## A very interesting Mathematical Paradox

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RonWessels
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### Re: A very interesting Mathematical Paradox

The problem is that the deniers don't recognize that 0.999... is just another way of writing [imath]\displaystyle\sum\limits_{n=1}^\infty \frac{9}{10^n}[/imath], probably because the "..." convention lacks the formalism of the infinite summation notation. So they end up creating nonsensical (from a notation semantics point of view) expressions like "0.000...1".

The bottom line is that 0.999... = 1 in the same way that 0.5 + 0.5 = 1. The first is just a longer expression requiring the understanding of limits to evaluate.

skullturf
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### Re: A very interesting Mathematical Paradox

charonme wrote:Real numbers don't have predecessors nor successors.

This is key.

The 0.999...=1 deniers sometimes describe 0.999... as the number immediately next to 1, so it's infinitesimally close but not equal.

The intuition that infinitesimals should exist is not, in and of itself, such a bad thing. That intuition, of course, plays a huge role in calculus.

But one intuition that the deniers are defiling is the intuition that the real line is "homogeneous", that every point is geometrically the "same". (The point corresponding to 1 isn't geometrically special; we can arbitrarily choose two distinct points and label them 0 and 1.)

If some decimal expansion represents a number that's infinitesimally close to 1 but not quite equal to 1 (the "previous" point, as it were), then because of "homogeneity", other points on the line should also have a nearby number that's "immediately" previous or "infinitesimally" close.

What's the point "immediately to the left" of 1/7? What's the point "immediately to the left" of pi? What do their decimal expansions look like?

Gwydion
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### Re: A very interesting Mathematical Paradox

skullturf wrote:What's the point "immediately to the left" of pi?

The point immediately to the left of pi is 2.99999... but then I'm from the Midwest.

Randomizer
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### Re: A very interesting Mathematical Paradox

I can prove that 2 =/= 2: I have 1 apple. I acquire 1 more. I now have 2 apples. IE, 1+1 = 2. I have 1 orange. I acquire 1 more. I now have 2 oranges. IE, 1+1 = 2. But apples (of which I have 2) and oranges (of which I have 2) are not the same. Therefore, 2 =/= 2 because then you're comparing apples to oranges.

Does 3 + 1 = 4? Not if we're talking about 3 mph, 1 meter, and 4 volts of electricity. Does 1 = 3? Yes, if you mean 1 yard = 3 feet. Which is why in science class you're always told to check your units.

But onto the .999... problem!

People get different answers because they're interpreting the question differently. Yes, the limit of .99... = 1. I'm not going to talk about the limit, though.

First, I'll say that whether an infinitesimal is > or = 0 is irrelevant to my conjecture that .999... < 1.

Let's say that 0 is a point (length = 0), and 1 is the line segment [0,1] (length = 1). Next, 0 =/= Ø, since Ø doesn't exist at all, but 0 does. If you take 1 and subtract Ø, you get [0,1] because you didn't actually subtract anything. If you take 1 and subtract 0 from the end, you get [0,1), which has fewer points than [0,1]. Therefore, 1 - 0 < 1. If 1 - .99... = 0, then 1 - 0 = .99... Since 1 - 0 < 1, So too must it be that .99... < 1.

But what about 1 - 1 then? Doesn't that equal 0? Eg, 1 - 1 = 0, therefore 1 - 0 = 1? Nope. 1 - 1 does not equal 0 (eg the single point [0]), 1 - 1 = nothing, ie Ø. There is a distinction between not existing, and a single point. 1 - Ø = 1 and 1 - 0 < 1.
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skullturf
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### Re: A very interesting Mathematical Paradox

Randomizer wrote:Let's say that 0 is a point (length = 0), and 1 is the line segment [0,1] (length = 1).

They're not, though.

Those subsets of the real line have some properties that are analogous to the integers 0 and 1, but not all the same properties. You can't "prove" unorthodox results about addition and subtraction of integers (such as 1 - 0 < 1) by playing around with those sets. At best you're just exploring an analogy that works in some ways and doesn't work in others.

douglasm
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### Re: A very interesting Mathematical Paradox

Randomizer wrote:Let's say that 0 is a point (length = 0), and 1 is the line segment [0,1] (length = 1). Next, 0 =/= Ø, since Ø doesn't exist at all, but 0 does. If you take 1 and subtract Ø, you get [0,1] because you didn't actually subtract anything. If you take 1 and subtract 0 from the end, you get [0,1), which has fewer points than [0,1]. Therefore, 1 - 0 < 1

Congratulations, you have demonstrated that your proposed definition for 0 does not match the one generally accepted in mathematics - namely, the additive identity. If the system you have come up with does not satisfy the requirement that adding or subtracting 0 has no effect, then it does not match the real numbers well enough for reasoning with it to be generally applicable to the reals.

charonme
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### Re: A very interesting Mathematical Paradox

Randomizer wrote:the limit of .99... = 1.
Limit: it's not what you think. If there was a meaningful way of speaking about "limit of .99...", it would be probably this: [imath]\forall x: \lim_{y\to x} 0.9... = 0.9...[/imath] because any limit of any constant exactly equals that constant. The constant 0.9... is not special in any way and we don't need to "put a limit around it" to "compute it's value". 0.9... is the value, which is just another way of writing 1.

But if you need limits this one would be more relevant to this topic: [imath]\lim_{n\to\infty}\displaystyle\sum\limits_{k=1}^n \frac{9}{10^k}[/imath]

Randomizer
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### Re: A very interesting Mathematical Paradox

Yeah, that's what I meant when I said the limit of .99...

douglasm wrote:Congratulations, you have demonstrated that your proposed definition for 0 does not match the one generally accepted in mathematics - namely, the additive identity. If the system you have come up with does not satisfy the requirement that adding or subtracting 0 has no effect, then it does not match the real numbers well enough for reasoning with it to be generally applicable to the reals.

Well, in my system, there's lengths and there's elements, and there's 0 and there's Ø. The null set Ø has 0 elements and does not have a length. [0], a single point, has a length of 0 and 1 element in the set. If you subtract Ø from 1 you still get 1. Just use my Ø where you would normally use 0 in the real numbers system and everything should match up.

I was actually expecting to get challenged on this part: If 1 - .99... = 0, then 1 - 0 = .99... Since 1 - 0 < 1, So too must it be that .99... < 1. with the question of whether I should be subtracting 0 or Ø.

But! I just realized something. If something with length 1 = [0,1], and something with length 2 = [0,2], then 1+1 > 2. Because, if we add [0,1] to [1,2], we have -two- points at 1, so we have to move the second point to the end, which means it ends at 2+0 giving [0,2+0], which is > 2. (I always thought measuring from 0 to 1 on a ruler to get a length of 1 seemed a little bit too long...)

So, a length of 1 must = the length of [0,1) so that adding 1+1 = 2. Eg. [0,1) + [1,2) = [0,2)

So, the only question remaining is, does .99... = [0,1] - 0, ie [0,1)? Um... I think so. So .99... = 1? Uh... yes? Dang this is confusing. My system was supposed to prove that they weren't equal. >_>

Wait, ok, I think I got it. .99... = 1 = [0,1), but .99... < 1+0, as 1+0 = [0,1]. So, my perception of .99... as being [0,1) was not incorrect, but my perception of 1 being [0,1] was.
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jestingrabbit
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### Re: A very interesting Mathematical Paradox

Randomizer, I think your contribution to this discussion is completely inane. Perhaps not as inane as the conversation in general, but very nearly.

Randomizer wrote:1 is the line segment [0,1]

You have two ones there, and you want to define the one of the left using the one on the right. If you want to talk about mathematics, you're generally going to have to use the one on the right, because no one is going to want to talk back using whatever weird system you invent.

Yes, there are ways to create systems where 1 and 0.999... are different things, but so what? In the real numbers, the ones that most mathematicians and physicists and economists and etc use, 1 and 0.999... are the same thing. Defining your way around the problem is as bad as the people who invent a means of communication to solve the blue eyes problem. It might be superficially smart, but its not genuinely intelligent imo, because it dodges the question instead of dealing with it.
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snowyowl
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### Re: A very interesting Mathematical Paradox

I still think the best argument is to ask an equality denier to draw a line 0.999... meters long and a line 1 meter long and then explain the difference.
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skullturf
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### Re: A very interesting Mathematical Paradox

I suppose one could also create a system where we're simply not allowed to have decimal expansions with infinitely many digits. Some deniers of 0.999... = 1 are also not quite happy with 0.333... = 1/3 or 0.142857142857... = 1/7.

This would, of course, be a very weak system. Most rational numbers wouldn't be representable. But one could probably adopt such a system and at least be consistent.

It's common elsewhere in mathematics to consider something as a single entity when it also, in some sense or another, consists of many parts. The single set {0,1,2,3,...} has infinitely many elements, and the single interval (0,1) has infinitely many points.

Really, I don't think the entity 0.999... or 0.333... is any more fishy than the entity {0,1,2,3,...}.

As others have said, if the expression "0.999..." is going to denote any mathematical object, it should denote the number 1.

Randomizer
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### Re: A very interesting Mathematical Paradox

I had another thought. I was watching a video about proving .99... = 1 and one of the statements was, "If .99... < 1 then there must be a number between them." and I paused the video because I thought, well, hell, that's easy to prove, and I don't even have to invent any new number systems for it. :p

First, take .99... in decimal. Next, convert it to vigesimal (that's the correct term for a number system with 20 digits, right?). Vigesimal does not loose any information vs. decimal, as in fact we are doubling the precision of our digits, so this should not be a problem.

Vigesimal uses 0-9 and A-J. 9/10 decimal = 18/20 decimal, so .9 decimal = .I vigesimal, and .999... decimal = .III... vigesimal. And there is at least one vigesimal point between .III... and 1; .J1 (actually, there's a lot more, but one will do for our demonstration).

.III... < .J1
.J1 < 1
.III... < .J1 < 1
.III... < 1

Being as .III... vigesimal = .999... decimal, therefore:
.999... decimal < 1

Unless someone wants to argue that .III... = .J1 ?

snowyowl wrote:I still think the best argument is to ask an equality denier to draw a line 0.999... meters long and a line 1 meter long and then explain the difference.

Yeah, I tried that and saw a gap. Well, not a real meter, a hypothetical one with infinite zoom. :p Didn't feel like explaining how I got that though and figured I'd tackle the problem a different way.
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phlip
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### Re: A very interesting Mathematical Paradox

0.999...10 != 0.III...20

Even with a shorter string, not recurring, 0.9910 != 0.II20

0.9910 = 9/10 + 9/100
0.II20 = 18/20 + 18/400 = 9/10 + 9/200

If you actually do it properly, 0.999...10 = 0.JJJ...20 = 1.0anything

For the record, 0.9910 = 0.JG20, not that it matters at all.

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### Re: A very interesting Mathematical Paradox

Oops.
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charonme
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### Re: A very interesting Mathematical Paradox

haha that's a good one

btw even if you created a system where you could find a number between 0.9... and 1, you still need to convert it back to decimal, otherwise you'll fool yourself the same way again

Randomizer wrote:Yeah, that's what I meant when I said the limit of .99...
But 0.9... is the value of that limit of the sequence of finite partial sums. You don't need to further put it inside some other limit. That would be like talking about "the limit of 2" instead of "2" just because there is some sequence that converges to 2.

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### Re: A very interesting Mathematical Paradox

phlip wrote:For the record, 0.9910 = 0.JG20, not that it matters at all.

I love how you edited your post to add something that matters not at all, but was exactly the thing I was idly wondering about (but too lazy to compute) when I read it originally.

arcane0wl
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### Re: A very interesting Mathematical Paradox

Both the 0.999... = 1 and the (1/9)*9 = 1 problems both suffer from rounding error because of the ill-defined limit at infinity. So what happens at infinity?

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x = 0.999....9999910x = 9.999....9999010x = 9.999....99990- x = 0.999....99999----------------------------9x = 8.999....99991x = 0.999....99999

Qaanol
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### Re: A very interesting Mathematical Paradox

arcane0wl wrote:Both the 0.999... = 1 and the (1/9)*9 = 1 problems both suffer from rounding error because of the ill-defined limit at infinity. So what happens at infinity?

Code: Select all

x = 0.999....9999910x = 9.999....9999010x = 9.999....99990- x = 0.999....99999----------------------------9x = 8.999....99991x = 0.999....99999

tw;dr
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xkcdfan
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### Re: A very interesting Mathematical Paradox

arcane0wl wrote:Both the 0.999... = 1 and the (1/9)*9 = 1 problems both suffer from rounding error because of the ill-defined limit at infinity. So what happens at infinity?

Code: Select all

x = 0.999....9999910x = 9.999....9999010x = 9.999....99990- x = 0.999....99999----------------------------9x = 8.999....99991x = 0.999....99999

Nope.

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### Re: A very interesting Mathematical Paradox

arcane0wl wrote:Both the 0.999... = 1 and the (1/9)*9 = 1 problems both suffer from rounding error because of the ill-defined limit at infinity.

There's no rounding, and nothing about the limit at infinity is ill-defined. Do you know what a limit is? Have you had any calculus?
So what happens at infinity?

Not this:

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x = 0.999....9999910x = 9.999....9999010x = 9.999....99990- x = 0.999....99999----------------------------9x = 8.999....99991x = 0.999....99999

That working presupposes a terminal 9 in the decimal expansion of x, and there is no terminal 9 in the number .999....

The best explanation for why .999...=1 (and really, the only satisfying one) involves limits. Essentially, the number .999... is by definition the limit of the sequence .9, .99, .999, .9999, etc., and one can show that the limit of this sequence is 1. Therefore, .999...=1.
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### Re: A very interesting Mathematical Paradox

Qaanol wrote:tw;dr

Loved this, by the way.
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### Re: A very interesting Mathematical Paradox

So basically, you have an infinite row of 9s, like, a row that never ends, and at the end of that, there is a 0. No, that's just silly.

Magicaltrevor81
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### Re: A very interesting Mathematical Paradox

I prefer this approach

1/11 = 0.0909090909.....
10/11= 0.9090909090....

11/11 = 0.999999999.... (0.9 recurring)

but

11/11 = 1

so

1 = 0.9 recurring

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### Re: A very interesting Mathematical Paradox

Randomizer wrote:you get [0,1), which has fewer points than [0,1].

Wait, this is also no good. Both of these sets contain an uncountably infinite number of points. You ought to take a class on set theory; it'd be good for you. Specifically, order([0,1)) = order([0,1]) even though [0,1) is a strict subset of [0,1]. This is why you can't do math on infinite sets via intuition. For finite sets, such a relationship cannot exist, but for infinite sets it can.

FWIW, the order of both [0,1) and [0,1] is larger than the number of integers. Now are you convinced that intuition doesn't work?

CyeKat
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### Re: A very interesting Mathematical Paradox

I can accept this. Now what I'm wondering is does:

0.8999... = .9

?

CyeKat
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### Re: A very interesting Mathematical Paradox

NVM...

I just posted a query as asking if .8999... = .9

I should have actually tried to work it out. It's correct.

8/9 = .888...
1/9 = .111...
1/90 = .0111...
.888... + .0111... = 0.8999...
8/9 + 1/90 = 81/90
81/90 = 9/10 = .9
.8999... = .9

Sorry!

jestingrabbit
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### Re: A very interesting Mathematical Paradox

That's right. Every number with a finite decimal representation, apart from 0, has a second representation ending with an infinite sequence of 9s.
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CyeKat
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### Re: A very interesting Mathematical Paradox

I like this, it kind of twists the mind.

So since 0.999... = 1, and intuitively 1 - 0.999... = 0.000...1, then does 1.000...1 = 1? Can't think of a way to test that.

CyeKat
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### Re: A very interesting Mathematical Paradox

I think I have it...not sure though:

1/E = 0.000...1
1+(1/E) = 1.000...1
1+(1/E) = (E+1)/E
1.000...1 * [E/(E+1)] = [(E+1)/E] * [E/(E+1)] = 1

Yes?

So 1 exists in at least three phases.

And this would mean 1.000...1 = 0.999... ?

If so...so weird.

But this CAN'T be right. I think all I did was multiply it by its reciprocal.

RonWessels
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### Re: A very interesting Mathematical Paradox

CyeKat wrote:So since 0.999... = 1, and intuitively 1 - 0.999... = 0.000...1, then does 1.000...1 = 1? Can't think of a way to test that.

0.999... is a short form for [imath]\sum\limits_{n=1}^\infty \frac{9}{10^n}[/imath]

What does 0.000...1 mean?

Qaanol
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### Re: A very interesting Mathematical Paradox

“Keep having 0’s forever, and they never end. There is absolutely no end to the 0’s. Then, after the 0’s end, which they don’t, add a 1.” No, that’s silly.
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### Re: A very interesting Mathematical Paradox

Yeah, it's important to notice that the two expressions

0.999... and 0.000...1

may perhaps superficially appear to be somewhat similar, but they're quite different.

In the first, you're just saying that every position has a 9. But in the second, you're somehow trying to say that there are zeroes in infinitely many positions, and then after you're "all done", you write down a 1. That's a very different thing, and a lot weirder (and harder to define) than merely having an infinite sequence of digits.

ibgdude
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### Re: A very interesting Mathematical Paradox

I think the only obvious definition for .000...1 is the limit as x goes to infinity of (1/10)x
This limit is just as obviously equal to zero, with an added bonus:
Since a number consisting of a decimal point followed by x 9's is equal to 1-(1/10)x, we can define .999... as the limit as x goes to infinity of 1-(1/10)x, and so .000...1+.999... = 1, canceling out the limits, but .999...=1 and .000...1=0 as well. The fundamental confusion here is the meaning of a limit. For all those confused to ponder, here is the definition of a limit:
The limit of f as x approaches infinity is L if and only if for all ε > 0 there exists S > 0 such that | f(x) − L | < ε whenever x > S.
Using the above limit definition for .999..., we can see that choosing L=1 and S=-log(ε)+1 satisfies the limit definition, and so the limit is 1.
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### Re: A very interesting Mathematical Paradox

Is there any hope for someone who believes the only 'real' numbers are those which one could physically work with (i.e. fractions and terminating decimals)?

I had another go of convincing my dad that .9 repeating did equal one and did a fair amount of digging and example giving, and it turns out he believes repeating numbers aren't even numbers (so 1/3 [which is a number] is only approximately equal to .3 repeating, because .3 repeating is a process and infinity isn't physical). Apparently pi isn't a number either since it has an infinite amount of digits, we can only ever approximate it.

I left him to chew on zeno's paradox (which he didn't view as a paradox because he knows where he'll be after any finite amount of time), and also challenged him to tell me how long it would take for someone who travels 1 meter in the first second, 1/2 meter the second second, 1/4 meter the third second, etc. to travel at least 5 meters, and how that time compares to the time required to travel 10 meters, since he believes the simple fact that know someone is moving at every finite time means they'll be able to travel any distance given enough time. (I of course know he won't be getting past 2.)

Limits are pretty fundamental to this whole thing, but limits involve infinity, and you can't measure an infinity of something, so progress is difficult. Is there a good argument for a layperson for why plugging the 'holes' in the real line is important? (Although I speak analysis with epsilons and such, that would be completely greek to him ( ) so such an approach isn't really viable.)

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### Re: A very interesting Mathematical Paradox

Well, 1/3 is repeating in base 10, but it's not in base 3. It's pretty hard for me to reconcile a point of view where 1/2 and 1/5 are numbers but 1/3 isn't.

What about sqrt(2)? I mean, if you can draw 1 I can draw sqrt(2) by just making a square of side length 1 and taking the diagonal.
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### Re: A very interesting Mathematical Paradox

Differing bases is a good point that I might possibly bring up, although I might worry about his ability to convert between them. Alternatively, 'real' numbers in one base may not be 'real' numbers in another base, and that might not be a problem in his mind.

His views on sqrt(2) might be interesting to get too, particularly since it's so easy to construct without cheating (e.g. trying to get pi from a circle, when physical circles aren't likely to be perfect mathematical circles). Explaining that it's definitely irrational (and thus involves infinity in some sense) might be tricky if I'm limited to words and can't just show him on paper, but I'll cross that bridge when/if it comes up.

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### Re: A very interesting Mathematical Paradox

It's entirely likely that when you try to bring more rigour to it, he'll come down on the side of "1/3 is a real number, but it can't be represented in decimal", which is a position that has come up in threads like this a couple of times. And really, I don't know of a satisfying answer to that one, short of "In mathematics, we have defined 'real numbers' and 'decimal expansions' in such a way that 0.333... recurring is a number, not a process, and is exactly equal to 1/3... and by the same definitions, 0.999... recurring is exactly equal to 1". Which isn't really an argument, just an assertion. Whereas the actual argument is complicated, since those definitions go back to infinite sums, and limits, and all the Archimedean property, and all those fun things.

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### Re: A very interesting Mathematical Paradox

As long as your dad isn't in some kind of field where it matters, I don't think there's any sort of point to arguing with him anymore. If his only reasoning is that "it doesn't feel like a number to me" but how he feels about what is a number doesn't matter... then simply telling him that according to the mathematics community, the definition of "number" includes those, might be all you can do.

Once I caught my grandfather trying to follow normal baking instructions at a higher altitude by reasoning that he kept his house warm enough and high altitude instructions were because it tends to be colder. I tried to point out that it has to do with air pressure which changes the boiling point of water, but all I could show him as proof was the formula for the change in boiling point of any liquid. He asserted that that water is not a liquid, because it's water (and liquids are apparently something else?). I tried to explain that according to physics and chemistry, liquid water follows pretty much the same rules as any other liquid, but he wouldn't have it. But seeing as how he's not a chemist, I just left it alone.

skullturf
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### Re: A very interesting Mathematical Paradox

When talking about "point nine repeating", I too have encountered the attitude that 1/3 isn't "really" equal to 0.333..., I guess because the person is imagining 0.333... as a kind of "process" that "gets closer" to 1/3 but "never gets there".

To a certain extent, one answer to the conundrum is as follows. We have two options: either we allow ourselves to treat an infinitely long decimal expansion as a single "thing", or we don't.

Mathematicians have chosen to allow infinitely long decimals, and to allow ourselves to treat something like 0.333... as a single object, which we "already have all of", so to speak.

It may seem weird to some people to treat an infinitely long thing as a single object, but it's perhaps no weirder than considering the infinite set N = {0,1,2,3,...} as a single set and giving it a name.

If we *do* allow infinitely long decimals, then the infinitely long decimal 0.333... is *exactly* equal to 1/3. If we *don't* allow infinitely long decimals, then 0.333... would, I guess, be an undefined expression that doesn't mean anything.

If we forbade infinite strings of digits, then the decimal system would be poorer. We wouldn't be able to write 1/3 or 1/7 in the decimal system, but we would be able to write 1/2 and 1/5. Since we believe that 1/3 is just as mathematically legitimate as 1/5, we've chosen to enrich the decimal system by allowing infinite strings of digits after the decimal point.

eulerIV
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Joined: Sun Nov 18, 2012 1:49 pm UTC

### Re: A very interesting Mathematical Paradox

Why is this 8 pages long, it was settled already in the second post.