A Mathematical Paradox?

Originally posted by FabianFnas
You say "infinitely close to 1[/i]" and not exact 1, right?

As this is mathematics, then this would be provable. Please give us the proof, along with the definitions needed for the proof to be stringent.

In fact you can't name an epsilon for the difference between the sum and 1 which can't be undercut by going far enough in n.

Originally posted by humy
Slightly confusing question:

Tim is twice the age of what Sue was when Sue was 8 years younger than the age Tim is now.
How old is Tim?

( just an attempt to lighten-up this atmosphere )

This is more a problem of putting down the problem than of mathematics 😉

Originally posted by Soothfast
Since you are now studying geometry and trigonometry, ...
...

What!?

Originally posted by Soothfast
Here's some arithmetic sleight of hand you see in "survey of mathematics" texts: what is the difference between 1 and the nonterminating decimal 0.999999…?

Let N = 0.999999…

Then 10N = 9.999999…

Now, 9N = 10N - N = 9.999999… - 0.999999… = 9.000000…

Hence 9N = 9.

So N = 1.

Therefore 0.999999… = 1.

Tah-dah!

(The argument is somewhat informal, but it is sound.)

Yes, I believe herein lies the problem.

Because 9N isn’t 9. 9N is 9.000000000.......

I believe it is the casual “let’s call 9.00000000000..... just 9, then it makes more sense” attitude which causes these problems. It’s basically the same “cheating” as before: an infinite string of numbers is reduced to being a finite number.

By the way, I realize this might result in the conclusion that 0.999... is 1.000.... which is arguably even worse...

Originally posted by FabianFnas
You say "infinitely close to 1[/i]" and not exact 1, right?

As this is mathematics, then this would be provable. Please give us the proof, along with the definitions needed for the proof to be stringent.

It is a fair question, however I do not have the proof. It should be obvious from my answers that I am not an expert at maths, and therefore lack the proof.

However, as I have stated, I believe the problem with these "solutions" lies in the fact that infinity is treated like a number.

9N isn't 9. It's 9.00000000... I believe this to be an important distinction which is brushed aside so as to reach an answer.

Originally posted by Soothfast
Yes, though it's much more common to formally regard complex numbers as ordered pairs: a+ib = (a,b). A plane is the natural habitat of the complex numbers, as opposed to presenting them as some subspace of a 4-dimensional space. Then the real numbers are simply identified with ordered pairs having second coordinate equal to zero: (a,0) = a. I'm sure in ...[text shortened]... s possible. Looking around, it seems that Arthur Cayley may have conceived of the idea in 1858.

Agreed, matrix representations weren't even mentioned in my formal education (theoretical physics). In the meantime it dawned on me what the paradox is. i is a unit vector in the sense that |i| = 1. Really one would expect |i^i| = 1, but we have i^i = exp(-pi/2) != 1. So I think it does count as paradoxical.

Originally posted by Great King Rat
The problem I have with 1/2 + 1/4 + 1/8 + ... = 1 is that it appears to me that on the left hand side we have infinity whereas on the right hand side that same infinity is actually given a number, namely 1.

You do not have infinity on either side. You have an infinite number of items on the left hand side, but not infinity.
And I agree that summing an infinite number of terms is problematic and that is why mathematicians do not actually do so, instead, we define the sum to be the number that the sequence converges to. We do not actually claim that the sum of an infinite number of terms is the given result.

Originally posted by Great King Rat
It is a fair question, however I do not have the proof. It should be obvious from my answers that I am not an expert at maths, and therefore lack the proof.

However, as I have stated, I believe the problem with these "solutions" lies in the fact that infinity is treated like a number.

9N isn't 9. It's 9.00000000... I believe this to be an important distinction which is brushed aside so as to reach an answer.

Most people have no problem accepting that

1/3 = 0.333333…

Thus 1 = 1/3 + 1/3 + 1/3 = 0.333…+0.333…+0.333… = 0.999…

In mathematics there is nothing "casual" about the idea that 0.999…=1, but the arguments I've given here are accessible to most laymen whilst inflicting only a minor hurt to rigor.

EDIT: So what number do you propose can fit "between" 0.999… and 1?

Originally posted by Soothfast
Most people have no problem accepting that

1/3 = 0.333333…

Thus 1 = 1/3 + 1/3 + 1/3 = 0.333…+0.333…+0.333… = 0.999…

In mathematics there is nothing "casual" about the idea that 0.999…=1, but the arguments I've given here are accessible to most laymen whilst inflicting only a minor hurt to rigor.

EDIT: So what number do you propose can fit "between" 0.999… and 1?

Yes, you first point (1/3 = 0.333...) is a good point that I do not currently have an answer for.

As to your question, I would say that between 0.999... and 1 lies an infinitely small number. I suppose that number would have to be 0.00000...

I understand that I'm fighting a lost battle, because there are many mathematical proofs for 0.999... = 1, and it would be silly to think that I could change that. I'm sure all the arguments that I will give have been given before.

I'll think about it some more.

Originally posted by twhitehead
You do not have infinity on either side. You have an infinite number of items on the left hand side, but not infinity.
And I agree that summing an infinite number of terms is problematic and that is why mathematicians do not actually do so, instead, we define the sum to be the number that the sequence converges to. We do not actually claim that the sum of an infinite number of terms is the given result.

Does the phrase "converges to" is this sentence mean "goes in the direction of"? Or something else?

Originally posted by DeepThought
Agreed, matrix representations weren't even mentioned in my formal education (theoretical physics). In the meantime it dawned on me what the paradox is. i is a unit vector in the sense that |i| = 1. Really one would expect |i^i| = 1, but we have i^i = exp(-pi/2) != 1. So I think it does count as paradoxical.

I haven't encountered matrix representation of complex numbers in my mathematical upbringing either. Anyway I always thought the "really" magical equation was e^(i*pi)=-1.

Originally posted by Great King Rat
Does the phrase "converges to" is this sentence mean "goes in the direction of"? Or something else?

No, it means it gets closer and closer the more terms you take. Its the same concept as used in calculus. In calculus infinities are also carefully avoided.

The post that was quoted here has been removed

Why does Duchess64 apparently not know when to stop embarrassing herself in public? I have told you my qualifications. Believe them or not, call me mathematician or not, I don't really care. I was merely sharing what little knowledge of mathematics I have with those who may be interested, and I am willing to learn from those who are willing to share. You on the other hand seem only interested in stroking your own ego.