What's the difference between pi and 0.999... ?

Question

According to math professors, the limit of the sequence (.9; .99; .999; ...) is 1 and thus the infinite sum 9/10+9/100+9/1000+... is defined by this limit. In fact, they claim that every real number has a limit. However, no mathematician can tell you what is the limit of pi, e, sqrt(2) or any other irrational number. One difference between pi and 0.999... is that we know the limit of the sequence (0.9; 0.99; 0.999; ...) but we do not know the limit of the sequence (3.1; 3.14; 3.141; ...). So what's the real difference?

Answer 1

What do you mean "every real number has a limit"? A number is a number. You can define any kind of sequence that will have a given number as its limit. The sequence related to pi is not 3.1, 3.14, 3.141, because that's not how pi is defined. Pi is the ratio of the circumference of a circle to its diameter, period.

However, pi IS the limit of the following infinite series:

pi/4 = 1/1 - 1/3 + 1/5 - 1/7 + 1/9 .....

as well as a few others one could define.

Answer 2

The only difference is that 0.999... has a repeating pattern and pi does not. Mathematicians will tell you that 0.999... is rational. This is untrue because there is no way to express 0.999... as a ratio of two integers.
So where do they go wrong? Well, for starters they 'define' a real number as the limit of a Cauchy sequence (or Dedekind cut or some other wrong method). This deep rooted misunderstanding of math professors starts in high school when they are taught (erroneously) that the sum of an infinite sequence is a/(1-r). The fact is that the limit of an infinite series is not equal to the actual infinite sum of the series nor can anyone just 'define' it to be so.

gimb1960 states that he knows the limit of the sequence (3.1; 3.14; ...) - he is lying (sorry it's a strong word). What is PI Mr. gimb1960? I am not interested that you are able to calculate pi to a large degree of accuracy (I can do this myself in over 153 different ways quite easily). This does not demonstrate that you know what the limit is. I can just as easily say that the limit of (.9; .99; ...) is POINT9RECURRING.

Limit does not mean that you can get arbitrarily close (how 'close' is 'close' and how many digits are 'enough'?) to some number. It means that the sequence/series is bounded from above (Archimidean Property). To say that you can find a number that is closer than any number I suggest does not say anything if you do not know what number cannot be exceeded. In the case of pi, fact of the matter is that you have no idea what this upper bound is and you cannot just call it PI.

Answer 3

Yes you know the limit of the sequence 3 , 3.1 , 3.14 etc
it is PI.

Why do you think that "However, no mathematician can tell you what is the limit of pi, e, sqrt(2) or any other irrational number."

Limit means that you can get arbitrary close to some number.
That is You tell me how close I should get to some number and I will tell you enough digits, or how to calculate these digits.

I am a mathematician and I just told you what the limit is...
I am wondering what you mean with the limit ?

The only difference I see between 0.9999 and PI is that 0.999 repeats and PI does not.

Answer 4

Short answer:

The difference between the two numbers (the limit of the series 0.9, 0.99, 0.999, 0.9999 ... and Pi) is that one is _rational_ and one is _irrational_.



Medium answer (grab a sandwich, this will take a paragraph or two):

Rational numbers can be expressed exactly as fractions:

0.25 = 1/4
0.4 = 2/5
0.666... = 2/3
0.9999... = 1/1
2.142857142857.... = 15/7
4.6 = 23/5

If it can be expressed as a fraction, it can also be expressed as either a whole number, a number with a decimal expansion that ends, or a repeating decimal.

Irrational numbers cannot be expressed exactly in fractional forms. But irrational numbers themselves come in a couple of flavours:

(1) Algebraic irrationals

This means numbers that are themselves the solutions to algebraic equations like (x)^2 - 2 = 0 ("x squared minus 2 equals zero") or 5(x)^3 -2(x) +27 = 0.

The best examples of these are the square roots of numbers that aren't square:

Square root of 2 = 1.41421356... (never repeats)
Square root of 41 = 6.403124... (never repeats)


(2) Transcendental numbers

These numbers usually come out of functions that have an infinite number of terms needed to figure them out precisely, like logarithms, or Pi, or e (the base of the natural logarthims).

These numbers cannot be expressed exactly as fractions:

Pi does not equal 22/7, for example. That's just an approximation.



Really long answer:

Take two years of post-high school algebra.

Answer 5

It's not correct to say that every real number has a limit. What is correct is to say every real number is the limit of a sequence of real numbers. Maybe what you meant to say was every real number is the limit of asequence of rational numbers, and this is correct.

What happens is the limit of 9/10 + 9/100....is 1, that is a rational number. And, in addition to being rational, 1 is integer. So we have a finite representation for 1 in our decimal basis.

On the other hand, pi is an irrational, and therefore it's decimal expansion contains infinitely many decimal places. And, what makes things even more difficult, this sequence of decimal places is not periodic, so we´ll never be able to determine precisely all the decimal places of pi.

Hope this makes things a bit clearer.

Answer 6

0.999999.... =1
and the decimal expansion is periodic, acutally it repeats itself from the first decimal

while pi has a decimal expansion that is not periodic, it never repeats itself

so the difference is that for rational numbers (including the integers) the decimal expansion is known, if it is not finite, then it is periodic, and therefore you can know all those decimals.
while for irrational numbers like pi or sqrt(2),
you can never know all the numbers in the decimal expansion, because they are infinite and there is no pattern to guess what the next one will be.

Answer 7

There are rational numbers, irrational numbers, and transcendental numbers.

pi is transcendental.

Transcendental means that the number is not the solution to any polynomial equation with rational coefficients.

Irrational means that the number cannot be expressed as a quotient of integers. All transcendental numbers are also irrational.

Answer 8

the difference between pi and .9999

pi:non-repeating ( never repeats the digits )
non terminating ( no matter how many times you slpit it you always have something left over)

.9999: is a reapeating number (repeats 9 a way to show it is using bar notation)

Answer 9

pi=22/7(non-repeating number)
0.999999(repeating number)
=9*10^-1+9*10^-2+9*10^-3+.......................

so, approximate differenceis

(22/7)-1
=22-7/7
=15/7
=2.142857

Answer 10

Tom, do you really think we're too stupid to figure out that Marianne M is one of your puppet accounts? I realize that this is an emotional issue for you, since the .[9] article just appeard on the front page of wikipedia without any of your ignorance attached, but you need to realize that the reason every professional mathematician disagrees with you is that you are wrong. For the benefit of everyone else here, I shall give a refutation of Tom's current rambling (not that I expect to recieve best answer for it, since obviously Tom will select himself):

"The only difference is that 0.999... has a repeating pattern and pi does not."

False. Aside from the fact that π exceeds 0.[9] by about 2.14159 26535 89793 23846 26433 83279 5, there's also the fact that π is not the root of any polynomial with rational coefficients, whereas 0.[9] is an integer (and thus both rational and algebraic).

"Mathematicians will tell you that 0.999... is rational. This is untrue because there is no way to express 0.999... as a ratio of two integers."

0.[9]=1/1

Next.

"So where do they go wrong? Well, for starters they 'define' a real number as the limit of a Cauchy sequence (or Dedekind cut or some other wrong method)."

Those methods are not wrong. These are in fact correct definitions of the real numbers. These definitions were chosen because they are necessary to ensure certain elementary and obvious properties of continuous functions. For instance, if f(x) is a continuous function, f(a) < 0 and f(b) > 0, then there should exist c such that f(c)=0 (i.e. you cant draw a continuous function from below the x-axis to above the x-axis without crossing the x-axis). Because of the way real numbers are defined, this is a provable theorem (see http://en.wikipedia.org/wiki/Intermediate_value_theorem). However, if you "define" a number system using only finite decimal expansions, then clearly this theorem is false (consider f(x)=x²-2, a=0, b=2. In the real numbers, f(√2)=0, but since Tom apparently only considers a number to be defined if its decimal expansion can be written exactly (which the square root of 2 cannot), that number doesn't exist in Tom's number system, and so f(x) continuously passes from one side of the x-axis to the other without ever touching the axis itself. This is absurd.)

"This deep rooted misunderstanding of math professors starts in high school when they are taught (erroneously) that the sum of an infinite sequence is a/(1-r)."

Sum of an infinite SEQUENCE? Sequences do not have sums. SERIES have sums. Please do not confuse the two.

Also, the sum of an infinite series is not, in general, a/(1-r). The sum of an infinite GEOMETRIC series with initial term a and common ratio r is a/(1-r), however most infinite series are not geometric.

"The fact is that the limit of an infinite series is not equal to the actual infinite sum of the series nor can anyone just 'define' it to be so."

Actually it is and we have. And the reason we can speak meaningfully about the sum of an infinite series is that the sum itself is NOT infinite.

"gimb1960 states that he knows the limit of the sequence (3.1; 3.14; ...) - he is lying (sorry it's a strong word)."

I find it ironic in the extreme that someone who uses puppet accounts to propogate their own brand of pseudomathematics would accuse others of lying. Actually, pseudomathematics gives us a good idea of what's going on in Tom's head. Consider the following excerpt from the wikipedia article (http://en.wikipedia.org/wiki/Pseudomathematics ):

"An appeal that a mathematical definition is in itself wrong (i.e., that primes were somehow poorly defined in the first place) is an appeal to an argument that attacks a well-established and well-understood definition: Primes are prime by definition, and such classes of numbers may or may not have properties that make them interesting. Pseudomathematics, however, sometimes appeals to change definitions to suit its claims. At this point, pseudomathematical arguments exit the world of mathematics altogether and even the appearance of following long-established mathematical models falls apart.

Any statement purporting to be a theorem must hold within the framework of the pre-existing definitions about which it purports to assert a truth. While new definitions may be introduced into a framework to substantiate a theorem, these new definitions must themselves hold within the framework addressed, without introducing any contradiction within that framework. Asserting that 33 is somehow prime because a flawed proof arrives at this eventuality, and then asserting that the definition of primes itself is flawed is pseudomathematical reasoning."

Tom, that least sentence should give you pause, because that is exactly what you are doing: giving a flawed "proof" that 0.[9] < 1 (which may be summarized as "it looks smaller") and then asserting, on that basis, that somehow the _definition_ of the sum of an infinite series is wrong, because it disagrees with a result that was incorrect to begin with.

Moving on:

"What is PI Mr. gimb1960? I am not interested that you are able to calculate pi to a large degree of accuracy (I can do this myself in over 153 different ways quite easily)."

I'd very much like to see Tom actually list those 153 different ways - it's very bad practice to make specific assertions about one's knowledge without being able to back them up. And as for the exact value of π, consider the following partition of Q:

A={x∈Q: [n→∞]lim [k=0, n]∑((-1)^k x^(2k+1)/(2k+1)!) > 0 ∧ x < 4} ∪ {x∈Q: x ≤ 0}
B={x∈Q: [n→∞]lim [k=0, n]∑((-1)^k x^(2k+1)/(2k+1)!) ≤ 0 ∧ x > 0} ∪ {x∈Q: x > 4}

Yes, I did just give you a dedekind cut for π. And nowhere in this construction did I make any reference to circles.

"This does not demonstrate that you know what the limit is. I can just as easily say that the limit of (.9; .99; ...) is POINT9RECURRING."

And I would then say that POINT9RECURRING=1. Remember, real numbers are defined in terms of dedekind cuts. Therefore, if you wish to claim that these numbers are different, show me a dedekind cut of 0.[9] which is not also a dedekind cut of 1. Just so everyone is clear, a dedekind cut is a partition of the set of RATIONAL numbers into nonempty sets A and B such that A is closed downwards, B is closed upwards, and A has no greatest element (i.e. A={x∈Q: x≤1}, B={x∈Q: x>1} is not valid, because A has a greatest element, namely 1. Whereas A={x∈Q: x<1}, B={x∈Q: x≥1} is valid, and is the dedekind cut of 1).

Now Tom, I already know how you will respond to this challenge, so let me preempt you: A={x∈Q: x<0.[9]}, B={x∈Q: x≥0.[9]}. Fine. Now demonstrate that {x∈Q: x<0.[9]} is a DIFFERENT set than {x∈Q: x<1}. To do this, you must find a RATIONAL number that is less than 0.[9], but not less than 1. I will request that you provide this number in the form p/q with p and q both integers. Unless you can do that, you have no right to go around claiming that 0.[9] and 1 are different numbers.

"Limit does not mean that you can get arbitrarily close (how 'close' is 'close' and how many digits are 'enough'?) to some number."

Actually, that is EXACTLY what limit means. And arbitrarily close means just that: arbitrarily close. You can name any positive number, as small as you like, and the sequence will eventually get closer than that (and stay closer than that). For the limit of a series, it means that the sequence of partial sums will eventually get arbitrarily close to the limit.

"It means that the sequence/series is bounded from above (Archimidean Property)."

Hooboy, three errors in one sentence. First:

Limit does not mean the sequence is bounded from above (although that is a necessary condition for it to have a limit, it is FAR from sufficient). Consider the sequence [-1, 1, -1, 1...] - The elements of this sequence clearly have an upper bound (specifically, 1), but the sequence itself has no limit.

Second: the archimedian property has very little to do with sequences. It has to do with the nonexistence of positive infinitesimals - specifically, the archimedian property asserts that there is no number x such that x > 0 and x < 1/n for EVERY natural number n. The archimedian property gets mentioned in the 0.[9]=1 "debates" because if 0.[9]<1, then 1-0.[9] would be a positive infinitesimal, and there is no such thing.

Third: judging from Tom's past postings, I'd say he's gotten the archimedian property mixed up with the least upper bound property, which states that every nonempty set with an upper bound in the reals also has a least upper bound in the reals. This is directly implied by the construction of real numbers from dedekind cuts (let A be the set of all rationals less than any element in your set, and B be Q\A). It is liberally used in the proof of the intermediate value theorem (see that article for details), and it directly implies the archimedian property (suppose a positive infinitesimal existed. Let S be the set of all positive infinitesimals. This set is clearly bounded above (by, say, 1), and therefore has a least upper bound in R. Let this least upper bound be x. Either x is an infinitesimal or it is not. If it is, then so is 2x, which is strictly greater than x (since x is positive), and so x is not an upper bound. But if it isn't, then neither is x/2, which is less than x, and so x clearly isn't the LEAST upper bound. Either way, we have a contradiction).

Now, I can somewhat understand confusing least upper bounds with limits, since all the sequences typically mentioned in the 0.[9]=1 "debate" are monotone increasing, and for monotone increasing sequences the supremum of the elements in the sequence is always the limit (though this is not true in general: the limit of [1, -1/2, 1/4, -1/8, 1/16, -1/32...] is 0, but the supremum of the elements is 1). However, I am totally at a loss to see how anyone could mix this up with the archimedian property, and can only assume that Tom doesn't understand any of them.

"To say that you can find a number that is closer than any number I suggest does not say anything if you do not know what number cannot be exceeded."

Here I have no idea what he's talking about, so here's a picture of a bunny with a pancake on its head:

http://heresabunnywithapancakeonitshead.com/

"In the case of pi, fact of the matter is that you have no idea what this upper bound is and you cannot just call it PI."

Of this, I can only assume that Tom thinks that knowing what a number is is the same as being able to write its decimal expansion. To which I say bollocks. It is entirely possible to know exaclty what a number is without knowing anything about its decimal expansion (see i, e, π, √2, ln 2, etc.). Conversely, it is entirely possible to know a number's decimal expansion and have no idea what the number is (consider 0.12345 67891 01112 13141 51617 18192 02122... - I know every digit in the decimal expansion of this number (in fact I created it by specifying its decimal expansion), but I have absolutely no idea what it is or what algebraic propreties it has, save that it cannot possibly be rational).

And finally, addressing Tom's original post:

"According to math professors, the limit of the sequence (.9; .99; .999; ...) is 1 and thus the infinite sum 9/10+9/100+9/1000+... is defined by this limit."

This part is correct, albeit somewhat restrictive. Specifially, we can remove the "according to math professors" part and not lose any validity.

"In fact, they claim that every real number has a limit."

Uh... no. To quote the first poster: "A number is a number." Sequences have limits. Functions have limits. Numbers do not have limits (although in some cases they ARE limits).

"However, no mathematician can tell you what is the limit of pi, e, sqrt(2) or any other irrational number."

Again, a number is a number. Numbers do not have limits. Perhaps, given Tom's history of confusing concepts, he is actually talking about least upper bounds. One might even, (interpreting his post in an especially charitable light) say that he's trying to grasp the concept that π, e, √2, etc. do not have least upper bounds in the rational numbers - i.e. there is no smallest rational number greater than π, e, √2, or any other irrational number. In this he would actually be correct, and this is one of the biggest differences between Q and R - in R, if you can say of any rational number p/q: "p/q < n" or "p/q ≥ n," then you are guaranteed that n is a real number, whereas it probably isn't a rational number. But then, since he disputes this, that would seem to indicate he has mistaken the reals for the rationals. Not surprising, since the set of numbers with finite decimal expansions is a proper subset of the rationals, and if Tom is convinced that you can only know what a number is by writing its decimal expansion, then it follows trivially that he wouldn't consider numbers outside of Q to be well-defined.

If I'm right about that, that might even explain his unease at 0.[9]=1 -- He feels that anything that is defined using an infinite series can never be grasped in its entirety, and that he can never truly know what 0.[9] is. But since he does know what 1 is, he thinks that 0.[9] cannot possibly be 1. Of course, if I'm correct about Tom's reasoning, this would amount to little more than an instance of the masked man fallacy (see http://en.wikipedia.org/wiki/Masked_man_fallacy ).

"One difference between pi and 0.999... is that we know the limit of the sequence (0.9; 0.99; 0.999; ...) but we do not know the limit of the sequence (3.1; 3.14; 3.141; ...). So what's the real difference?"

Acutally, IF I'm right about the generating function, I do know the limit of the second sequence - it is π (and no, I do not have to write π as a finite decimal to know what π is). And of course there are the obvious differences between π and 1 that I listed at the beginning of this megapost, which even he should be aware of, so clearly he's not asking about the differences between 1 and π. All I can surmise is that Tom is asking "why do we give the limit of one infinite series the name of a familiar object (namely 1), when the general practice with infinite series is to give them symbolic names independent of familiar objects, so that we can keep the unkowable infinite seperate from the knowable finite?" And the answer is "for the same reason that we give the function (ln (-1))/i the familiar name π instead of keeping it in the realm of imaginary objects by denoting it continually with (ln (-1))/i - because we already have a name for that number." There is, Tom, no magical barrier between the sums of infinite series and ordinary, finite numbers, no border between the rational and irrational, no partition keeping the imaginary and real seperate -- there is nothing stopping two functions, one incredibly simple, and the other inordinately complex, from having the same result. The statement 0.[9] = 1 is not due to some conspiracy of mathematicians to pretend they know the unkowable (infinity, far from being unknowable, is actually pretty well defined), or some prejudice against the use of symbols instead of explicit numbers, but simply due to the fact that, by chance, we already have a name for that number.