A question of numbers (one for the maths fans)

Well then what is it?While I'm at it I think part of your problem stems from your using a personal definition of infinity. What is your mathamatical definition of infinity?Btw do you accept that this is a well established proof in mathematics? i.e. recognized and accepted by mathematicians? So why would they do this if it's wrong? I suggest you need to update your math knowledge rather than try to change mathematics to fit your intuitive sense of how it should be.lfen

RR,I had heard that Pythagoras was so upset with irrational numbers that he pledge his students to secrecy but I'm not sure about this story: Yes, the incommensurateness of the rationals and irrationals is a puzzling even counter intuitive finding. Let say you have a line length of 1 and another line length of pi. It seems like you could find a small enough length that would evenly measure both of them. Pi would have three something times as many of those tiny lengths than would the line of 1, but no way. It can't be done. There is no length however tiny that can be used to evenly measure the two lines.Here is a simple proof by contradiction (I believe that is what it's called. It's been quite a while since I've done math) They are called irrational numbers because they can't be expressed as the RATIO of two integers: .3333 is 1/3 but pi is not a repeating decimal. The decimal is only getting closer and closer to pi. It will never equal pi.I still find this a strange thing about everyday life. Incommensurability is a mystery to me. It is one of many hints that reality is not what I think it is.lfen

Rat,Take a look at this wikipedia article:
Zeno's paradoxes - Wikipedia You are having the same problem Zeno had. He proved that Achilles could never catch up to a tortoise because in order to catchup he would have to first go half way. Once there he would have to go half of the next distance and then half of that and he would always be some fraction of the distance behind the tortoise.This brilliant (but ultimately fallacious proof) demonstrates something we know not to be the case. But if I recall correctly the mathematics to handle this were developed much later by Newton and Leibniz..999... is shorthand for an infinit sum .9+.09+.009+ ... and calculus has proved that the value of the sum is 1.It's good to think about these things but naive assumptions can be false. There are some good books on mathematics written in a popular style. Why not read one?lfenABE: Here is another link to a overall math view of this:
http://www.andrews.edu/~calkins/math/webtexts/numb13.htmEdited by lfen, : see ABE above

That is elementary algebra. If you multiply both sides of the equation by the same number the equality still holds that is:
w=y and aw=ay
so x=.999... multiply both sides by 10 you have10x=10*.999... to multiply .999... by ten you move the decimal place right and
10x+9.999...This is basic math RR. It's not controversial. It's not debated it done with proofs. Mathematical proofs are probably the most valid deductions you can make. Stop arguing this and instead try and understand it. The {...} is a mathematical convention. It has been proven over and over again that .999... = 1.Now there may be some other way to define this in some special mathematics just like you can assume that parallel lines meet, but in dealing with ordinary world stuff .999... has to equal 1 or we end up with a Zeno's paradox that you can't move. You could never cross a room. Since you can the mathematical concept fits with our everyday experience. You are willing to believe the most unsupported religious things and then you turn around and argue endlessly about well developed mathematics! What gives? You want to live in total subjective fantasy or what?lfen

We don't lose any number at all. It's an infinite series and that means it has no last term. If you added 1=1 an infinite number of times you would have an infinitely large number. But the series of .9+.09+.009... approaches a limit and that limit is one.It can never grow larger than one and as you calculate more and more terms the sum gets closer and closer to 1. So the sum of the infinite series can be shown to be 1. That is in the basis of the calculus and it's been too many years for me to recall those proofs.There are infinities but infinity itself is not a number. It may not be intuitive that an infinite series of numbers is bounded but it is a mathematical commonplace.lfen

Well, there is no last digit. If there were a last digit it wouldn't be infinite. Infinite means you can go on adding digits without ever ending.Like for example the set of integers is infinite. There is no last integer. What ever you take to be the last integer call it n, you can always add 1 to it and then you can add 1 to that. It's infinite and unbounded.An infinite series like .999... is bounded. The bound of that series is 1 and btw did you read any of the web pages I gave you links for? The thing about math is that it is not philosophy. You have to set down and rigorously develop the arguments step by step. It's hard to do in this forum because well, it's one a lot of work, two hard to coordinate, and hard to type math symbols.Just think about there not being any last digit. You know if you multiplied .999... times a thousand you would have 9999.999... and .999... part of that number would be just as infinite as .999...Infinity has that property so to speak. The number of points in a line an inch long is infinite, the same infinite as the number of points in a line a mile long. Infinity equals infinity (well there are infinities that aren't equal but they don't play a role in this).lfen

You've heard of the calculus? That is one of the things it deals with. If you are dealing with unbounded infinities then they just keep getting larger like adding 1 to 1 an infinite number of times.But if you, sticking to this example, add 9/10+9/10^2+9/10^3 and so and on for an infinite number of integer values of the power of 10 and we could write that as .9+.09+.009... or write is as .999... then that sum won't grow huge. If you calculate it for any value you notice that the further you go the closer it gets to one but will never exceed one. so at 10 places it's .9999999999 which is larger than .9 but less than 1. More places gets even closer .999999999999999999999999999.There are proofs which it's been too many years so I don't recall that show that the sum of the infinite series is 1.Take a room that has a finite distance call it 10 feet. There are an infinite number of points in that 10 feet and yet you can walk across the room. You can go .9 of the way, and then .9 of that distance which is .99. Do you wish to claim that you can't walk across the room? clearly the sum of .9+.09 etc in the end is the distance across the room otherwise you would never be able to walk across the room, but you know you do.The above is not a proof but indicates a way to approach a proof.lfen

another source for you on infinity.lfen

It sounds like you are confusing concept with the universe. Infinity is a mathematical concept, just as a point and a line are concepts. A line is defined as having two dimensions. It has no width. You won't find actual lines out in the world that have no width. They don't exist. As far as I know at this point, well, at least for the known universe matter is assummed to be finite. There is a very large number of electrons but not an infinite number.But we are talking about mathematics. Do you accept that between 0 and 100 there are an infinite number of numbers (you can call them points as well) and that between .99999 and 1 there are also and infinite number of numbers? Infinite is not a number it means that no matter how far you count, or how many times you divide the line you can still keep counting, still keep dividing, there is no end to it. Now in the universe there may be a limit to how small you can make a division but we are talking mathematics here, pure concept.lfenABE: I'm sure they prove infinity in number theory every day. Some one here can do this better than I but the proof is along the lines of pick some number as large as you want, call it n. You claim there is no larger number. But by adding 1 to it we have n+1>1. If you say that is the largest well we make another number by adding 1 to that. Edited by lfen, : after thought added

Two errors here. First there is no such thing as an infinite number. There are infinite sets, infinite series, etc. But infinity is not a number.Two the last number of an infinite number doesn't make sense. If you are referring to the last digit of a decimal representation of pi for example there is no last digit. There is not a last digit to .999... If there were a last digit it wouldn't be an infinitely repeating decimal, it wouldn't be an infinite series, it would be finite, that is to say countable.So there is no last digit, so the last digit, which never existed in the first place, can't change. There is no problem. The calculus works. You can cross a room.lfen

.999...=1 is not wrong. .999 does not equal 1, nor does .99999, but
.999... is another way to represent 1. So is 1/1, or 99999/99999 they are all different representations for ONE.Are you claiming to be a better mathematician than Newton, Leibniz, Cauchy, Cantor, Godel, Euclid, on and on and on? This is not Bible interpretation, it's mathematics. You can have you own private math if you want but who will use it? Well, it's a different way of representing things. Do you have a problem with Zeno's paradoxs or not?lfen

No it's not proof of anything. I don't know how you got
-0.5x=.166multiplying by -1 would give you -.5x=-.166...x= .166.../.5 =
1.666.../5 =
.333... = 1/3lfen

It's not acceptable for a rigorous claim for the calculus and there are proofs. Engineers used to get it the theory wrong but the application right by saying "it goes to the limit". It's just that the proofs take time. You have to define what a limit is and a bunch of things. That is why I've tried to use Zeno's paradoxs to give you a more easily visualized solution to the problem. You keep confusing notation with concept.In the number 123, 1 is digit, 2 is a digit, 3 is a digit.
They are not the numbers 1, 2 , or 3 they are the digits of the number 123. How can you add "the last number to an infinite set"? if it is infinite then by definition there is no last number! Oh this is not about rounding decimals up! no, no.
.4999 rounded to the nearest tenth is approximately .5 but that is not at all what is happening here. No rounding is going on. The sum of an infinite series is not arrived at by rounding!!!! It may look that way but that is the diffence is as I stated .499 rounded up is APPROXIMATE but .4999... is an EXACT representation, that is to say is EQUAL to .5. Big difference.

I suspect you are conflating physics and math. The universe maybe finite. Infinity doesn't exist in the real world. Neither do numbers. Things exist and we count them. But if you go looking for 1 or 2 they don't exist except as math concepts.I find an apple, I find another apple. If I can count I say I had no apples, found 1 then found another so I have two apples. So we count 1,2,3,... What does the [...] mean? It means in my mind I can go on adding 1 without ever stopping. Now if we could count all the apples in the world we would reach an end. They are finite.But in math the set of natural integers is infinite. Is this true of the universe? It is not neccessary that we know that. In math what we know is that we can add 1 to n, n+1 without limit. That is what we mean by infinity. It's not real, it doesn't exist, it's not a number, it's a property of some sets that they aren't limited. They are limitless. You can go on counting them forever and never reach an end.Now take the series of halving a distance. 1/2 then half of that is 1/4 then half of that is 1/8 and on and on. You can do this with out end because 1/2n for the natural numbers means n is not limited. It can get huge and 1/2n gets very very small. If you sum this infinite series up it will equal 1.This is an intuitive solution to Zeno's paradox. Zeno saw that you would have to go half way across a room, then half way across the remaining distance, then half of that, and the series was infinite so he stated you can't ever cross the room. But you can. Why? because intuitively the sum of 1/2+1/4+1/8... to infinity equals 1.Now the calculus does this rigorously. Decades ago I took a course for a year where the professor proved the entire calculus. Just wrote proof after proof on the board. I can't recall all that and I can't begin to reproduce it here. You would need to take a college course on calculus. But intuitively you should be able to take this and see what is happening.The other thing is you sometimes are confusing a notation with an operation. Notation in math is variable as you have seen 1/2=.5=.4999... or even 500/1000 or 1x/2x etc. etc. Don't confuse notation with what is notated. Although 1x/2x doesn't look like .5 that is what it is.lfen

Bingo! Infinity is not a quantity. As for the convention of [...] it
is well defined and presents no problems when used by mathematicians. I think the problem is you are trying to use very simple algebra and are getting some of the conventions wrong. I like your idea of taking some math classes. Just Do It!lfen