MATH questions

Wow, this is a confusing thread indeed. :-)

@Remellion: my apologise for getting you wrong. I was not clear in the fact that I was talking about an infinite number too. I just did not take the time to write it all out. :-)

That infnite number is by definition unequal to 1, whatever number base someone is looking at.

If someone says on the one hand that a number is infinite, yet say that it is equal to the next number in line (in an arbitrairy chosen base number) on the other hand, then is there an implicit change in definition applied. That is the process of rounding up (if that is the correct term in English). 999 grams equals 1 kilo. Just as 998 grams.

If a domain can not have infinitesimals, then can it not have infinite numbers as well, because the difference between two discrete infinite numbers (which sounds quite absurd) is a inifinitesimal.

If 1/3 is not exactly 0.33333 (the infinite number) then is it more or less 0.3333. There is mathematical symbol for it: ≈.

3 * 1/3 should be written as ≈0.9999..., where 3/3 = 1.

If R can not have infinite numbers nor infinitesimals, then must it be a discrete collection of numbers and can the question be answered: what is the smallest unit of measurement for R? I am all ears. :-)

@LoekBergman: Ah, and here lies several problems (it's seriously hard to argue my case with my limited mathematics knowledge and by trying to pick words extremely accurately to avoid causing confusion.) I see that you're actually on the other side of the fence, and I must convert you! :P

Starting with terminology, the number 0.999... is not an "infinite number". It is a number with an infinite decimal representation. (That said, those with finite decimal representations can also be represented by, e.g. 1.00000... you get the idea.)

Now the argument I make is not that 0.999... is being rounded up to 1. Far from it; I am asserting that the two are one and the same; they are equal; they are two different ways of writing the same value.

To try and convince you (this topic is tricky enough, easier to convince one person at a time than throwing out a generic argument for everyone) we will calculate the value of 0.999... (infinite nines.) It is: 0.9 + 0.09 + 0.009+ 0.0009+ ... = 9/10 + 9/100 + 9/1000+ 9/10^4 + ... with the sum having infinite terms. This infinite series converges, and the value of the infinite sum is 0.9/(1 - 1/10)) = exactly 1. This is to say, the exact value of 0.9999... = 1. It does not "tend to", "approximate" or "approach" 1, it is "equal to" 1, since 0.9999... is a precise number with a precise value, not a sequence or series. We calculate its value by writing an infinite sum that can be calculated to have an exact value, which happens to be 1.

Next, we deal with the nature of the real numbers. The real numbers contain no infinitesimals (refer to what waffllemaster posted: if one assumes there exists a "smallest" infinitesimal, legal manipulations allow us to find an even smaller one, contradiction, therefore no infinitesimals exist.) This same argument also answers your question: There is no smallest positive real number in R.

However, R does contain numbers with infinite decimal representations, the most familiar would be 3.141592653589... pi. sqrt(2), phi (the golden ratio) and other irrationals also exist within R. One can also say, for instance, sqrt2 (1.414...) < phi (1.61803...) < pi (3.14159...). One can also add, subtract, take log or other things with them. However, one cannot ask "what is the next biggest irrational number after ___?" because infinitesimals do not exist.

I could try to explain a little more than this, but that might just end up creating more confusion unless there's a specific query to answer. That, and my mathematics knowledge isn't exactly deep or rigorous, but I'm pretty sure it's accurate where it stops.

@Remellion: no, let's agree to disagree, because what you say does not make sense to me. If you think about what has been said in this thread before that 1 + 1/2 + 1/4 + 1/8 + .... ends up equal to 2. That is incorrect, because 2 is the limit value it approaches, but never reaches. The same applies to .99999. 1 is the limit value, it is never the exact value.

1/x is never 0, it has the limit of 0.

About the question of R: that is exactly what I tried to say and why I am all ears when people say that two different values are equal to each other. The way I used infinitesimal is indeed not inline with the mathematical definition, although I understood it that way. An inifinitesimal is for me the smallest unit of measurement possible. The way I use it, is it a paradox, because you can not reach the inifinitesimal. Once you think you have reached it, new depths arise, just like the vain effort to write out an infinite number. Any numerical system that has a definite end is a quantum like set. That has discrete values and that is why I wrote about the discrete collection of R.

We should agree, irrespective of how you formalize infinitesimal, that when there is no infinitesimal, then are all different values unequal.

When you take a look at that standard part of x in *R, then will you read about rounding off values. The standard part of 0.99999... is 1, but so is 1 the standard part of 0.5 and 1.23456 and 1.499999.... and some more values in between.

I'll say my other set piece, although it probably won't do much in the way of changing your convictions, since all this is really tricky to even formalise coherently. Forgive me if the bolding and italics appear rude or forceful; those are simply to highlight the key points to my argument.

The main thrust of the below is that these are mathematical definitions and facts I'm basing it on. The conclusions may not be pretty or comfortable for some people, but they are deductively valid.

The word "limit" also has a strict mathematical definition. The limit of a function (of some variable say x) at a point x = a is the value that the expression approaches as x approaches a. Intuitively this is easy to understand; the key thing to understand is that the limit IS a value, and it is the value that the function approaches. It is not greater or less than the value it approaches; it is exactly equal to it.

Now, a limit, if it exists (there are criteria irrelevant to our current discussion) is completely irrelevant of the function actually being defined at that point. [A famous example is the topologist's sine curve, y = (sin x) / x, where (lim x->0) (sin x)/x = 1, despite it being undefined at that point.]

To use your example, (lim x -> +infinity) 1/x = 0; it is an equality. We can write the statement despite 1/+infinity not actually being defined; the limit of 1/x as x tends to +infinity is well-defined, and it is equal to 0.

So, to take 1 + 1/2 + 1/4 + ..., it is a sum to infinity of a convergent ("its partial sums of n terms approach some finite value as n tends to infinity" if you want a definition) series. Now by definition, the sum to infinity of a convergent series exists, and it is the limit of the partial sums of n terms as n tends to infinity, i.e. (lim n -> +infinity) (sigma k = 0 to n) (1/2^k). (Pardon the lack of mathstype.) Now as previously explained, the limit of the expression is equal to the value it approaches, and in this case, the sum 1 + 1/2 + 1/4 + ... to infinity = 2. Exactly equals, since by definition the limit is the value.

It also follows that the sum of 9/10 + 9/100 + 9/1000 + ... to infinity = 0.9999... = 1, using the same method as the above paragraph. (Details left as an exercise to the reader.)

I understand that this may not be entirely acceptable to some people. It may seem confusing, unacceptable, illogical, fallacious even. However, this result is a natural consequence of the formalisations mathematicians have chosen; I assure you it is true, regardless of whether you yourself like it or not. Whether or not we like it is irrelevant; from a rational point of view, the logic is sound and the conclusion is valid. Coming to terms with it I suppose is up to the individual (and most non-mathematicians don't actually make that step.) What I can do here is merely try to phrase the logic in hopefully more digestible manner, but I cannot force someone else to accept it.

As for your intuitive interpretation of an infinitesimal, as you point out it is not mathematically consistent in R (the set of real numbers.) From what you wrote, I think you can also see why it was so difficult to formalise this concept, as a generalised yet rigorous definition only came centuries after the idea was first conceived (by Archimedes actually, no less.)

When there is no infinitesimal, unfortunately it's not as simple as saying then different representations of values must therefore represent different values. Two representations can represent the same value, and this is again unrelated to infinitesimals.

If you want a discrete (quantised is the word you want?) subset of R, I suggest Z (the set of all integers) and N (the set of natural numbers), as even Q (the set of all rational numbers) is still uncountably infinite (aaargh more maths terms) and therefore still will bring up the problems earlier discussed.

R* is notation for the hyperreals, a topic I am completely unfamiliar with, so is left as extra reading for the interested student (it is a constructed set that extends R to include infinitesimals; there are infinite infinitesimals in that system, and it's honestly quite beyond the casual reader, myself included.)

Standard parts and rounding at no point come into play. It's not part of the argument, and there should be no contention with their definition. It is completely unrelated to the idea of limits, sums to infinity and infinitesimals.

It does not follow from your definition of limit that the sum of 0.9999... equals 1 (I argue it says the opposite, because only a finite process could be made to stop at a specific value). Equality between a function and its limit is never implied in the expression of a limit. The strict definition of a limit (let's say the "epsilon-delta" or "neighbourhood" expression) describes the approach to a limit and no more than that.

A limit is a property of a process which never comes to an end. The classical definition of a limit is that, when one exists, it can be approached within an arbitrary distance; it does not say the limit can be reached.

Eh. Perhaps it's my phrasing that seemed muddled: At no point did I say that the value of a function at a point and its limit at the same point are equal. In fact in the paragraph immediately after the one you quoted I even note that it's possible for a limit to exist where the function is undefined (citing that classic sin x/x.)

Epsilon-delta is a rather unfriendly way of putting it, so I just gave the more intuitive "approach" definition. More rigorously (but still poorly phrased) it would have been "a limit at a point exists if and only if within some distance delta of the point, (implies =>) the expression adopts a value within epsilon of the limit." (And to make it more intuitive, add the completely unnecessary "as delta decreases, epsilon decreases" to show an approach.)

And even though the classical definition of the limit makes no mention of whether it can actually be reached, it is still defined as that value it approaches (irrespective of whether it's reachable.) It is a value; taking the limit gives you a nice concrete value (where the limit can be defined, of course.)

It's important to keep the topic of argument well-defined, so let's try:

Given that (lim[sum{i=1 to inf}of{9*10^(-i)}]) exists and is "1", is it true that the sum and its limit are equal?

Any objections? Corrections? Improvements?

Aha. Yes. They are equal, because the sum to infinity of a convergent series is defined as: the limit to infinity of the sequence of its partial sums. The notation, though:

i.e. sum(i=1 to inf)of(9*10^-i) = lim(i=1 to inf)of[sum(j=1 to i)of{9*10^-j}].

Let's not rush :). Is the problem description correct? It's been a long time since I have had to think about this kind of problem and I'm suspicious of my own recall.

If you're talking about the bolded line you gave, then there's an issue with notation, since both the limit and sum need to have their separate bounds (limit to and sum to) defined. That, and the sum you gave takes on a specific value, so you can't take a limit of it as it's not dependent on any variable.

I think the argument should go 0.9999... = sum(i=1 to inf)of(9*10^-i), and then (a) does the sum exist and (b) if it does, is it = 1? Whereupon the expression I gave in my previous post comes in. (Which features taking the limit of a [sum to a variable].)

Why has nobody adressed i^i yet I think that's intereseting too

http://www.wolframalpha.com/input/?i=i%5Ei

i^i is not quite unexpected if you're familiar with the exponential and natural logarithm function. It follows naturally by plugging exp(i * ln i). ln i gives a single imaginary number (i * pi/2), so when multiplied by i, it gives a real number (-pi/2) and hence e raised to this real number is also a real number. However, ln is multi-valued (ln i = i * 5pi/2, i * 9pi/2, i * -3pi/2, and so on), so this is not the only one.

Now one for math-amateurs : I wonder who gives me the correct answer on the Height H of my ladder against my house...

I got 202.83, but I'm not sure if that's right.

I had to plug a rather big polynomial into my calculator to get the answer after quite a bit of work, but I guess that's what I get for thinking algebraically (there is probably some very easy geometric way to solve the problem, but I'm terrible at geometry, algebra and calculus are more interesting to me).

I have almost 139 as the height and just like shepi13 I do not know if it is correct either.

I got 202.83 as well.

Similar triangles and Pythagoras' theorem.

  h           =              (h-60)      =       height    
 220                √((h-60)² + 60²)      hypotenuse

 Biggest                    Top
Triangle                 Triangle

By the way, you can confirm easily enough by running the results back through Pythagoras' theorem to get the rest of the unknowns.