Obscure & Useless

[Resha Caner]

If N is an arbitrarily large number less than infinity then sum is a partial sum. But you can sum an infinite series. It is done all the time in the integral calculus.

If the partial sums of an infinite series converge to a limit then the sum of the infinite series is the limit. That is the whole point.

OK. I get your point now. I'm not saying I agree with it, but I understand you. This is getting fun ... at least for me.

When I learned L'Hopital's rule, I took it as a legitimate and formal procedure. Many years later I was told that it is really only a math trick. The "formal" version is much more involved and requires a delta-epsilon proof (if I recall correctly). So, just because the calc teacher says it don't necessarily make it true!

This is not something that admits of scepticism. This is a matter of formal proof. In mathematics a formal proof trumps intuition and scepticism every time.

So, you're saying that Richman isn't rigorous enough for you? Be careful. If that's going to be your position, then we must indeed become rigorous.

We have proven, using the formal rules of arithmetic that 0.999... = 1. Thus you cannot make a cut on the number line between 0.999... and 1.

As such, this is not rigourous either. Rather, it comes across as a bit of circular logic.

Restate your example please. I am not sure which you are referring to.

The original "proof" used the operation 10*x, where x = 0.9...

I am claiming that the "proof" should hold for a*x, where a is any integer. So, I picked 9 and tried to show (though not rigorously) that it didn't work. So, you have several options:

1) Show that the "proof" can be done with 9.
2) Prove that an arbitrary integer, a, is not satisfactory, but rather that the integer multiplier must be consistent with the base (i.e. 10).

With that said, I'll note that I took on your challenge that Richman is not rigorous enough. I found a paper that seems to agree with you. It then tries to develop a more formal proof. Note that it calls "liberal" those mathematicians who accept 0.9... = 1, and claims there are problems with the way they try to formalize the multiplication of a number with infinite decimals - exactly the point I was trying to make.

So, I guess I have another bulwark to throw up:
arxiv.org/pdf/0910.5870

(Hmm. Not sure why the link isn't working. The paper is: Klazar, Martin, "Real numbers as infinite decimals and irrationality of sqrt(2)". Maybe you can find it by searching for that.)

 

[driewerf]

I'm intrigued that 0/0 ≠ 1.

Consider this.
Any real number a*0= 0. Agree?
cvonsider then

equation (1) a* 0/0=?

If 0/0=1 then (1)1 can be written as a*1=a

But it can also be written as (a*0)/0= 0/0=1

As (1) can not be a and 1 at the same time 0/0 is undefined.

 

[driewerf]

Question 1: Do you have an obscure science or math problem that you find fascinating - even though it's most likely useless?

Mine would be the question: Does 0.999 repeating = 1?

Reminds me of a joke a heard.

Some private company has a vacancy and is looking for a mathematician. There are three applicants for the job. So the recruiter interviews each of them.
The first is a pure mathematician. When asked to calculate 1*3/3 he writes the equation, bars the 2 3's and has 1 as an answer.

The second is an engineer. He pulls out an impressive calculator. Types first 1/3, and then multiplies the result by 3 to get 0.9999999999999.

The third applicant is a statistician. He looks the recruiter is the eyes and asks:"What do you want the result to be?"

 

[GrowingSmaller]

Every number is a point on the number line. Pi is on the number line and it is a non-repeating infinite series. A repeating infinite series is a rational number, but all irrational numbers are also on the number line.

So 0.9999.... is at the point "10"?

I don't get it. If Pi (or 0.999999....) is described a point on the number line shuldn't there a point at which the series ends? But if there is a point then I would imagine the series is not infinite precisely because the series ends somewhere.

 

[Resha Caner]

I wanted to be careful what I was conceding. Things were moving fast and I needed a little time to think. So, let me clarify a few things.

First: convergence. Upon further thought I will concede that a convergent infinite series equals a number. However, proving convergence is a finite process (as per the definition given earlier). Therefore, proving it converges only proves that there is some number to which it is equal. Proving the exact value of that number is a different matter.

Second: Gracchus answered my comment that the symbolic representation of 0.9... is unique by showing some that are not unique. I would distinguish his example from mine as follows. It is true that 0.5 can be represented as 1/2. It can also be represented as 2/4, 3/6, 4/8, ... an infinite number of ways. That is not the point. Using fractions changes the system for representing the number. If one changes the system, of course the number can be represented a different way. What is interesting is that in the base 10 system, 0.9... is unique (as all numbers are unique). There is no other way to represent 0.9... There is only one way. Likewise, 1 is the only way to represent 1. There is no other way. Each number is unique, which is one of the beauties of the system.

Now, Gracchus would argue that there are two ways 0.9... and 1. But that begs the question: can every number be represented 2 ways or is 0.9... unique in that regard? So, we are led back to trying to prove it one way or the other.

 

[juvenissun]

Question 1: Do you have an obscure science or math problem that you find fascinating - even though it's most likely useless?

Mine would be the question: Does 0.999 repeating = 1?

Question 2: What do you think is the most underappreciated aspect of science or math? In other words, little known but very important.

Mine would be the assumptions behind space-time measures.

My experience is: whatever idea comes into your mind, it is not new. A bunch of people have already done a lot of work on it.

So, to your two questions, my answer is: none.

 

[hasone]

9.999... = 1 is one of those results in mathematics that isn't intuitive but that we have to put up with because otherwise very useful and basic assumptions have to be discarded.

There are other examples. The Banach-Tarski paradox (which isn't obscure from the perspective of a mathematician) is one of those things where an intuitive assumption (the axiom of choice) leads to a highly unintuitive result. The axiom of choice is so basic and useful that I'll accept what craziness may result.

 

[Gracchus]

9.999... = 1 is one of those results in mathematics that isn't intuitive but that we have to put up with because otherwise very useful and basic assumptions have to be discarded.

To pick a nit (as lovers of math are wont to do), 1 = 0.999... not 9.999...

 

[Gracchus]

I wanted to be careful what I was conceding. Things were moving fast and I needed a little time to think. So, let me clarify a few things.

First: convergence. Upon further thought I will concede that a convergent infinite series equals a number. However, proving convergence is a finite process (as per the definition given earlier). Therefore, proving it converges only proves that there is some number to which it is equal. Proving the exact value of that number is a different matter.

Uh no! It is the same matter.

Second: Gracchus answered my comment that the symbolic representation of 0.9... is unique by showing some that are not unique. I would distinguish his example from mine as follows. It is true that 0.5 can be represented as 1/2. It can also be represented as 2/4, 3/6, 4/8, ... an infinite number of ways. That is not the point. Using fractions changes the system for representing the number. If one changes the system, of course the number can be represented a different way. What is interesting is that in the base 10 system, 0.9... is unique (as all numbers are unique). There is no other way to represent 0.9... There is only one way. Likewise, 1 is the only way to represent 1. There is no other way. Each number is unique, which is one of the beauties of the system.

The number is unique. The representation is not.

Now, Gracchus would argue that there are two ways 0.9... and 1. But that begs the question: can every number be represented 2 ways or is 0.9... unique in that regard? So, we are led back to trying to prove it one way or the other.

The number one is unique, but there are a great many ways to represent it. 1 = 1! = 0! is true by the definitions of factorial notation. But that is not, I think what you are trying to get at. e^0 = (pi)^0 = 1 probably isn't either.

Consider 0.777... = 7/9 or

x = 0.131313...
100x= 13.131313...
99x = 13
so x = 13/99 = 0.131313...

or x = 0.363636.
100x = 36.363636...
99x = 36
x = 36/99 = 12/33

Of course nine has a unique position in base ten arithmetic as the highest single digit. So does one in binary arithmetic, or fifteen in hexadecimal, or two in base three arithmetic. Think about it. Change any of the above problems to a new basis, as you will, the answers remain the same.

Consider the base eight equation: x = 0.777...
10x = 7.777...
7x = 7
x = 1

Pay around with stuff like this and you will get a feel for it.

 

[leftrightleftrightleft]

Question 2: What do you think is the most underappreciated aspect of science or math? In other words, little known but very important.

Not that it is little known, but definitely under appreciated: Calculus. Calculus is the building block of the modern world. You can probably trace nearly all modern inventions back to some science which can be traced back to some theory which can be traced back to some equation involving calculus.

Another one is the Rydberg Constant. This constant is used to represent the limiting value of the highest wavenumber of a photon that can be emitted from the hydrogen atom. The crazy thing is that it was determined empirically and then later identified as a mathematical equation containing five seemingly unrelated physical constants: the speed of light, the mass of the electron, the elementary charge of an electron, the permittivity of free space and Planck's constant. It remains the worlds most accurately determined physical constant.

 

[Resha Caner]

Uh no! It is the same matter.

Hmm. OK. Since you haven't proven that 0.9... = 1, then I guess you are saying that you haven't shown the series converges either. If you plan to present a proof for both of those, I will wait for it.

Consider the base eight equation: x = 0.777...
10x = 7.777...
7x = 7
x = 1

If the proof didn't convince me in base 10, it won't convince me in base 8. You need to address the issue of multiplication that I have mentioned several times. Richman was not rigorous enough for you, so I gave another reference.

But there is something interesting about the different bases.

For base 10 (say the index is i) the series is: 9/10 + 9/100 + 9/1000 + ...

For base 8 (say the index is j) the series is: 7/8 + 7/64 + 7/512 + ...

Though I didn't look for it, I expect there might be some <i,j> where the 2 series have the same value. But if that is so, then they do not have the same value for the near cases <i-1,j>, <i+1,j>, <i,j-1>, and <i,j+1>. I think that is always true. That is very interesting ... especially the implications it has for their limits. It makes me wonder if the two series represent different kinds of infinities - akin to (though not the same as) what you mentioned about Cantor's levels of infinity. If that were correct, it could mean that there are numbers between 0.9... and 1, but the system does not allow their representation. Maybe base 11 (using "A" as the additional numeral) has a number denoted as 0.A... which is between 0.9... and 1. It's not something I can prove at the moment, but it might be fun to try it.

Pay around with stuff like this and you will get a feel for it.

What makes you think I haven't? Because I'm willing to admit when I make a mistake?

 

[Resha Caner]

Not that it is little known, but definitely under appreciated: Calculus. Calculus is the building block of the modern world. You can probably trace nearly all modern inventions back to some science which can be traced back to some theory which can be traced back to some equation involving calculus.

Oh, yeah. I love calculus. Are you familiar with Archimedes' "method of exhaustion", or Grossman & Katz's "non-Newtonian calculus"? Very cool stuff.

 

[Gracchus]

Hmm. OK. Since you haven't proven that 0.9... = 1, then I guess you are saying that you haven't shown the series converges either. If you plan to present a proof for both of those, I will wait for it.

But I did prove it!

First I defined x = 0.999...
Then I multiplied both sides by ten. They must be equal by the axioms of arithmetic. This is apparently the step you can't get your head around. But fif you don't accept the axioms of arithmetic then you should be able to demonstrate that they are in error.

I think that you probably probably know that 0.999... = 1. I have reached the conclusion, that beyond a reasonable doubt, ... that is, it is a conclusion reached by a combination of non-mathematical induction and deduction that you are a troll, acting from mean spirited malice, which you tell yourself is humor, to produce frustration. Good comedians know that the truth is always funnier than lies. Ha. Ha. You are on ignore.

 

[Resha Caner]

I have reached the conclusion ... that you are a troll ...

An ad hominem approach gains you nothing and does not reflect well on you.

Then I multiplied both sides by ten.

Nor do you answer my questions. If you have lost interest or do not have the time, I understand. But if you plan to continue, I would appreciate you answering my questions.

This is the step I have challenged - that you "multiplied both sides by ten." It's not an original challenge. It is one mentioned by Richman (which you dismissed as lacking in rigor). So, I supplied a second paper claiming more rigor than Richman (the paper by Klazar). This is the challenge you need to meet.

In simple terms, the problem is illustrated as follows:

All the proof really does is multiply by "a" and then divide by "a"-1. That is a common technique used for preserving a certain amount of precision in fixed-point math (such as the control systems I use in my job). But it is always considered a method of approximation. Why should it be considered any differently in the supposed "proof"? As mentioned before, I can replace "a" with any value. The original proof used 10. But if I use 9, the issue is better demonstrated.

When multiplying by 10, we assume we end with the same repeating sequence with which we started. When we multiply by 9, that doesn't happen.

0.9 * 9 = 8.1
0.99 * 9 = 8.91
0.999 * 9 = 8.991
0.9999 * 9 = 8.9991
...

What "rule" allows us, in the limit, to conclude that 0.9... * 9 = 8.9...? Saying "because it does" is not a proof. I have not yet seen any reason given. This (as I understand it) is the issue that Klazar tries to formalize, i.e., how does one multiply numbers with an infinite number of digits?

The result, as I read it, does not support your conclusion.

Further, as I said, I even have the inklings of another type of proof. However, at the moment, I am not confident enough in my understanding of certain properties of infinite series to say it actually works. I've started talking to my dad about it, so we'll see where that goes.

 

[[serious]]

An ad hominem approach gains you nothing and does not reflect well on you.



Nor do you answer my questions. If you have lost interest or do not have the time, I understand. But if you plan to continue, I would appreciate you answering my questions.

This is the step I have challenged - that you "multiplied both sides by ten." It's not an original challenge. It is one mentioned by Richman (which you dismissed as lacking in rigor). So, I supplied a second paper claiming more rigor than Richman (the paper by Klazar). This is the challenge you need to meet.

In simple terms, the problem is illustrated as follows:

All the proof really does is multiply by "a" and then divide by "a"-1. That is a common technique used for preserving a certain amount of precision in fixed-point math (such as the control systems I use in my job). But it is always considered a method of approximation. Why should it be considered any differently in the supposed "proof"? As mentioned before, I can replace "a" with any value. The original proof used 10. But if I use 9, the issue is better demonstrated.

When multiplying by 10, we assume we end with the same repeating sequence with which we started. When we multiply by 9, that doesn't happen.

0.9 * 9 = 8.1
0.99 * 9 = 8.91
0.999 * 9 = 8.991
0.9999 * 9 = 8.9991
...

What "rule" allows us, in the limit, to conclude that 0.9... * 9 = 8.9...? Saying "because it does" is not a proof. I have not yet seen any reason given. This (as I understand it) is the issue that Klazar tries to formalize, i.e., how does one multiply numbers with an infinite number of digits?

The result, as I read it, does not support your conclusion.

Further, as I said, I even have the inklings of another type of proof. However, at the moment, I am not confident enough in my understanding of certain properties of infinite series to say it actually works. I've started talking to my dad about it, so we'll see where that goes.

Another approach is to come at it this way:
.9999... and 1 are both real numbers
a property of real numbers is that between any two real numbers there exists another real number
There is no real number that falls between .9999... and 1
Hence, .9999... and 1 must be the same real number.

Here is a proof again in standard form with justifications at each step:
.9999...=a Variables can be set to represent any real number (and at least some non real numbers)
9.999...=10a S You can do the same thing to both sides. in this case multiply by 10
9.999...-a = 10a-a subtract a from both sides.
9=9a substitution property
1=a divide both sides by 9
1=.9999... substitution property.

proofs aren't about what sounds or feels right. They are about what can be proven. In order to reject a proof, you must reject some step of the proof. For example, in the common "proof" that 2=1, you can reject the step where both sides are divided by (x-y) because that only holds for nonzero numbers and since x=y, x-y=0.

 

[sandwiches]

I'm not a mathematician but this is what finally convinced me:

1/3 = .333...

1/3 + 1/3 + 1/3 = 1

.333... + .333... + .333... = .999... = 1

 

[sandwiches]

Another approach is to come at it this way:
.9999... and 1 are both real numbers
a property of real numbers is that between any two real numbers there exists another real number
There is no real number that falls between .9999... and 1
Hence, .9999... and 1 must be the same real number.

Here is a proof again in standard form with justifications at each step:
.9999...=a Variables can be set to represent any real number (and at least some non real numbers)
9.999...=10a S You can do the same thing to both sides. in this case multiply by 10
9.999...-a = 10a-a subtract a from both sides.
9=9a substitution property
1=a divide both sides by 9
1=.9999... substitution property.

proofs aren't about what sounds or feels right. They are about what can be proven. In order to reject a proof, you must reject some step of the proof. For example, in the common "proof" that 2=1, you can reject the step where both sides are divided by (x-y) because that only holds for nonzero numbers and since x=y, x-y=0.

He's already been shown that proof several times. He still hasn't gotten it.

 

[SithDoughnut]

When multiplying by 10, we assume we end with the same repeating sequence with which we started. When we multiply by 9, that doesn't happen.

0.9 * 9 = 8.1
0.99 * 9 = 8.91
0.999 * 9 = 8.991
0.9999 * 9 = 8.9991
...

There's your problem; you're trying to count to infinity. Skip to the end, where you multiply 0.9 recurring by 9, and tell me where the repeating sequence ends.

Here's another way of looking at it. Every individual real number is separate from other real numbers. As the number line is continuous, each real number therefore has other real numbers between it (for example, between 4 and 5 is 4.1, between 4 and 4.1 is 4.01, between 4 and 4.01 is 4.001, and so on). If 0.9 recurring and 1 are separate real numbers with different values, what is a real number between 0.9 recurring and 1?

 

[Eudaimonist]

One of the most fascinating things is addition: that 1+1=2, for instance.

Why? It's interesting from a meta-mathematical perspective, a philosophy of math.


eudaimonia,

Mark