Every number we know makes up exactly 0% of numbers

[Farzan]

Not sure if this is the correct forum, but here is a very simple concept I realized about numbers.

Let's take an example number to be 1,564,003

Now let's say the variable "x" represents the quantity of numbers in our number system... so the amount of numbers.

To make it simple, consider only positive integers.

The equation

(x - 1564003) / x

gives the percent of numbers that are NOT 1564003.Since there are an infinite amount of numbers...

lim x-->infinity (x - 1564003) / x

lim x-->infinity = 1

So 100% of numbers are greater than 1564003.

To me, this answer seems a little strange.

Of course, there are numbers that are less than 1564003, for example... 1564002 or 23 or 792.

But to say that there are numbers less than 1564003 is like saying 0.999~ DOES NOT equal 1, when it certainly does!
Edit:

Now what is really strange to me is that this applies to every number ever used and every number that will ever be used. So all the numbers from 1 to 99999^99999999999 are 0% of numbers.

 

[slider142]

See The Frivolous Theorem of Arithmetic
For a more interesting result, see the book "Meta Math! The Quest For Omega" by Gregory Chaitin.

 

[bob1182006]

well if it says 100% of numbers are > 1564003 you could also do that 100% of all numbers are < 1564003.

But I think what you're trying to say is that if you have a set of all numbers real/complex/etc... which has infinite numbers in it. you remove a finite amount of numbers you still have an infinite amount of numbers in that set.

So even if you take every number from 1-99999^99999999 out of that set you haven't decreased the amount of element's in the set.

You could also think about every number between [0,1]. if you remove every number from [0,.5] there's still an infinite amount of numbers you could find between [0.5,1].

 

[mathwonk]

some of us know a lot of numbers. i myself am personal friends with (well almost) all numbers between 40 and 41.

 

[Hurkyl]

well if it says 100% of numbers are > 1564003 you could also do that 100% of all numbers are < 1564003.

Sure you could apply the same idea. But by assumption, we know all 1564003 numbers that are less than 1564003 -- so the end result is that we are missing 0% of the numbers.

 

[arildno]

"Of course, there are numbers that are less than 1564003, for example... 1564002 or 23 or 792.

But to say that there are numbers less than 1564003 is like saying 0.999~ DOES NOT equal 1, when it certainly does!"

Not at all!

In the latter case, we show that the DIFFERENCE between 0.999... and 1 is zero, and hence, that they are representations of the same number.

In the earlier case, you show that the RATIO between the number of positive integers below some specified one and the "number" of positive integers as such is zero.

Difference and ratio is not the same thing.

For any finite x, we can say roughly speaking, $$\frac{x}{\infty}=0$$
But that relationship does NOT imply that x is equal to 0!

 

[HallsofIvy]

It is also relevant to this topic that, in continuous probability distributions, a probability of 0 does NOT mean "impossible" and a probability of 1 does NOT mean "certain".

 

[Hurkyl]

It is also relevant to this topic that, in continuous probability distributions, a probability of 0 does NOT mean "impossible" and a probability of 1 does NOT mean "certain".

To wit, mathematicians have adopted the phrase "almost never" and "almost certain" as the appropriate technical term for these cases.

(i.e. a probability of 99.99999999% is not almost certain, but a probability of 100% is)

 

[HallsofIvy]

I'll have to keep that in mind when I play the lottery!

 

[Chris Hillman]

See my thread on the Cramer-Euler-McLaurin paradox for an instance where "almost every" comes up in a geometric setting (look for my suggestion that readers compare the notion of "an r-tuple of points which stand in general position wrt each other" with the notion of "choosing r points at random"; my challenge was essentially this: "does it make sense to say that almost every r-tuple is in general position"?).

 

[wysard]

It is most certainly not zero...

It IS incredibly and (for an infinite x) incalculably small.

So I believe the correct description is that for any given number and all numbers less than that number and greater zero compared to infinity, ie: 0 < n < infinity the percentage is finite, but small.

Note: I made the equation 0 < n < infinity on purpose. If as the original post implies that we choose and N and all numbers less than N without a lower limit like zero or some other point we include all numbers below N including those that approach negative infinity. At which point the percentage of numbers would still not be zero, but 50% plus some finite, but small increment. :)

 

[Hurkyl]

It is most certainly not zero...

For a given natural number N, the asymptotic proportion of natural numbers that are less than N most certainly is zero.

 

[Crosson]

See my thread on the Cramer-Euler-McLaurin paradox for an instance where "almost every" comes up in a geometric setting

I recommend "The Strange Logic of Random Graphs" by Joel Spencer. Using logic to prove things like "Almost all graphs contain a triangle" and (the punchline) "if P is a property which can be expressed in first order quantifier logic, then almost all graphs are P or almost no graphs are P".

 

[wysard]

For a given natural number N, the asymptotic proportion of natural numbers that are less than N most certainly is zero.

Then you didn't use enough decimal places. A rounding error to zero isn't the same thing as zero. It's just "close enough that it makes no practical difference". A position which I would certainly uphold, but not to be confused with "is actually the value". Besides as I pointed out, the original post didn't specify natural numbers. Could be integers, could be reals or something other than naturals any of which would produce a result of 50%.

 

[Hurkyl]

Then you didn't use enough decimal places. A rounding error to zero isn't the same thing as zero.

I mean what I said. The question is "what proprtion of the entire set of natural numbers is less than 1,564,003?", not "what proportion of a very large, but finite, sets of natural numbers is less than 1,564,003?". Are you sure you know precisely what that means? It is not a trivial matter to generalize the notion of "proportion" to the infinite.

 

[wysard]

I mean what I said. The question is "what proprtion of the entire set of natural numbers is less than 1,564,003?", not "what proportion of a very large, but finite, sets of natural numbers is less than 1,564,003?". Are you sure you know precisely what that means? It is not a trivial matter to generalize the notion of "proportion" to the infinite.

Yup. Know what proportional means. Even Googled it just to be sure...lol

Reread the original post, and sure enough it is about natural numbers, so the number is close to zero.

As you pointed out, the asymtotic curve approaches zero as the number approaches infinity. No question. But there is an enormous difference between "approaches zero" and "is zero". Now, that being said I certainly know that for all intents and purposes there is no engineering reason to make the distinction at some arbitrary number of significant digits, but the difference is not semantic, nor philisophical, but one of precision. Let me clarify. By definition the asymptotic curve never cuts the axis, in this case infinity. The puzzle is like a mathematicians version of Zeno's Paradox.

Just out of curiosity you said it is not a trivial matter to generalize the notion of "proportion to the infinite", and yet that is precisely what happens when you extrapolate an asymtotic curve to the point it, by definition, can never reach. I suspect that means that you were just using the wrong equation.

Put it this way, can you give me an equation that shows that some arbitrarily large non-zero number that is not infinity, divided by some other arbitrarily smaller non-zero number equals zero instead of "approaches zero"?

 

[CRGreathouse]

Put it this way, can you give me an equation that shows that some arbitrarily large non-zero number that is not infinity, divided by some other arbitrarily smaller non-zero number equals zero instead of "approaches zero"?

But we *are* talking about an infinite number.

 

[Hurkyl]

But there is an enormous difference between "approaches zero" and "is zero".

Yes. Similarly, there is an enormous difference between "a sequence" and "the limit of a sequence" -- this difference is the one you seem to be missing. The usual measure of proportion is defined to be

$$m(S) := \lim_{n \rightarrow +\infty} \frac{\# (S \cap \mathbb{N}_n)}{n}$$

where $\mathbb{N}_n$ is the set of natural numbers less than n. And even when the individual terms are all positive, that does not imply the limit is positive.

 

[Hurkyl]

Put it this way, can you give me an equation that shows that some arbitrarily large non-zero number that is not infinity, divided by some other arbitrarily smaller non-zero number equals zero instead of "approaches zero"?

I assume you meant "larger" not "smaller"?

Anyways, such a quotient cannot be zero.

More importantly, it doesn't even make sense to ask if it "approaches zero" -- the ratio of two numbers is merely a number.

 

[robert Ihnot]

In some cases in Number Theory the term density is used. Thus while the multiples of 7 are infinite, their density is 1/7. Of course, in the case under discussion, the density is 0.

 

[arildno]

1. In the real number system, "infinity" is not a number.*

2. From this, it follows that it is meaningless to talk about the ratio of a number and infinity, since ratios can only be formed between numbers.

3. What we CAN do is to establish that number which is the limit some sequence of ratios approaches when the denominators of the ratios increase without bound.
That number, i.e, the limit, IS zero, and not anything else, since limits can be proven to be unique.


*In the extended reals, infinity IS a number, and the ratio between a non-infinite number and infinity is by definition equal to zero.

 

[wildman]

some of us know a lot of numbers. i myself am personal friends with (well almost) all numbers between 40 and 41.

So you have an infinite number of irrational friends? I knew it!

 

[wysard]

I assume you meant "larger" not "smaller"?

Anyways, such a quotient cannot be zero.

More importantly, it doesn't even make sense to ask if it "approaches zero" -- the ratio of two numbers is merely a number.

You are correct sir, I meant larger. It doesn't make sense to ask? The ratio of two numbers is merely a number. True. Like, say, a percentage?? In any event, such a quotient cannot be zero, so I am correct. Thanks.

I wholeheartedly agree that at some point there is no engineering difference between the infinitely small and the zero state and to quible about it without a specific problem is meaningless. To me this whole thread tenor seems to bring about where people are comfortable with what parts of their working skillsets become innacurate. As aldarino points out, the refuge of the embattled is dogma, not fact or science. Before that gets quoted out of context, in the positive integer set is infinity a member of the set?

 

[arildno]

Can somebody please lock this thread now?

 

[Hurkyl]

To me this whole thread tenor seems to bring about where people are comfortable with what parts of their working skillsets become innacurate.

Then correct that -- instead of belittling everyone you should seek to learn what's going on so that this topic is no longer outside of your "working skillset".

For example, the problem is to compute a limit, as I told you in this post. This is a very easy limit to compute exactly, there is no approximation involved.

If you've gotten it into your head that a limit is an approximation, then you should get that out of your head as soon as possible. While limits are related to the idea of approximation, they are exact expressions. e.g.

$$\lim_{x \rightarrow 0^+} x^2 = 0$$

is an exact statement... and it follows that if x is a sufficiently small positive number, then $x^2$ is approximately zero. Also

$$\lim_{x \rightarrow +\infty} \frac{1564003}{x} = 0$$

is also an exact statement... and it follows that if x is a sufficiently large number (i.e. 'approximately' infinity), then 1564003/x is approximately zero.



Your "working skillset" appears to have an engineering bent; and I appreciate that in engineering situations, you are very frequently (almost exclusively?) working with approximations and estimates.

But we are doing mathematics here -- in mathematics, everything is exact unless explicitly stated otherwise.

in the positive integer set is infinity a member of the set?

No. And on a closely related note, in the real numbers, there is no positive infinitessimal number. (zero is the only infinitessimal)

Can somebody please lock this thread now?

Im a glutton for punishment. I still have a glimmer of hope that wysard is not a lost cause.

 

[CRGreathouse]

If it helps, you can think of the limit in a game-theoretic way.

$$\lim_{x \rightarrow +\infty} \frac{1564003}{x} = 0$$

means that in the following game, the second player has a winning strategy (despite the first player's strategy):

Player 1: Choose a number. (Strategy: make it really, really small, but positive.) Call it $\varepsilon$.
Player 2: Pick a number x in response to player 1's number $\varepsilon$.

Player 2 wins if $1564003/x<\varepsilon$ or if $\varepsilon\le0$ and loses otherwise.


This is a precise characterization: either player 2 can always win, or there is some number $\varepsilon>0$ that player 1 can pick that will make player 2 lose no matter what she does.

Here's a winning strategy for player 2: if $\varepsilon>0$, choose $1000000000/\varepsilon$, otherwise you win regardless of what you pick.

Therefore
$$\lim_{x \rightarrow +\infty} \frac{1564003}{x} = 0$$

If player 1 had a winning strategy, that would have shown that the limit was not 0 (either the limit was something else, or the limit didn't exist).

 

[slider142]

To Wysard,
Note that limits are how we define real numbers. Treating them as approximations shows that you have not had a firm grounding in what real numbers are. Without real numbers, the hypotenuse of a triangle with legs of equal length has no fixed value. Of course, you could just throw in all algebraic numbers in addition to rational numbers, but then you lose the basis of Euler's number and pi, both of which show up everywhere, entangled in the algebra and geometry of the most disparate topics. And that's just a tiny bit of the world that breaks without the precision of limits.

 

[Hurkyl]

To Wysard,
Note that limits are how we define real numbers.

This is one of the inutitive motiviations for the real numbers, but it is not a definition.

Without real numbers, the hypotenuse of a triangle with legs of equal length has no fixed value.

I'm not exactly sure what you meant to say, but what you actually said doesn't really make sense. My best guess is that you meant to say that the notion of length in Euclidean geometry requires one to use the real numbers.

 

[slider142]

This is one of the inutitive motiviations for the real numbers, but it is not a definition.

More precisely, every (non-empty) bounded subset of R has a supremum that is an element of R. If we drop Q into this set, this statement automatically generates R. This is the essence of limit methodology that the OP mistakes for approximation. By this view, all of R is full of approximations. :D

I'm not exactly sure what you meant to say, but what you actually said doesn't really make sense. My best guess is that you meant to say that the notion of length in Euclidean geometry requires one to use the real numbers.

Indeed. More precisely, the length of the hypotenuse... . Where by having no fixed value, I refer to attempting to find the value without limiting procedures. Of course, this makes no sense in the ideal view, where mathematical objects exist independently of the ability to fully understand them.