Proving 0.999... Is Equal To 1

eldavojohn writes "Some of the juiciest parts of mathematics are the really simple statements that cause one to immediately pause and exclaim 'that can't be right!' But a recent 28 page paper in The Montana Mathematics Enthusiast (PDF) spends a great deal of time fielding questions by researchers who have explored this in depth and this seemingly impossibility is further explored in a brief history by Dev Gualtieri who presents the digit manipulation proof: Let a = 0.999... then we can multiply both sides by ten yielding 10a = 9.999... then subtracting a (which is 0.999...) from both sides we get 10a — a = 9.999... — 0.999... which reduces to 9a = 9 and thus a = 1. Mathematicians as far back as Euler have used various means to prove 0.999... = 1."

Comments

(0.999...)st Post!

[Anonymous Coward]

(0.999...)st Post!

Re:(0.999...)st Post!

[derrida]

2+2=5 for sufficiently big values of 2.

or for sufficiently small values of 5.

Time to Update my SLA

[Anonymous Coward]

Now I can replace my SLA with 100% uptime.

Re:Finally

[Vectormatic]

just as long as no-one proves 0 = 1 we computerpeople are safe...

Cribbed, Since My Memory for Jokes Sucks

[Anne_Nonymous]

In the high school gym, all the girls in the class were lined up against one wall, and all the boys against the opposite wall. Then, every ten seconds, they walked toward each other until they were half the previous distance apart. A mathematician, a physicist, and an engineer were asked, "When will the girls and boys meet?"

The mathematician said: "Never."
The physicist said: "In an infinite amount of time."
The engineer said: "Well... in about two minutes, they'll be close enough for all practical purposes."

Re:When you add/subtract/multiply/divide infinite

[BlackPignouf]

Wrong, wrong and wrong.

First off, you're not talking about sets, but separate finite numbers.

Then, infinity is neither rational nor irrational.

Then, all numbers that have "infinite repeating decimals" are rational. See : http://en.wikipedia.org/wiki/Rational_number

The decimal expansion of a rational number always either terminates after finitely many digits or begins to repeat the same finite sequence of digits over and over. Moreover, any repeating or terminating decimal represents a rational number. These statements hold true not just for base 10, but also for binary, hexadecimal, or any other integer base.

So that means 0.999999..... is rational. Which rational you ask? Why! 9/9 :D

Finally, if you say 0.99999999..... is less than 1 : what is the difference between both?
We know it's less than any positive epsilon (0.1, 0.01, or 0.00000.....00001).
Which means it's nil.
There's no place for a single mosquito fart between 0.999999... and 1.

Corrected, Since My Memory for Jokes Sucks

[Anne_Nonymous]

In the high school gym, all the girls in the class were lined up against one wall, and all the boys against the opposite wall. Then, every ten seconds, they walked toward each other until they were half the previous distance apart. A mathematician, a physicist, and an engineer were asked, "When will the girls and boys meet?"

The mathematician said: "Never."
The physicist said: "Eventually, they will come to a point where they would be required to move less than 1.616252(81)×1035 meters closer together. From the uncertainty principle, we know we cannot measure position more accurately than that. So either they will not move at all, or they will superimpose at that point."
The engineer said: "Well... in about two minutes, they'll be close enough for all practical purposes."

Re:Cat and Mouse

[blueg3]

An infinite number of mathematicians walk into a bar. The first one orders a beer. The second one orders a half a beer. The third orders a quarter of a beer. The bartender says, "You're all idiots," and pours two beers.

Re:This is second place

[Chapter80]

I remember being told this in highschool. There was much objection, but the teacher shut us up by simply saying "give me a number in between them."

Duh. 0.9999... and a half!