2=1

[Guv]

Noble has the right idea.

0.9999 recurring forever equals 0.9999 recurring forever and 1 equals 1. 0.9999 recurring forever "approaches one and for simplicity can be represented as 1".

The definition of limit does not claim that a function is equal to it limit. It merely states that the difference between the function and its limit can be shown to be smaller than any finite value.

You must be careful about setting a function equal to its limit. For example.

• Limit[ (1 + 1 / n)^n ] approaches e (2.71828...) as n grows without bound.
• Limit[ 1 + 1 / n ] approaches one.

You cannot say that 1^n = e for infinitely large n, which is what you get if you set functions to their limits here and try to specify what is happening for infinite values. Mathematicians avoid statements about what happens at infinity.

Fried Egg: I have never heard that a proof must be in some sense reversible in order to be valid. Is there a text somewhere that makes this claim? How do you define the reverse of a proof? Such a definition would have to be on a case by case basis. Certainly a single definition would not cover all the possible methods of proof. It seem particularly difficult to define the reverse of a Reducto Ad Absurdum proof. What is the reverse of Wiles proof of Fermat's Last Theorem? What about disproving a statement by showing a counter example? Can such a disproof be reversed? I do not remember any book or text that showed a proof, and then went on to validate it by showing it in reverse.

I can prove that a particular number is not a prime by showing that it is the product of two particular numbers. 7 * 13 = 91, therefore 91 is not prime. What does it mean to reverse this proof? Can you determine the factors from the conclusion that 91 is not prime? The forward proof is simple, how do you do it backwards without using information from the forward proof? 91 / 7 = 13 is the only approach I can think of and that uses data from the forward proof. Godel proved that certain axiomatic systems are either inconsistent or incomplete. What would be the reverse of such a proof?

Most ordinary geometric proofs seem reversible. Given the Pythagorean relationship, I suppose you could prove the triangle to be a right triangle. Many other proofs are probably reversible. After all, if a forward going proof seems valid, you would hardly expect a reverse proof to be invalid. It seems reasonable to assume that most valid proofs are in some sense reversible. It does not seem reasonable or necessary to require reversibility for validity.

Others: The proof that recurring .9999 equals one seems reasonable, but does not seem valid to me in spite of what some calculus text might say. I do not question its validity due to lack of reversibility.

I think the proof is invalid because of the multiplication of recurring .999 by 10 and subsequent subtraction of the recurring nines. These operations seem reasonable, but they also seem undefined. It does not seem valid to do arithmetic on infinitely long decimal fractions. There must be some subtle unstated assumptions about such operations. Reasonable assumptions about transfinite and infinitesimal numbers are often incorrect.

The proof seems valid, especially since there is a valid proof showing that the limit of recurring .99999 is one (this proof does not rely on decimal notation). Without the other proof, I would be nervous about accepting the one involving arithmetic on infinitely long decimal numbers. I would worry about the possibility of such methods leading to erroneous results in another context.

BTW: In base 16, you can similarly prove that recurring .FFFFF equals one. Does this prove that recurring .99999 in decimal equals recurring .FFFFF in hex? For all finite examples, the two recurring fractions are unequal. Note that you can certainly prove that they approach the same limit (namely one).

[simonm]

Fried Egg,

You may have a point. Perhaps your reversal of the proof indicates that an infinitely small quantity is lost thereby allowing you to 'prove' 1 = 0.9999...
Probably in the step that multiplies 0.9999... by 10 to get 9.999...

At the end of the day though, 1 and 0.9999... can still be considered, for all intensive purposes, to be equivelant (even if they aren't, in the strictest sense, equal).

[marnitzg]

Well, I still say that is isn't a limit. Plain and simple maths shows this to me. I believe that something is wrong but I can't find anything wrong with wither proof.

Originally posted by Guv

BTW: In base 16, you can similarly prove that recurring .FFFFF equals one. Does this prove that recurring .99999 in decimal equals recurring .FFFFF in hex? For all finite examples, the two recurring fractions are unequal. Note that you can certainly prove that they approach the same limit (namely one).

No because they are different bases.

Sorry, this one of those brainless days again. I take back the above statement.

[Starman]

marnitzg,

>No because they are different bases.


I may be wrong but I have always thought that 1 was 1, no matter what base you are counting it in.

[simonm]

Guv,

The definition of limit does not claim that a function is equal to it limit. It merely states that the difference between the function and its limit can be shown to be smaller than any finite value.

The statement above is saying the same thing that Fried Egg was saying (albiet in a different way).

Fried Egg said that there is an infintely small quantity lost in the proof that 1 = 0.9999(...) and not a small fininte quantity.

In this specific case, the reversability of the proof highlights the loss of this infinitely small quantity. He did give this impression that 'reversability' is a requirement of a proof (which is obviously not) but I think what he meant was that, in this case, reversing the proof demonstrated a flaw in the proof.

[parksie]

Originally posted by Starman
I may be wrong but I have always thought that 1 was 1, no matter what base you are counting it in.

What if it's base 1/2?

[Guv]

Serious mathematicians try to avoid proofs based on the notation used for expressing numbers.

The above is one reason why I question the validity of the proof involving the multiplication of recurring .9999 by ten. Another reason to distrust this proof is the requirement for arithmetic on infinitely long decimal numbers, which surely requires some supporting assumptions and/or definitions. The use of intuitive notions (rather than formal methods) in the 17th, 18th, and 19th centuries led to problems in mathematical logic, which is the reason mathematicians tend to be very picky and formal.

Note that specifying recurring .9999 as the sum of the series 9/10 + 9/100 + 9/1000. . . is independent of the radix. It is equivalent to the hex series 9/A, 9/64, 9/3E8. . . The proof could be done using hex notation followed by showing that the infinite series is equivalent to recurring nines in decimal notation.

Similarly recurring .FFFF in hex can be proven to approach one as the limit independent of the radix used for numerical notation.

[marnitzg]

I never knew you got decimal points in hex. How do you represent it? My HP won't accept F.FFFFF (or any other hex fraction)

[Guv]

Marnitzg: My HP calculator will do Hex, octal, and binary arithmetic on integers only. I think there are special calculators which do general purpose radix arithmetic, but I do not know of any regular calulator which will do radix arithmetic on fractional values.
However, such values exist in the Mindscape of Mathematics.

Depending on the country you favor, a dot or a comma separates the integer and fractional parts of Hex (or other radix) numbers. In Hex: 3.2FA represents 3 + 2/16 + 15/256 + 10/4096

If you kept track of the radix point yourself, you could use your calculator to do hex arithmetic on fractional values. On my calculator, division would be a problem because it gives an integer quotient and no remainder. Perhaps you could do division if you added some trailing zeros to the dividend.

If you wanted to multiply 3C.F by 2.F5, you could use integer multiplication of 3CF by 2F5, getting B431B. Since there are 3 digits (total) to the right of the radix point in the multiplier and multiplicand, there must be 3 digits right of the radix point in the product. B4.31B is the product.

In this day and age, I would how many people know how to do multiplication and division by hand methods, keeping correct track of the decimal point.

[Sam Finch]

0.999999............. = 1


Remeber this axiom

For any 2 real numbers A and B exactly one of the folowing is true.

A = B

or

there exists a real number C s.t.

A < C < B

or

there exists a real number C s.t.

A > C > B



And as there is no real number between 0.9999999... and 1 it must be true that 0.999999.... = 1.


QED.

[Guv]

Sam: You do come up with interesting ideas.

Remember this axiom

For any 2 real numbers A and B exactly one of the following is true.
A = B
or
there exists a real number C s.t.
A < C < B
or
there exists a real number C s.t.
A > C > B

My answer to the question is that I do not remember such an axiom. There could very well be such an axiom, but I do not remember it. It looks valid for any real number whose value has a finite expression.

I do wonder about the context of such an axiom. Is it really applicable to real numbers which cannot be expressed in a finite manner? I would be surprised if there are not some caveats with that axiom.

What about the following?

• Recurring .99999 is a representation of the geometric series: 9/10 + 9/100 + 9/1000.....
• In hex, recurring .FFFFF is a representation of 15/16 + 15/256 + 15/4096....

For a given number of finite terms, the sum of the second series is greater than the sum of the first, and is also less than one. Hence there is always a finite real number between recurring .99999 and one. Hence, they are not equal. This argument could be dressed up with some picky-picky mathematical language, but I think you get the picture.

The sum of both series is limited by one. However, I still agree with Noble.

0.9999 recurring forever equals 0.9999 recurring forever and 1 equals 1. 0.9999 recurring forever "approaches one and for simplicity can be represented as 1".

[Sam Finch]

I figure this line of argument can only continue with increacingly complicated nit picky maths, so I suggest we take a different route.


What is 1 - 0.99999999..... ?

What is the reciprocal of this number?

[Guv]

Sam: Variations on this argument have occurred several times now.

If you tell me how many nines we are talking about, I will tell you what 1 - .99999... is equal to, and I will estimate the reciprocal. I will almost always insist on being picky, because I believe in avoiding sloppy terminology and sloppy methods. In practice, I will use one as a replacement for .999999 and 1/3 as equivalent to .333333, but I will not accept statements like " recurring .33333 equals 1/3," except as convenient shorthand notation.

This thread started with a proof that recurring .999999 = 1.0000 using some questionable operations on "infinitely long" decimal numbers. I objected to the proof for various reasons and suggested viewing the recurring .99999 as a geometric series. It is fairly easy to prove that the limit of the series is one, without using questionable arithmetic operations on infinitely long decimal numbers.

I agree with Noble

0.9999 recurring forever equals 0.9999 recurring forever and 1 equals 1. 0.9999 recurring forever "approaches one and for simplicity can be represented as 1".

Limits are defined by terminology similar to the following.

The number a will be called the limit of the sequence X1, X2, X3 . . . Xn . . . provided that, given any positive number h, no matter how small, there exists a corresponding term of the sequence Xn such that every succeeding term of the series lies between the numbers a-h and a+h.

The wording is slightly different for the limit of a function and the limit of the sum of a series, but the idea is always the same. As some number or variable grows without bound, some value gets closer to the limiting value.

The formal definitions use terms like increases indefinitely or there exists or grows without bound. The formal definitions refer to an extremely small difference between the value and its limit. They avoid referring to infinity and avoid the statement that the value equals its limit.

The use of the limit as a value is viewed as a convenience which avoids lengthy terminology.

Every serious mathematical text I have ever read avoids claiming that a function equals its limit. When not dealing with transfinite numbers ala Cantor, they also avoid using the term infinity. I have a book (Fundamentals of Mathematics by Moses Richardson) which specifically warns against the use of such a term.

The notation "Limit(Xn) = a as n —>lazyeight" is commonly used, but it often misleads the unwary student into believing that there is a peculiar number called infinity which n approaches. This is, of course, not true.

In a later chapter of this book

. . .It would be even more misleading to write 1/0 = infinity, although some books do. . . .they do not mean that infinity is a number which you obtain by dividing 1 by zero. As we have seen 1/0 is a meaningless symbol and infinity is not a number. "Infinity" in this sense is merely a way of describing the manner in which certain function behave.

In a footnote it is mentioned that historically the statement 1/0 = infinity was taken literally before mathematicians were familiar with modern mathematical logic.

[jmcswain]

1 / 9 = .1111111...
2 / 9 = .2222222...
3 / 9 = .3333333...
4 / 9 = .4444444...
5 / 9 = .5555555...
6 / 9 = .6666666...
7 / 9 = .7777777...
8 / 9 = .8888888...
9 / 9 = 1

It does seem like a breach of the progression.

9 / 9 = .9999999...

would fit so much more naturally.

[jmcswain]

Isn't there some rule about all real numbers can be expressed as

a / b

where both a and b are real numbers? If so, then .999... is not a real number.

[parksie]

But then what about 1/9?

[jmcswain]

.111... is fine because it fits the "find the a / b" mold

 10x = 1.111...
-  x =  .111...
-------------------------
  9x = 1

   x = 1 / 9 = .111...

[Guv]

JmcSwain: Rational numbers can be expressed as a / b, where both a and b are integers. I think numbers like SquareRoot(3) are called surds (I am not sure about this). Numbers which are the roots of polynomials are called algebraic numbers. Numbers like e (2.71828) and Pi (3.141592653589793) are called transcendental numbers.

[jmcswain]

Guv,

Ahhh, I knew there was some rule like that. It was starting to bug me what the rule actually was; thanks for the explanation.

[Ajai]

until now 2 wasnt 1

but now 1 and 1 which is 2 got married and became 1!!

[simonm]

Can you point out the flaw in the supposed proof that 1 = 0.9999... ?

[Guv]

Simonm: From a common sense or intuitive point of view, there is nothing wrong with the proof that recurring .99999 equals one. Almost every serious mathematician up to the end of the 19th and many in the early 20th century would have accepted it as valid. For all practical purposes it is valid. After all, there is a valid proof that recurring .99999 has one for a limit. How far off base could the proof's conclusion be?

A serious mathematician would not claim that the proof is erroneous. He would claim that it is invalid due to undefined operations. In particular, the multiplication of recurring .9999 by ten and the subsequent subtraction of recurring .9999 from the product. These steps require doing operations on decimal fractions of unbounded length, which are undefined operations.

For any finite number of recurring nines, the proof breaks down. As some posters have already mentioned, perhaps then is a nine missing or unaccounted for in the proof. For a finite number of nines, you merely prove that recurring nines equals recurring nines. It requires a leap of faith to assume that for infinitely many nines the proof would work.

Some time starting in the late 19th or early 20th century, mathematicians became very picky about what constituted a valid proof and what are acceptable operations. The new attitude came about because some extremely subtle errors crept into the mathematical discipline. Proofs which later turned out to be invalid (and contradictory to other proofs) were discovered.

The problems were attributed to the use of common notions rather than formal logic.

Concepts relating to infinity are primarily (perhaps exclusively) dealt with in Set Theory. In analysis, calculus, algebra, analytical geometry, et cetera only finite operations are defined. Terminology like increases without bound is used to avoid words like infinity. When dealing with limits, a concept like the following is used.

• Consider the difference between the sum of a series and its proposed limit.
• Choose any small finite amount which we will call Delta.
• If for any arbitrarily small value of Delta a numbers of terms can be specified which makes the difference smaller than Delta, then the series sum is said to have or to approach the proposed limit.

The above definition does not say anything about an infinite number of terms. It does not refer to some value becoming infinite. It does not require any operations involving an infinite number of terms. It merely says "You choose a small finite Delta and I will tell you a finite numbers of terms that will result in the difference between the sum and the limit being less than Delta."

[nathantkd]

guys, that whole thing with .9999... is only true because you are treating infinity as a number. Infinity is not a number, rather, it is a mathematical animal in and of itself that describes an action. Of course regular mathematic breaks down when you stop using actual numbers. Dammit. I'm turning into math geek. i'm leaving before i start going to geek parties and asking for some Pi.

[bver]

1 - .9999~ = 0 is flawed logically. With an infinite set of 9's there would never be a "Last" 9 to borrow from.