The Identity Function Paradox

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Whatever happened to taking the limit of the partial sums of the series?

A line, by definition, contains an infinite number of points. Suppose we have line A, and line B which is twice as long as line A. Line A by definition has an infinite number of points. line B also has an infinite number of points by definition. Even if, conceptually, line B, being twice as long as line A, should contain twice as many points as line A, this is not so. Infinity multiplied by two is still infinity.

In fact, if I'm remembering my calculus correctly... we could say that if line A has x points, and line B has 2x points, being twice as long, then the ratio of the number of points in line B to line A would be 2x / x. Taking the limit as the number of points approaches infinity, then according to L'Hopital's Rule, the ratio of points in Line B to Line A is 1, not 2 as one might expect.

??? You might want to think about that a while.

This proof doesn't stand on its own, since it doesn't address whether we should think of infinitely repeating decimals as a consistent, well-defined idea. If we take as a given that two decimals are distinct if and only if they have at least one digit that is different (a very intuitive idea), then this set of manipulations becomes a proof-by-contradiction that infinitely long decimals are inconsistent (or that the 9.999... - 0.9999... step is not justified). It works if we take as a given the definitions mathematicians ordinarily use, but if we don't understand those definitions, this doesn't help at all.

It was implicit, as I think anybody can understand.

I did mention that there was an equation that proved it to be true, I just didn't want to look it up. Thanks for going out of your way guys to make me look stupid

Intuition fails because it’s ill-suited to the properties of infinite sets. For example, they have proper subsets with the same cardinality as themselves, which can‘t happen with finite sets.

one does not need to look it up. it is obvious that that 1 - .9 (repeating) is 0.0 (repeating).

That is totally incomprehensible.

That proof is as easy to understand and simple and intuitive and clearly correct as anything I have ever seen in mathematics.

If you can point to the source of your confusion, perhaps I can explain it better.


The manipulations are simple, but repeating decimals are not intuitive. You were answering someone who considered it perfectly clear that 0.999... could not possibly equal 1, but you didn't bother trying to explain it. Can you not see how unintuitive it is for a number to have two different decimal expansions, neither of which have a single digit in common? Can you not see why someone who considered the impossibility of 0.999.... = 1 to be obvious would reject the manipulations you presented as wrong?

There are "proofs" that 0 = 1 that look obvious and intuitive and simple, and have the same sort of simple algebraic manipulations you were using.

Would you agree that 1/3 = .33333...?

If so, multiply each side by 3. Then 3*1/3 = 3*.33333... or 1=.99999...

If not, then what do you believe the decimal representation of 1/3 to be? (Please justify your answer mathematically.)

note that 1/3 = .3 + .1/3 = .33 + .01/3 = .333 + .001/3 and so on.

If you multiply each of those in turn by 3, you get

3*1/3 = 3*.3 + 3*.1/3 = 3*.33 + 3*.01/3 = 3*.333 + 3*.001/3 and so on, or

1 = .9 + .1 = .99 + .01 = .999 + .001 and so on.



Those aren't proofs. They are, at best, brain teasers of the "spot the bad mathematics" sort. Every one of them contains a flaw -- in all that I remember seeing, they involved a division by 0. That hardly counts as "algebraic manipulations".

Nothing is absolute. You can indeed construct a mathematical structure which gives 0.99999 … and 1 different meanings, but you have to be consistent. It would probably be much messier than the concept of real numbers commonly used by mathematicians.

According to the latter, 0.9999 … is defined as the limit of the sequence (0, 0.9, 0.99, 0.999, … ), which is the same as (1-1, 1-1/10, 1-1/100, 1-1/1000, … ). The axioms defining real numbers entail that there is such a limit, and it’s unique. Any real number less than 1 is a non-zero distance away from 1. As the inverses of powers of 10 approach zero, they will sooner or later be less than that distance, so any real number less than 1 will eventually be overtaken by the sequence. Considering no term of the sequence will ever be greater than 1, either, it follows that 1 is the limit.

The property that two real numbers whose decimal representations differ in at least one digit are different is a consequence of how decimal representations are defined, and it has precisely this one exception: numbers of the form a1 a2 … an . b1 b2 … bp 1 0 0 0 … and a1 a2 … an . b1 b2 … bp 0 9 9 9 … (where a1, a2, b1, b2, etc., are digits, and the zeroes and nines keep repeating forever) are the same. Consider it an artifact of decimal notation.

From the viewpoint of someone who questioned infinitely repeating decimals, the obvious questions are: "Why do you think that 0.333... is a number?" and "What makes you think 1/3 has an exact decimal representation?".

To answer them, you need the concept of limits.


Your proof looks like them, and it has a result that looks like a contradiction to the person you posted the proof in response to. Why should she not respond in the same way you did to the proofs that 1 = 0? Why shouldn't she say, "That's not a proof, at best it's a brain teaser..."?

If you're dealing with someone who thinks the result you're trying to prove is a contradiction, they're either going to think that what you really have is a proof by contradiction of something else, or that there's a flaw in the proof somewhere, whether or not they can see it.

What's to think about? Infinity is, by definition, a number that is too large to quantify. Doubling any such number also results in a number that is too large to quantify.

A point is, by basic geometric definition, something which has no size or has a size that is infinitely small. Therefore, the number of colinear points making up a line segment of any length, even a finite length, is infinite as well. Thus the number of points making up a line segment that is twice as long as another line segment is not double - they are both too large to quantify, and as such, a ratio between them cannot be quantified either. For all practical purposes they are the same number. Any attempt to quantify the number of points to a value other than infinity means that the points must have some finite size - which by definition they do not.

Are you familiar with rational and irrational numbers?

Rational numbers are numbers that may be expressed as a fraction. Such fractional representations may have a limited number of non-zero digits or they may have an infinite number of digits after the decimal point but with an infinitely repeating sequence after some point in the representation.

Irrational numbers are those numbers that cannot be represented as a fraction. They always have an infinite number of digits after the decimal point with no infinitely repeating sequence.

For example, the following are rational numbers: 1, .75, .156769147, .33333... (3 infinitely repeats), and .512379797979... (79 infinitely repeats).

Some examples of irrational numbers are pi, e, and the square root of 2. None of these can be expressed exactly as a fraction.

Perhaps you would be interested in how to determine the fractional representation for any rational number.

For rational numbers that don't repeat forever, it is trivial.

x = .75 = 75/100
x = .9734 = 9734/10000
x = .2 = 2/10

For rational numbers that do repeat forever, let x equal the number and then multiply each side by 10^n where n is the length of the repeating sequence of digits and m is the length of the nonrepeating sequence of digits. Then subtract the smaller from the larger and solve for x. Then multiply the top and bottom by 10^m where m is the number of digits to the right of the decimal point, if any, for the top.

x = .7575757575... (75 repeats forever)
100x = 100*.7575757575... = 75.75757575...
99x = 75
x = 75/99

x = .275757575... (75 repeats forever)
100x = 100*.275757575... = 27.575757575...
100x-x = 27.575757575 - .275757575
99x = 27.3
x = 27.3/99 = (27.3*10)/(99*10)=273/990

It's simple high school mathematics. At least, that's where I learned this.



Limits are not necessary but can be useful for some ways of looking at the problem, for example the way Spiderpig explained it above.



For one thing, in the 1=0 type of problems, there is always a flaw. There is no such flaw in what I presented earlier.



Then they have a problem. At some point, one generally gives up trying to explain it to them.

I was thinking more along the way you used L'Hopital's rule and your limits.