Infinite Of Infinates, whats the solution?

[Harrybarlow]

Ok I am 16 and the other day my mind started wandering until i started thinking about this, there was no inspiration for it and I've not seen anything like it elsewhere, so tell me what you think :), sofar I've not been able to get a solution for it, I've never done one of these before so I am not exactly sure where to start, I am not even sure if I am the first and only one to think this up, I wish, but probably not.

Ok so it should be fairly obvious that there are different kinds of infinite, for example, an infinite in whole numbers from zero to +infinite and from zero to –infinite
There would also, then, be an infinite in decimals from zero to 0.1 and from 0.1 to 0.1001, its easy to think that you can just throw more and more digits into get closer and closer to the, but without quite ever reaching it.

But let's try and think more practically, ill give an example of heat, a calorie is defined as the amount of energy required to raise the temperature of one gram of water by one degree, this does not change weather your heating it from 1ºC to 2 ºC, or from 98 ºC to 99 ºC, therefore, as long as a constant heat is being applied to an object, its raise in temperature is linear

Say you are raising the temperature of an object from 39 ºC to 40 ºC, if you imagine plotting the line on an axis of temperature versus time, the line would pass through 39 º and 40 º, it is safe to assume that the object would pass through 39.1, 39.2, 39.3 and so on

Now here is where the infinite idea comes in, if it passes through all the points from 39-40 it passes through 39.9, 39.99, 39.999, 39.99999999999, if you can keep throwing in more nines into the temperature, where does it cross over the boundary into 40 º?

We know that it is easily possible to heat an object from 39 º to 40 º, but how can it be, where does it end, unless infinite is only a theoretical concept with no practical applications?

Thoughts?

 

[Grogerian]

It's unreasonable to think of infinite, many people adjust that .999_ repeating is simply 1.00 and can be proved by 1. / 3. = .33333333 (.333333 * 3) = .999999 which is essentially 1 by (1/3) * 3 where the 3's cancel

infinite in the world. there is a + and - infinity, clearly in math, but... in the real world very difficult to say, i feel infinity is finite, and can only be expressed in mathematical relations such that we have proven time travel, teleportation, etc. through math.

i hope that answers your .999999 problem. if it ever happens again remember x is element of the integers :D

 

[gmax137]

Hi Harry - I don't think I've heard this explained the way you did (temperature change) but it is very similar to Zeno's Paradox - Google or look on the Math forum here. Basically Zeno said, to walk from here to the door, first you must get to halfway to the door, and then halway from there to the door, and then halfway from *there* to the door... so how can you ever get to the door? Very similar to your question, "how do you ever get to 40 degrees?"

 

[Grogerian]

Hi Harry - I don't think I've heard this explained the way you did (temperature change) but it is very similar to Zeno's Paradox - Google or look on the Math forum here. Basically Zeno said, to walk from here to the door, first you must get to halfway to the door, and then halway from there to the door, and then halfway from *there* to the door... so how can you ever get to the door? Very similar to your question, "how do you ever get to 40 degrees?"

i actually have heard this before and have seen lots of things on it.

you get to the door when you run into it moron. - seriously.

however it is theoretically impossible to reach something completely you may feel the object but i am willing to bet there is at leas 1x10^-99 millimeters of distance between your hand and the wall, or your electrons and the walls. wait quirks, hmmm... your w/e that is 10^-99 small.

 

[Harrybarlow]

It's unreasonable to think of infinite, many people adjust that .999_ repeating is simply 1.00 and can be proved by 1. / 3. = .33333333 (.333333 * 3) = .999999 which is essentially 1 by (1/3) * 3 where the 3's cancel

infinite in the world. there is a + and - infinity, clearly in math, but... in the real world very difficult to say, i feel infinity is finite, and can only be expressed in mathematical relations such that we have proven time travel, teleportation, etc. through math.

i hope that answers your .999999 problem. if it ever happens again remember x is element of the integers :D

Yeah its like pi is infinate, but when you keep adding digits, it gets closer and closer to the exact answer, but never quite reaches

the .99999 was just an example, same thing can be shown with, say 39.1-39.2
passes through 39.11178263 or whatever, there is no limit to it, but it will never get there

 

[Grogerian]

Yeah its like pi is infinate, but when you keep adding digits, it gets closer and closer to the exact answer, but never quite reaches

the .99999 was just an example, same thing can be shown with, say 39.1-39.2
passes through 39.11178263 or whatever, there is no limit to it, but it will never get there

for Pi however it is more complicated, but i myself believe that at one point ie maybe 10 trillion more digits down the road pi will once again start over. and Pi isn't infinite, it is however

$$\pi \approx$$3.1415

it is less than 4, and greater than 3 and thus is contained in the boundary x [3,4]

x$$\in\Re\left[3,4\right]$$
Edit: LaTeX was struggling bad xD
More Edit:
or use limits ^.^

i may be wrong but isn't the definition of infinity something along the lines of - can not be confined in a boundary?

 

[Lojzek]

Now here is where the infinite idea comes in, if it passes through all the points from 39-40 it passes through 39.9, 39.99, 39.999, 39.99999999999, if you can keep throwing in more nines into the temperature, where does it cross over the boundary into 40 º?

The number of steps of adding nines is indeed infinite, but that does not mean that you need an infinite amount of time to perform them. In the experiment you described, the time needed for each step is not a constant. You don't have to stop at each temperature, wait for a while and then continue. The time to needed perform each step decreases exponentially with the number of the step, so the sum of times needed to finish all steps is finite.

 

[Harrybarlow]

The number of steps of adding nines is indeed infinite, but that does not mean that you need an infinite amount of time to perform them. In the experiment you described, the time needed for each step is not a constant. You don't have to stop at each temperature, wait for a while and then continue. The time to needed perform each step decreases exponentially with the number of the step, so the sum of times needed to finish all steps is finite.

i thought of that myself, and it seems like the most probable answer, so what your saying is that it passes through all those points, but in the finite amount of time that it takes to raise the temperature of the object, so it must, then, cross the boundary at some point?

 

[Lojzek]

Correct

 

[Harrybarlow]

i think what you said is right, but then it raises a few more questions
so having already established it crosses it at some point, the questions is, where?
brain-ache :P

 

[Tac-Tics]

Infinite is a word that gets thrown around a lot by people who are interested in math, but for whom it hasn't quite "clicked".

Forget repeating decimals and pi entirely for right now. Let's do some counting. How do we count discrete objects? Answer: one at a time. If there are two puppies, we can point at one, then move our hand and point at another. If there are twenty students, we can go along the rows of the classroom pointing to each one, one, two, three..., all the way up to twenty.

However, what if we had an infinite number of stones to count? What would be different. If we put them all in a line, we could, of course, count them like anything else: one, two, three, ... There's a big difference when we count something that's infinite, though: we will never stop counting. Ever. Even if you lived forever. Even if the world was not destined to end. No matter how many stones you've counted, you've still got infinitely many more to go. You'd never reach the end.

The basic principle behind infinity is a never-ending process.

Let's talk math. If you're in high school, you're probably missing out on a lot of cool mathematics. Most schools teach math only if it's useful for all disciplines later in life. Arithmetic is just as useful for a lawyer as an engineer. Algebra is useful for almost everyone, too. But at the high school level, you will learn hardly anything about set theory, linear algebra, or analysis.

In these higher disciplines, you'll come across two new concepts: theorems, which are statements we have found to be true about numbers, and proofs of those theorems.

One such famous proof is that the square root of two is irrational. If you don't know it already, check it out. It's almost like a poem, albeit, written in a language older than Latin.

http://www.grc.nasa.gov/WWW/K-12/Numbers/Math/Mathematical_Thinking/irrationality_of_2.htm

Proof are the bread and butter of mathematics. They are what make mathematics distinct from physics and the sciences. "Truth" in mathematics is not about showing beyond all reasonable double something is true. For something to be "true" to a mathematician, you have to be absolute. Physical laws may some day be found to be incorrect through experiment. (This happened with Newton's laws... twice!) Mathematical laws are true forever. They are true, even in alternate universes without gravity or inertia!

One property of proofs stands out. They must be of finite length. To prove something is true, you must be able to explain to someone else *why* it is true. Proofs cannot be infinitely long!

So, let's start over with infinity. You might come across a cute equation like this:

$$1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \dots = 2$$

That is, the sum of all fractions of the form $$\frac{1}{2^n}$$ is equal to two.

How can you add an infinite number of fractions together? The answer is surprising to some people: "You can't". If you try, you will never stop. And if you never stop, you can't prove to anyone you are correct.

So a mathematician's job is to take these statements and prove them in a finite amount of time. In this particular case, and many, many like it, what we are interested in is actually something we call a "limit". What is the "limit" as we add more and more of these terms together?

The concept of a limit is probably the most important one to understand problems about infinity. Unfortunately, it's technical definition is a bit hard to understand. It will have to wait for a separate post. But the idea is very easy to see.

A limit is a number to which we can "approximate" to any degree of accuracy. In our infinite sum problem above, we are approximating the number "2" with the partial sums of the sequence $$1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \dots$$. A partial sum is simply taking a finite number of the terms and add them together, instead of all infinity of them. Using "~" to mean "approximately equal", we arrive at a series of statements:
$$1 ~ 2$$
$$1 + \frac{1}{2} ~ 2$$
$$1 + \frac{1}{2} + \frac{1}{4} ~ 2$$
$$1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} ~ 2$$
$$1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} ~ 2$$
etc...

If you use a calculator, you can see that the more terms you add together, the better an approximation you get. The first, 1 ~ 2, is clearly a very poor approximation. But the last is more accurate than you can draw freehand on a small piece of paper.

Again, this doesn't fully explain how to calculate the limit or even show how to verify this, in fact, IS the limit of this sum. But if you want to know the details, we'd be happy to explain them.

Knowing about this limit trick, let's talk about a silly looking paradox that math majors like to play on their less math-oriented friends. Consider this sum:

$$1 - 1 + 1 - 1 + 1 - 1 + ...$$

What is this sequence equal to? Some people say, "clearly, it's 0":

$$1 - 1 + 1 - 1 + 1 - 1 + ... = (1 - 1) + (1 - 1) + (1 - 1) + ... = 0 + 0 + 0 + ... = 0$$

Well, not quite. Other people say "clearly, it's 1":

$$1 + (-1 + 1) + (- 1 + 1) + (- 1 + 1) + ... = 1 + 0 + 0 + 0 + ... = 1$$

What? It can't be equal to two different things! If it is, then 0 = 1!

Well, fear not. Let's use our limit approach on your calculator. What are the partial sums?

$$1 (= 1)$$
$$1 - 1 (= 0)$$
$$1 - 1 + 1 (= 1)$$
$$1 - 1 + 1 - 1 (= 0)$$
$$1 - 1 + 1 - 1 + 1(= 1)$$
$$1 - 1 + 1 - 1 + 1 - 1(= 0)$$
etc.

The partial sums simply alternate between 0 and 1. Do they get closer to any single value? No. Whatever value the limit might be, we can't approximate it very accurately. We'll always be at least 1 off! It doesn't appear to grow closer to any number.

When this happens, we say the series *diverges*. It *has* no limit.

All infinities are handled using special tricks like this. I hope this has been helpful, or at least interesting to read. You'll soon be under way to understanding infinity!

 

[Hurkyl]

The basic principle behind infinity is a never-ending process.

Emphatically no!

One of the major obstacles some people face when understanding calculus is that they fail to comprehend the difference between a 'never-ending process' and the 'limit' of that process. That confusion leads to people making nonsensical statements like "pi doesn't have an exact value because you can just keep adding more digits", or "0.999... is not equal to 1, because it never quite reaches it", or "Achilles cannot finish the race because he has to keep crossing those midpoints".

Furthermore, while using 'never-ending processes' in analysis is certainly one application of infinite objects, it is by no means the only one. And objects called 'infinity' generally have much simpler 'principles' behind them.

For example, the 'principle' behind completing the real line R by adding the two points $+\infty$ and $-\infty$ to form the extended real line is exactly the same as that of completing the open interval (0, 1) by adding to it the two points 0 and 1 to form the closed interval [0, 1].

So, let's start over with infinity. You might come across a cute equation like this:

$$1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \dots = 2$$

That is, the sum of all fractions of the form $$\frac{1}{2^n}$$ is equal to two.

How can you add an infinite number of fractions together? The answer is surprising to some people: "You can't". If you try, you will never stop. And if you never stop, you can't prove to anyone you are correct.

So a mathematician's job is to take these statements and prove them in a finite amount of time.

You're missing a critically important part of the story -- part of the mathematician's job is to figure out what the heck we mean by expressions like
$$1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \dots.$$
The rules of arithmetic tell us that if a and b are numbers, then a+b is a number. This can be repeated: if c and d are also numbers, then ((a+b)+c)+d is a number. The notation can be simplified by adopting conventions like the one that a+b+c+d is just shorthand for ((a+b)+c)+d. However, the rules of arithmetic cannot be applied to state that the infinite expression above sum is at all meaningful, let alone a number!

However, infinite sums can be effectively used to solve problems -- so it's the mathematician's job to come up with an entirely new function that can be used to give meaning to the infinite expression above.

 

[robert Ihnot]

Grogerian: for Pi however it is more complicated, but i myself believe that at one point ie maybe 10 trillion more digits down the road pi will once again start over. and Pi isn't infinite, it is however

If the tail of pi repeated, we would have, for some a, n, $$\sum^{\infty}_{i=1}\frac{a}{(10n)^i}$$ This is just an geometric series resulting in a rational number. Thus pi would be rational.

 

[Lojzek]

i think what you said is right, but then it raises a few more questions
so having already established it crosses it at some point, the questions is, where?
brain-ache :P

This is not a difficult problem. If you heat the water with constant power, then the temperature will increase lineary with time: dT=C*dt, where C is a constant (independent of time). So to cross temperature differences 0.9K+0.09K+0.009K+... you need a time:
0.9K/C+0.09K/C+0.009K/C+...=(0.9K+0.09K+0.009K+...)/C=1K/C

As you see the sum of times needed to cross an infinity of smaller and smaller temperature intervals equals the time needed to cross the sum of all intervals, which is finite (1K).

 

[Harrybarlow]

my friend posted this on another forum, and one of the replies he got was that time is continuous, not discrete, I've come across the term before and i think i sort of get it

but if someone could clarify what he means, that would be great

=D

 

[Almanzo]

Infinite expressions, like 1 + 1/2 + 1/4 + 1/8 + ... , may or may not refer to existing mathemathical objects. If one wants to reason about such an object, one must first ascertain its existence. (For a finite expression, such as 3 * 1/5; the existence of the object is usually "prewired" in the rules of the mathematical system; in this case arithmetic. But not always: 3 *1/0 does not refer to an existing object.)

An example may clarify this.

Suppose that x = 1 + 1/2 + 1/4 + 1/8 + ... is an existing number. Then we can prove that it must be equal to 2, because 2x -2 = x. (After all, 2x = 2 + 2/2 + 2/4 + 2/8 + ... = 2 + 1 + 1/2 + 1/4 + ...) If 2x - 2 = x, then x- 2 must be 0. But only provided that x exists.

Now, suppose that y = 1 + 2 + 4 + 8 + ... is an existing number. Then we can prove that it must be equal to -1, because 2y + 1 = y. (After all, 2y = 2 + 4 + 8 + 16 + ..., so 2y + 1 = 1 + 2 + 4 + 8 + ...) If 2y + 1 = y, then y + 1 must be zero. But only provided that y exists.

In the second case it is clear that y does not exist. If it did, a negative number could be produced by adding positive numbers.

But in the first case, the nonexistence of x is less obvious. In fact, x will exist if we allow the limit of a converging sequence of numbers to exist (i.e. to be a number). That the sequence 1, 3/2, 7/4, 15/8, ... converges can be proved because it is monotonously increasing, and has an upper bound. So, it is reasonable to state that x = 2.

By the way, there is a famous paradox in Set Theory, about sets which include themselves as members. (Such as the set { 1, 2, { 1, 2, { 1, 2, ...}}}, which could be written as X = { 1, 2, X }.) This is the Cantor Paradox, which states that the Set of all Sets which do not include themselves as a member must include itself as a member if it does not, and cannot include itself as a member if it does. In my opinion, the solution of this paradox is that the Set of all non-self-including sets does not exist. It has an infinite expression to which no valid object happens to correspond. Just like y = 1 + 2 + 4 + 8 + ...

 

[Changbai LI]

Ok I am 16 and the other day my mind started wandering until i started thinking about this, there was no inspiration for it and I've not seen anything like it elsewhere, so tell me what you think :), sofar I've not been able to get a solution for it, I've never done one of these before so I am not exactly sure where to start, I am not even sure if I am the first and only one to think this up, I wish, but probably not.

Ok so it should be fairly obvious that there are different kinds of infinite, for example, an infinite in whole numbers from zero to +infinite and from zero to –infinite
There would also, then, be an infinite in decimals from zero to 0.1 and from 0.1 to 0.1001, its easy to think that you can just throw more and more digits into get closer and closer to the, but without quite ever reaching it.

But let's try and think more practically, ill give an example of heat, a calorie is defined as the amount of energy required to raise the temperature of one gram of water by one degree, this does not change weather your heating it from 1ºC to 2 ºC, or from 98 ºC to 99 ºC, therefore, as long as a constant heat is being applied to an object, its raise in temperature is linear

Say you are raising the temperature of an object from 39 ºC to 40 ºC, if you imagine plotting the line on an axis of temperature versus time, the line would pass through 39 º and 40 º, it is safe to assume that the object would pass through 39.1, 39.2, 39.3 and so on

Now here is where the infinite idea comes in, if it passes through all the points from 39-40 it passes through 39.9, 39.99, 39.999, 39.99999999999, if you can keep throwing in more nines into the temperature, where does it cross over the boundary into 40 º?

We know that it is easily possible to heat an object from 39 º to 40 º, but how can it be, where does it end, unless infinite is only a theoretical concept with no practical applications?

Thoughts?

It is a complex problem. If you are interising,there is some paper.but it was write by Chinise. www.lcbmaths.com

 

[CRGreathouse]

If you are interising,there is some paper.but it was write by Chinise.

I noticed that the same page has one English paper, a crackpot piece asserting that $0.999\ldots\neq1.$

 

[csprof2000]

I'd say, simply and without appeal to calculus, that nothing is continuous... space, time, temperature, are discrete, and therefore there is no paradox in walking to the door or in adjusting the knob. You could enumerate the different temperatures, if you were quick enough.

Why not?

 

[matticus]

I'd say, simply and without appeal to calculus, that nothing is continuous... space, time, temperature, are discrete, and therefore there is no paradox in walking to the door or in adjusting the knob. You could enumerate the different temperatures, if you were quick enough.

Why not?

the same reason you cannot enumerate all of the real numbers, no matter how fast you are. there is a one-to-one correspondence between a time interval and an interval of real numbers, and every interval of real numbers is uncountable.

 

[csprof2000]

I think you misunderstood my point, matticus.

The real numbers are not "real" in the general sense. There is not a one-to-one correspondence between mathematics and nature. Math is an approximation... it's a language for describing nature. Mathematics is true in its context. But there's no reason to expect it to apply *exactly* to the real world.

You said:
"there is a one-to-one correspondence between a time interval and an interval of real numbers"

Prove that, and you'll be the most famous mathematician who ever walked the face of the earth.

 

[csprof2000]

Furthermore, just for kicks, tell me about a real number you can't enumerate. Describe it for me. Give me an example... tell me what its value is.

 

[matticus]

i'll tell you what. enumerate them, and i'll show you one you missed :)

 

[csprof2000]

Simple.

Using the decimal expansion of every number, start by filling the ones place with 0-9, then the 10s and 10ths place with 0-9, etc. You'll never reach the end, but you wouldn't reach it if you enumerated the natural numbers either.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9
00.0, 01.0, 02.0, ..., 10.0, 11.0, ... etc.

Perhaps a more disturbing result of my above enumeration scheme is that it produces only a countable infinity of different numbers. (unless I missed a few...)

 

[matticus]

Perhaps a more disturbing result of my above enumeration scheme is that it produces only a countable infinity of different numbers. (unless I missed a few...)

your enumeration scheme produces all the decimals, which is not countable, but has the cardinality aleph-1. To be countable there must be a bijection (1-1 correspondence) between the set and the set of naturals. Cantor's diagonalization process shows that no such bijection exists.

 

[Hurkyl]

your enumeration scheme produces all the decimals, which is not countable, but has the cardinality aleph-1.

While R is indeed uncountable, its cardinality is aleph-1 if and only if you adopt the continuum hypothesis.

 

[csprof2000]

But it is impossible to give an algorithm which produces all real numbers. The simple reason: there are only countably many algorithms, and as such, there are more real numbers than algorithms. There must exist some real numbers for which there is no algorithm which can tell you the digits... some examples of numbers for which there are algorithms to give you the decimal digits:

integers:
Put the integer x into a variable k.
Let n = 0.

Divide k by 10.
Put a zero into the place corresponding to the power 10^-n.
Put the remainder after division into the place corresponding to the power 10^n.
Increase n by 1.
Put the quotient back into k, and repeat (forever, or until k = 0).


rationals:
see Long Division


special irrationals (roots, etc.):

For square roots, for instance...

1.) Let k = 0, delta = 1, steps = 0
2.) if k^2 < n, add delta to k and go to step 1
3.) if k^2 = n, return this value of k
4.) if k^2 > n, subtract one from k
5.) divide delta by 10
6.) If steps > max, return the approximation k
7.) go to step 1, add 1 to steps



The interesting thing to notice is that even if you take the natural numbers, the rationals, and the "special irrationals" (such as roots, logs, exponentials, sines and cosines, special constants, etc.) then you have a countably infinite set. How so?

Well, consider an ordered 5-tuple of rational numbers.

(a, b, c, d, e)

a can be any rational number
b can be any power to raise a to... assuming it's valid for that a
c can be either 0 or 1. if it's zero, then evaluate the 5-tuple as b*exp(a). Otherwise, don't.
d can be either 0 or 1. if it's zero, then evaluate the 5-tuple as b*log(a). Otherwise, don't.
e can be either 0 or 1. if it's zero, then evaluate the 5-tuple as b*sin(a). otherwise, don't.

I could get really, really technical, and make it an ordered 50-tuple, and wipe out about any numbers you could imagine. Do you see the rub?

There are as many rational numbers as there are ordered 50-tuples of rational numbers. So I can't possibly compute (or even "define", if by define you mean give the value of) the majority of the real numbers. Which also means you will never see one or use one (directly).

 

[HallsofIvy]

Simple.

Using the decimal expansion of every number, start by filling the ones place with 0-9, then the 10s and 10ths place with 0-9, etc. You'll never reach the end, but you wouldn't reach it if you enumerated the natural numbers either.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9
00.0, 01.0, 02.0, ..., 10.0, 11.0, ... etc.

What you are doing here will only give positive integers won't it?

Perhaps a more disturbing result of my above enumeration scheme is that it produces only a countable infinity of different numbers. (unless I missed a few...)

If, however, you meant .0, .1, .2, .3, .4, .5, .6, .7, .8, .9, .10, .11, .12, .13. etc. that will give only terminating decimals, a small subset of the rational numbers (specifically those that can be written as fractions with only powers of 2 and 5 in the denominator) and so countable.