# 1=0.999... infinities and box of chocolates..Phliosophy of Math...

Discussion in 'General Philosophy' started by Quantum Quack, Nov 2, 2013.

1. ### Billy TUse Sugar Cane Alcohol car FuelValued Senior Member

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The procedure given below works for ALL Repeating Decimals (all RDs) to produce the exactly equivalent rational fraction, but some stubborn and /or ignorant people posting here make the extraordinary claim that it does not when the RD = 0.9,999,999,999.... and offer zero proof of their claim when extraordinary proof is required!

i.e. People who think that 0.9999... is not 1.0 are not only wrong, but irrational (pun intended) when postulating, with no proof, that the validity of algebraic rules fails in the proofs showing that 0.999... = 1. They make the extraordinary claim that general truths, true in an infinity of different cases, that ALL repeating or terminating decimals all have equivalent rational fractions like a/b where a & b are positive integers, with a < b are not true in one special case (the case when a = b) without giving ANY, much less the extraordinary proof, required for their extraordinary claims!

I first illustrate, several of the infinite numbers of examples, of true statements concerning terminating or infinitely Repeating Decimals (I.e. about the rational fractions equal to RDs):

1/3 =0.333333.... and 1/1 = 0.99999.... are rational fraction numbers with a "repeat length" of 1 in their equivalent decimal versions.

12/99 = 0.12121212... and 19/99 = 0.1919191919... and 34/99 = 0.343434... are rational numbers with a "repeat length" of 2 in their equivalent decimal versions.
and in general, any integer less than 99 divided by (and not a factor of) 99 will produce a decimal repeating with length 2. Some of the factors will too. For example 3/99 = 03/99 = 0.03030303... does but not 11 or 33. I.e. 11/99 =0.111... and 33/99 = 0.3333333333... but even in these cases the obvious pattern (decimal repeats the two numerator numbers) is still true. I.e. the two digit numerator repeats in blocks of two.

Likewise any integer less than 999 divided by 999 will be a decimal fraction with repeat length not more than 3 and always will repeat in blocks of 3, but for somecases, like 333 /999 = 1/3 the least long repeat block is less than 3. Check with your calculator if you like. Etc. For example, 678 /999 = 0.678,678,678, .... and that is slightly larger than 678 /1000, which equals 0.678 and should given you a hint of the proof to come.

However, any integer divided by a factor of the number base (1, 2 & 5 for base 10) or any product of these factors (like 4, 16, 2^n, 5 or 5^m, {2^n x 5^m} ) will terminate, not repeat. For example 17 /(1x4x5) = 0.85

The proof I and others have given that 1 = 9/9 = 0.99999.... is just particular case of the fact ALL rational fractions like a/b or a/9 (both a & b being integers and a < b) are equal to an infinitely repeating decimal (if they are not a finite decimal when b is a factor or product of factors of the base).

For example, the general proof of this goes like:
Rational Decimal, RD = 0.abcdefg abcdefg abcdefg .... Where each letter is one from the set (0,1,2...8,9) and the spaces are just to make it easier to see the repeat length in this case is 7.
Now for this repeat length 7 case, multiplying RD by 10,000,000 moves the decimal point 7 spaces to the right. I. e. 10,000,000 RD = a,bcd,efg . abcdefg abcdefg ... Is a 2nd equation with comas for easy reading the integer part.

Now, after noting (10,000,000 - 1) = 9,999,999 and subtracting the first equation from the second, we have the integer:
a,bcd,efg = 9,999,999 x RD. Note there are no infinitely long numbers here and 9,999,999 certainly is not zero so we can divide by it to get: RD = a,bcd,efg / 9,999,999 the rational fraction of integers exactly equal to the infinitely long repeating (with repeat length =7) decimal, RD.

Now lets become less general and consider just one of the repeat length = 7 cases. I. e. have a=b=c=d=e=f=g = 9 and recall RD was DEFINED as 0.abcdefg...so is now in this less general RD = 0.9,999,999,... and from green part of line above, RD = 9,999,999 / 9,999,999, which reduces to the fraction 1/1 which is unity as the numerator is identical with the non-zero denominator. I.e. the least numerator rational fraction equal to 0.999,999... is 1/1.

By exactly the same procedure the RD = 0.123,123,123,.... a case with repeat length of 3, can be shown to be equal to 123/ 999, which happens to reduce to 41/333 as your calculator will show as best as it can.
To prove this one multiplies this repeat length 3 RD by 1000 to get the 2nd equation and then subtract the first from it to get: RD = 123 /999.

In general when the repeat length is "n" one always multiplies by 1 followed by n zeros (an integer power of 10) the DR defining equation and then subtracts the RD defining equation from the results of the multiplication to eliminate all infinitely long number strings and get an easy to solve equation for the DR now as a rational fraction.

Then objectors to 1 =0.99999 need not only to give extraordinary proof for their objection but also need to explain why one of the other infinite number of successes of this procedure fails to produce the rational fraction that is exactly equal to the infinite Repeating Decimal , RD.

3. ### someguy1Registered Senior Member

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Billy, are you defending the 10 * .999... = 9.999... types of proof? It's hard to tell from your post. The reason those types of proofs are fallacious is that the fact that you can multiply an infinite series by a constant is a theorem that must be proven before you can use it. But by the time you've proven that theorem (which involves understanding and manipulating the formal definition of a limit), you have already understood why .999... = 1.

The ring axioms only allow finite distributivity: a(b + c) = ab + ac. This can be extended by induction to any FINITE sum:

c(a_1 + a_2 + ... a_n) = ca_1 + ca_2 + ... + ca_n

In order to extend this to the "infinite distributive law" you have to

* Have a rigorous construction of the real numbers, such as by Dedekind cuts or equivalence classes of Cauchy sequences;

* Define the limit of a sequence;

* Define the limit of a series as the limit of the sequence of partial sums;

* And then PROVE the theorem that the term-by-term multiplication of a convergent series by a constant has the same limit as multiplying the limit of the original series by that constant.

That's a much more sophisticated chain of reasoning and formal proofs than the mere fact that .999... = 1.

So all such "10x" proofs are actually no more than heuristics for high school students. They are not formal proofs.

If you like I can provide references. Any undergrad text on Real Analysis will have these proofs, for example Principle of Mathematical Analysis by Walter Rudin.

5. ### Quantum QuackLife's a tease...Valued Senior Member

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Tach and co have repeatedly stated that 1/infinity = 0. Which is essentially true IMO except that it trivializes the differences between zero and the infinitesimal to the extent that renders them not only equal but the "same" as well.
Zero and 1/infinity are not of the same "class" so to speak. To consider the infinitesimal to be a maximum boundary of zero would be better than to consider it equal to zero.

If one draws a circle and labels it as follows:

You can see that zero has a dimension of 1/infinity.

So does this mean that 1/infinity = zero?

If we presume that the center of the circle is absolute center of zero then the infinitesimal diameter circle is "filled with zero.
However until the size of 1/infinity is gained then how can zero be considered as infinitesimal?
Is the infinitesimal the largest zero can get before gaining a value? But if it gains the value of infinitesimal is it still zero? ~ a paradox yes?

Last edited: Nov 25, 2013

7. ### someguy1Registered Senior Member

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Mr. Quack, May I ask you, what drawing package do you use for these cool diagrams? I'm looking for a good math drawing package.

Also, did anyone ever tell you that you look just like Dick Cheney?

8. ### Billy TUse Sugar Cane Alcohol car FuelValued Senior Member

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I appreciate your well informed reply, but really I only described the act of moving the decimal to x spaces to the right as "multiplying by 10^x."

What I assert is the after such move the new series is larger than unity and the original infinite Repeating Decimal series (my RD = 0.abcdefg...) was not. My proof is more geometrical than algebraic as all were in Newton's The Principles of Mathematics. For your more exacting standards, I will try to restate it, in more geometrical terms.

The original RD is a point on the number line and RDsn is another point on that line with larger value (more to the right) as it is the same RD but with the decimal point shifted n places to the right. The length of the difference RDsn - Rd is 10^n - 1, which is not described or expressed by an infinitely long numerical string, but finite. I.e. is 999 for n =3 then the rest of the proof follows without doing any multiplying of an infinitely long string of numbers.

My proof is really built into the meaning of the notation using decimal points, not multiplication.

I. e. that any an all positive numbers, rational or irrational, correspond to only one point on the number line to the right of 0, and their distance from 0 increases by a factor of 10 if the notation specifying them has the decimal point shifted one space to the right. This is NOT multiplying by 10, but built into the notational system we use. Without this understanding of the notational system, one could not tell where any number was on the number line.

Last edited by a moderator: Nov 25, 2013
9. ### Quantum QuackLife's a tease...Valued Senior Member

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The software I have been using is really simple and actually a free download
started using it about 6 years ago... does animation gif slides as well.

I use it mainly for creating background images/graphics for my web design work.

But lacks the precision needed for proper schematics/graphics.
or
http://www.serif.com/free-graphic-design-software/
No , no one has mentioned this Dick Cheney resemblance before, but having a look at a few of his images I guess I could be similar...

10. ### someguy1Registered Senior Member

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You should practice sneering!

I truly commend your courage in posting an actual photo of yourself. Believe me, I haven't got the stones for that!

11. ### Quantum QuackLife's a tease...Valued Senior Member

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Being involved in the IT industry, publishing and security, told me that when it comes to the web, anonymity is an illusion. So I thought well, I am on face book, twitter, Linkedin etc so why not publish here. It has raised the standard of my posts and that of others quite significantly as well too I might add. [except for Tach of course (chuckle)]
It depends on why the member is using sciforums. What their main motivation is I guess. Some rely heavily on the delusion or illusion of anonymity some don't.
I guess some people take their intellectual investments more seriously than others who are more about entertainments.

12. The form, yes. The content, no .

13. ### Quantum QuackLife's a tease...Valued Senior Member

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hee hee you are funny Tach .. you know that?

14. ### Quantum QuackLife's a tease...Valued Senior Member

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@Tach, have a go if you dare at: post#303
and maybe you can share your conceptualizations of what a paradox is?

or are you scared others may make fun of you if you make a mess of it?

edit: Tach have you ever considered studying how to detect a person who is operating under the influence of severe conditioning?
Often referred to as "mind control techniques offered by some pseudo religious type cults such as ... go on guess...

15. The "paradox" exists only in your mind. You should really take a break from trying to pass BS as science, you aren't fooling anyone. Well, maybe yourself. Wow, 13,007 posts of nothing, that's a record.

16. ### Quantum QuackLife's a tease...Valued Senior Member

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23,320
Tach have you ever considered studying how to detect a person who is operating under the influence of severe conditioning?
Often referred to as "mind control" techniques offered by very extreme parenting, top end military and some pseudo religious type cults such as ... go on guess...

Go on you can do it.....

edit: btw it is 13008 now

17. Not really, what are you taking?

18. ### Quantum QuackLife's a tease...Valued Senior Member

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23,320

Can't do it.... can you?

19. ...stuff that the Quack make up in order to boost his post count.

20. ### Quantum QuackLife's a tease...Valued Senior Member

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23,320

Can't do it.... can you?

21. ### someguy1Registered Senior Member

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727
I bet Tach and Quantum Quack are really the same person.

22. ### YazataValued Senior Member

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5,802
It seems to me that

0.9 =/= 1.0

There's a difference between their values.

Adding another 9 doesn't eliminate the difference, it only shrinks it by 90%. There's still a difference between the values.

So 0.99 =/= 1.0

Adding another 9 doesn't eliminate the difference, it only shrinks it by 90%. There's still a difference between the values. And on it goes...

Maybe somebody could formalize this into a proof by mathematical induction. It certainly seems that a decimal point followed by some finite number n of 9's won't equal 1.0. Changing that to (n+1) 9's isn't going to make things any different. It seems, intuitively a least, that it won't change things no matter how many 9's are added to the decimal expansion, which is always going to be inexact, a closer and closer approximation.

Yet there's this idea that if we can just add enough 9's, an infinite number of them, things will be different somehow. Things will finally be exact. I don't have a clue what justifies that particular leap. The magic word in this case seems to be 'infinite', and things always start to get weird in mathematics when infinities are involved...

I'm just a layman in mathematics and claim no special expertise at all. I'm not making any claims and am not trying to convince anyone else of anything. The only reason why I said the things that I just wrote is because I'm trying to give some account of why I'm still unconvinced. That's the way I'm going to remain, until I have some convincing reason to be convinced. That's just the way it goes.

If any of you really do think that you understand this stuff, I'd appreciate you helping the rest of us understand why you see things your way.

That doesn't mean insulting us, sneering at us, calling us names or putting us down. It doesn't call for ex-cathedra statements of your own infallible authority. It means TEACHING.

Seriously, one of the reasons why a larger and larger proportion of the general public every year is becoming estranged from and even hostile towards math and science is because teaching in these subjects is often so appallingly bad.

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