1 = .9999.....

[yuno]

you dont even need calculus to prove this.i teach this in my algebra classes.we have to first agree that "0.999..." is in fact a real number (this makes sense,since 0.999... is indeed a repeating decimal),and then all we need are the axioms of equality.

proof:let x=0.999...
=> 10(x)=10(0.999...) (by multiplication property of equality or MPE)
=> 10x=9.999...
=> 10x-0.999...=9.999...-0.999... (by addition property of equality or APE)
=> 10x-x=9.999...-0.999... (here we replaced the 0.999... on the left side by x,since they are equal)
=> 9x=9
=> x=1 (again,by MPE)
now,by transitivity of equality,since x is equal to both 0.999... and 1,then 0.999...=1.

QUOD ERAT DEMONSTRANDUM.

a few remarks:in this proof,please note that the addition and multiplication properties of equality only require that we add/multiply real numbers,regardless of whether these real numbers are terminating,infinitely repeating or non-terminating,non-repeating decimals.

and i have made this thread just a tiny bit boring.

[Raistlin]

Math is fun, not boring. Also, yeah, there are algebraic proofs which were discussed earlier this thread (and all the previous threads on this), and have the benefit of being much easier to understand for a layperson. The calculus proof just allows you to demonstrate how an infinite string of 9's after a decimal does, in fact, equal 1.

This debate was apparently so ubiquitous on the WoW forums that Blizzard even made an announcement a few years ago on its website that .999... does, in fact, equal 1.

[Tavrobel]

QUOD ERAT DEMONSTRANDUM.

By multiplying .9... by 10, that would give it an end with 0, even though you can't technically get there, the resulting number is no longer properly infinite.

This debate was apparently so ubiquitous on the WoW forums that Blizzard even made an announcement a few years ago on its website that .999... does, in fact, equal 1.

It didn't even take WoW for Blizzard fans to go nuts with it; it's been around on the WCIII and DII boards long before WoW was released in 2004. Sometime shortly after the release of Frozen Throne, one of their April Fools jokes was replacing the whole of the website with a giant plaster stating that .999... = 1.

[I Took the Red Pill]

EDIT 2:

According to Arithmetic: 0/0 = 1, 0, undefined (infinity) too...

Only one of those is right (undefined). And undefined does not equate to infinity, it means just what it says: undefined.

No, it's indeterminate. Quite different from being undefined. If you define division in terms of multiplication (18/3=6 because 6x3=18), and then look at something like 3/0, you can clearly see that we cannot define an answer for this because there is absolutely no number that, when multiplied by zero, will result in the number three. Hence the term undefined. 0/0, however, is different. Let's look at it in terms of multiplication then. Is the answer zero? Well, 0x0=0, so that would certainly work. But what about the identity that anything divided by itself is 1? Would 1 work here? Well, 1x0=0, so yes. It now becomes clear that the answer works for all real numbers. 0/0 could be equal to sqrt(2), because sqrt(2)x0=0. Whereas an expression being undefined means we literally cannot define an answer with our current definition of multiplication, the indeterminate form is quite in its own category and becomes integral (pun intended) in laying the foundations for differential calculus.

Yeah I'm not here to jump into the whole .999...=1 argument, because anyone denying the fact is, well, thickheaded and wrong. Just here to clear up some mathematical jargon.

[Raistlin]

By multiplying .9... by 10, that would give it an end with 0, even though you can't technically get there, the resulting number is no longer properly infinite.

No, the algebraic proof is valid. If the nines are infinite, where's the end that you'd stick the 0? Infinity minus one is still infinity. The end result would simply be moving the decimal point over so it'd be 9.999... (still goes onto infinity).

[Tavrobel]

No, the algebraic proof is valid. If the nines are infinite, where's the end that you'd stick the 0? Infinity minus one is still infinity. The end result would simply be moving the decimal point over so it'd be 9.999... (still goes onto infinity).

Well, as it turns out, Blizzard said it to be true, so therefore, I must believe to also be true.

In case you cared, their .999... = 1 spiel was 2004, several months before WoW was released.

[Raistlin]

Yeah, I saw it in your previous post. I just didn't see it until after WoW was released. Regardless of when it was, that action shows it was a hot topic of debate.

Also, you all fail for not noticing my calculus error earlier in the thread, which I just caught. I had lim (x->0) of (1/x) = 0 (when it is actually infinity), when it should be lim (x->infinity) of (1/x) = 0.

[Tavrobel]

Woah man, math and computer notation don't mix, okay?

And I figured that you had seen my comment, but not known the precise time when it occurred. I've been trying to find a screenshot of the whole spiel, but to no avail. I believe that it had Kel'Thuzad's picture as a background for the text, but I can't be precisely sure.

[Darkswordofchaos]

not in math but in real life there is no such thing as .999.... cause you cant have a never ending part of somthing in the real world so it has to be that .999... equals 1 .666666..... =2/3 and .3333333 equalls 1/3 simple as that

[eestlinc]

we want to believe that .9999... < 1 because there is that receding tiny difference that goes on and on. But there is only a difference if we stop and assess at a specific number of 9s. Once we go on infinitely, there is no difference and .9999... = 1 because you never stop at any set of 9s.

[Ultima Shadow]

Infinity/infinity = infinity

infinity - infinity = infinity

These two can't be evaluated, actually. The problem with using infinity interchangeably with large numbers is that it doesn't work. You can;'t use it like that.

Well, let's say you have an infinite line of red and yellow sticks (yes, sticks. Because sticks are cool). Now, take away all the yellow ones. Since the amount of coloured sticks total is infinite, there's an infinite amount of yellow sticks and also an infinite amount of red sticks. The amount you take away (all the yellow ones), is infinite, yet what remains (the red ones) is also infinite. There, just took away an infinite from an infinity, yet infinity remains. In other words, infinity - infinity = infinity is 100% possible.

As for dividing, it's exactly the same. There's just an infinite amount of "different coloured sticks" that you divide into an infinite amount of infinite lines.

If you do it that way, it's possible.

[I Took the Red Pill]

Infinity/infinity = infinity

infinity - infinity = infinity

These two can't be evaluated, actually. The problem with using infinity interchangeably with large numbers is that it doesn't work. You can;'t use it like that.

Well, let's say you have an infinite line of red and yellow sticks (yes, sticks. Because sticks are cool). Now, take away all the yellow ones. Since the amount of coloured sticks total is infinite, there's an infinite amount of yellow sticks and also an infinite amount of red sticks. The amount you take away (all the yellow ones), is infinite, yet what remains (the red ones) is also infinite. There, just took away an infinite from an infinity, yet infinity remains. In other words, infinity - infinity = infinity is 100% possible.

As for dividing, it's exactly the same. There's just an infinite amount of "different coloured sticks" that you divide into an infinite amount of infinite lines.

If you do it that way, it's possible.

No. It's undefined. Lrn2Cardinality. The problem arises because there can be different "kinds" of infinities. For example, are there more whole numbers or perfect squares? The answer, to both, is infinite, even though that would defy intuition; there has to more whole numbers than perfect squares, right? Nah, they're just different infinities. That's one reason why ∞ - ∞ is undefined. Not to mention the fact that infinity is not a number and doing operations on it is silly.

If you want something more concrete and with an actual example, check this out.

edit: what you're saying is basically the same as the example in the link I posted.

Assume ∞ - ∞ = 0

Add infinity to both sides: (∞ + ∞) - ∞ = 0+ ∞

Since infinity plus infinity is infinity, and 0 plus infinity is infinity, you get

∞ - ∞ = ∞

In the same way that you can assume that ∞ - ∞ = 0, add pi to both sides, getting (∞ + pi) -∞ = pi, and since ∞ + pi = infinity, you get ∞ - ∞ = pi.

It all has to do with the Cardinality of infinites.

[KentaRawr!]

Hey guys, what do you call a bird that doesn't eat?

A polynomial.

Hahahaha.

[Ultima Shadow]

No. It's undefined. Lrn2Cardinality. The problem arises because there can be different "kinds" of infinities. For example, are there more whole numbers or perfect squares? The answer, to both, is infinite, even though that would defy intuition; there has to more whole numbers than perfect squares, right? Nah, they're just different infinities. That's one reason why ∞ - ∞ is undefined. Not to mention the fact that infinity is not a number and doing operations on it is silly.

If you want something more concrete and with an actual example, check this out.

edit: what you're saying is basically the same as the example in the link I posted.

Assume ∞ - ∞ = 0

Add infinity to both sides: (∞ + ∞) - ∞ = 0+ ∞

Since infinity plus infinity is infinity, and 0 plus infinity is infinity, you get

∞ - ∞ = ∞

In the same way that you can assume that ∞ - ∞ = 0, add pi to both sides, getting (∞ + pi) -∞ = pi, and since ∞ + pi = infinity, you get ∞ - ∞ = pi.

It all has to do with the Cardinality of infinites.

Oh, what the heck. Fine then. >_>

[yuno]

infinity minus infinity is not undefined,it is indeterminate.more specifically,it is what we mathematicians call an indeterminate form.indeterminate is very much different from undefined.undefined means we cannot assign any real number to be the answer.on the other hand,indeterminate means the answer may still be any number;we just are not sure yet.

consider the following:the expression x+1 blows up to infinity as x goes to infinity.the expression x+2 blows up to infinity as well if we let x go to infinity.therefore,the expression (x+2)-(x+1) represents infinity minus infinity as x approaches infinity.but here,we know that (x+2)-(x+1)=1,so here,even if x approaches infinity,infinity minus infinity is equal to 1.

next,consider the following two expressions x+1 and x+3.again,both blow up to infinity as x approaches infinity.so (x+3)-(x+1) represents infinity minus infinity again.but now,infinity minus infinity is 2.

another!how about 2x+1 and x+1?again,both approach infinity as x approaches infinity,so this is also infinity minus infinity.but (2x+1)-(x+1) is x,so here,infinity minus infinity is infinity!zomgwtfbbq!

to put it simply,indeterminate means the answer cannot be determined yet.as i tell my students,indeterminate means we are not yet done.

Math is fun, not boring.

i absolutely (not conditionally I cried aloud with mirth and merriment) agree.i cant say much about other people though.