0.999... = 1

Thank you wikipedia.  Math is make believe.

I felt the same way when I was first shown the light.

It took me almost half a century to understand this...  Now I teach it to fifth-graders.  I get to blow away their little minds with the 1/3 + 1/3 + 1/3 proof.   Thankfully, 1/3 of them still don't believe me.

"Math is make believe." I think this is the soundest philosophical position to take on the nature of the subject! I might say it as "Math is a very useful fiction." (or, because I'm a nut, "Math is a beautiful fiction.")

BTW - I love the wiki and the struggle people have had with this. "The students' reasoning for denying  the equality is typically based on their intuition that each number has a unique decimal expansion..." The intuition is correct if you restrict to finite decimal expansions only. It is the infinite expansions that are counterintuitive! 

If you sit and think infinitely about math then it ceases to exist.

Yertle wrote:

0.999... = 1

Thank you wikipedia.  Math is make believe.

To be more accurate, Math isn't make believe, but the systems that we use to explain it are.

The problem in this particular proof is actually a problem with the Base 10 number system. It stems from the fact that you can not express every fraction as a decimal. 1/3 does not equal 0.333...

Yes, you heard correctly, .3 repeating is not the equivalent of one third. That is simply the closest representation we can get in a Base 10 decimal system. People have accepted this approximation as being exact and in doing so have come to incorrectly create the theoretical proofs showing that 1 = 0.999...

By teaching this as fact, what individuals are actually doing is perpetuating the errors that exist within infinitesimalisms.

Seige, Your logic may be way above my head here, but everything I read and understand about infinitesimals is in conflict with what you are saying.

For instance, if your supposition is true, calculus doesn't work.  The first proof most of us learned that shows how a derivative is found requires that 0.999... = 1.  Otherwise every answer in calculus would be a "closest representation".

Ugh... It's like sci.math of the 90's all over again.

Mathematicians have very precisely defined the set of real numbers and of the notion of a decimal representation of a real number. Once you accept those definitions (and standard ways of constructing mathematical objects and standard ways of making inferences), then you have accepted

1) .999... is shorthand for the limit of the sequence of numbers a_n = sum_{i=1}^n 9*10^{-n}.

2) Using a simple (finite!) formula, a_n = (10^n - 1)*10^{-n}.

3) The limit of that sequence is 1.

4) When the limit of a sequence exists, it is unique, which implies the equality of .999... and 1. Uniqueness of limits also trips up many a Calculus student.

So, you might well be refusing to work in the real number system. That is fine. When I say real number system, I mean the set of equivalence classes of Cauchy sequences of rational numbers or anything known to be equivalent. This is what has been called "the real number system" for a long long time, and if you want to either dismiss that nomenclature, or refuse to work with that system, or not accept methods of construction from a philosophical point of view, that is fine. The point is that once you have accepted such constructions, these are theorems that have NO logical flaws in the proof. That is right, you heard me, ZERO flaws. Pure acceptance by the mathematical community (again, assuming you aren't opposed to standard ways of constructing mathematical objects or standard forms of reasoning). 

You might also be rejecting the standard notion of decimal representation. That is fine also.  Just keep in mind that some form of limit of partial sums is what you will find in standard real analysis textbooks. These definitions have been around for years and accepted for years. You want to insist on a different convention, fine. That's your business.

Perhaps you favor an infinitesimal system, something like the hyperreals. Fine, your .999... is not equal to 1 statement might make sense there. But, within the real number system, it is a genuine theorem.  No flaws. I'll joke about the reality of the objects we reason about, but once we set down the definitions, theorems are theorems.  DEFINITION-THEOREM-PROOF!!! If mathematics is nothing else, it is the definition-theorem-proof game. If nothing else, we can verify that the statements .333...=1/3 and .999...=1 can be derived in accordance to the rules of this game. Please don't take that away from us.

Pfew.  I'm glad you're back Hugh.

Now, help me figure out how to make an infinite WarGear board based on Hilbert's Hotel.

He's got my vote for the most air-tight forum post of all time. /highfive

Hugh for President (of the AMS)

Lets get a board up about molecular genetics and protein chemistry, for it is those topics which are interesting.  Math is a bore.  : )

I have a solution. We'll change to a base-3 "tricimal" system - then 1/3 = 0.1, 2/3 = 0.2, and 3/3 = 1. When we need to express other "x-imal" values, we'll just switch bases on the fly.

It's foolproof!

Oatworm wrote:

I have a solution. We'll change to a base-3 "tricimal" system - then 1/3 = 0.1, 2/3 = 0.2, and 3/3 = 1. When we need to express other "x-imal" values, we'll just switch bases on the fly.

It's foolproof!

I... can't find anything wrong with this line of reasoning...

Did you know the human head weight 8 lbs?

Weight?  Who's the moron that can't spell late at night?

Math logic and common sense logic are at odds within my brain right now...

Something I never understood, is Ln still in base 10? i thought its base was e.  And also I just wanted to say that I know how to augment complex matrices so if anybody needs help with augmenting matrices I can augment your matrix.

From Russia with love, bullet holes, arsenic toxicity, hypothermia and blunt force trauma to the head

The Real Rasputin

Evan

Yay for Hugh! Well said.

RealRasputin wrote:

Something I never understood, is Ln still in base 10? i thought its base was e.

I think the confusion here may be that "base" has more than one meaning.  The natural logarithm, usually denoted by ln, is the logarithm base e in that it is the inverse function of the natural exponential function, e^x. Similarly, the logarithm base 10, usually denoted simply log or log₁₀, is the inverse function of the exponential function 10^x.  That said, when doing computations and calculations, we usually express numbers like ln(2) in base 10 (or decimal, note the somewhat different usage of the word "base") as ln(2) = 0.693147… as we are using 10 distinct digits to represent the number.

haha, this is fun... i'm going to go find more unsolvable math problems that show math is make believe!