as if they were done on the fractions themselves.

Can you show this hypothetical to be internally inconsistent?

If we axiomatically decided 1/7 = .142857? Ok. What does keep us from axiomatically deciding 1/7 = 0.14285714285714285714285714285714 then?

[septimus]

@netzweltler

To make the last equation true you need to show that 1/7 = .142857.

You don’t know how to show that 1/7 = .142857 ? Do you know that
1 + r + r^2 + r^3 + … = 1/(1-r)
(This was discovered before Euclid.) Substitute that into the definition of .142857 to get
.142857 = .142857 (1 + .000001 + .000001^2 + … ) = .142857 / .999999 = 1/7

HTH.

[naita]

@netzweltler

If we axiomatically decided 1/7 = .142857? Ok. What does keep us from axiomatically deciding 1/7 = 0.14285714285714285714285714285714 then?

We could do that, but if we did, previously well defined operations such as addition and multiplication would give different answers depending on our choice of representation of 1/7.

If you accept the operations of doing multiplication and addition on infinite series, then axiomatically equating non-terminating repeating decimals with the fractions that produce them gives an internally consistent mathematics.

[naita]

@septimus

You don’t know how to show that 1/7 = .142857 ? Do you know that
1 + r + r^2 + r^3 + … = 1/(1-r)
(This was discovered before Euclid.) Substitute that into the definition of .142857 to get
.142857 = .142857 (1 + .000001 + .000001^2 + … ) = .142857 / .999999 = 1/7

HTH.

This whole thread has for several pages been about netzwelter not accepting that infinitely repeating decimals gives one a number, by that logic (1 + .000001 + .000001^2 + … ) isn’t a number, so whatever you do with it doesn’t matter. Let’s try not to repeat that ground.

[netzweltler]

@naita

We could do that, but if we did, previously well defined operations such as addition and multiplication would give different answers depending on our choice of representation of 1/7.

If you accept the operations of doing multiplication and addition on infinite series, then axiomatically equating non-terminating repeating decimals with the fractions that produce them gives an internally consistent mathematics.

Ok. But how does that show that 1/7 = .142857 exactly. It would also work if we axiomatically define it to be “closest possible” decimal representation.