0.999~ = 1

[newfangle]

Will the teacher ever give out a mark of 3.49999999~?

 

[Hitro]

He might give a 3.5...

 

[newfangle]

Can we then assume that 3.5 = 3.49999999999~ and award the student the lower mark?

 

[Hitro]

I think the 3.5 gives the higher mark.

 

[newfangle]

I would give the student the lower one because he wasted my time.

 

[Hitro]

Well, he didn't have a 3.4999~ anyway, he had a "3.4999 whatever" (or in fact complained about the possibility of getting only Silver if he had it), and for that he deserves no Gold.

 

[pboily]

@MTSK: so what did you get anyway, the Gold or the Silver?

 

[MSTK]

I got silver

 

[general_kill]

i'm sure most people have heard of the theory that 1=0.999 because:
(...)=inifinite repeating unit
let x = 0.999...

10x = 9.999...

10x-x=9.999...-0.999...

9x=9

x=1

well i was just wondering if someone can explain to me why something that seems so wrong can be proven right. Is it because of a flaw in the system of proof that allow loopholes in cases of infinity? or is a flaw in the decimal system? or is it that we do not understand the theory of infinity? This post was inspired by my math class today which really made me start thinking.

 

[North King]

I believe it is because of the fact that mathmatics cannot handle infinity. .9 repeating is infinitely close to 1, but not 1. It is just that our mathematics has no real way to express infinity, or infinite closeness, so it comes with the extremely close approximation .9 repeating = 1.

 

[Sanaz]

Round-off error. You forgot about that last "9" at the far right side when you subtracted, the one so far away it looked infinitely small but was still there. This is not a proof, it's just a game. I'm sure one of the mathematicians here can explain the difference, I sure don't know. And for all purposes outside of theory, 0.999999... does equal 1. I'm an engineer, not a mathematician.

 

[general_kill]

@sanaz

well 10x=9.999...(infinity)

10-x=9.999...(infinity)-0.999...(infinity)

therefore there is no last 9 to consider since .999 is infinite

 

[pboily]

@sanaz

well 10x=9.999...(infinity)

10-x=9.999...(infinity)-0.999...(infinity)

therefore there is no last 9 to consider since .999 is infinite

There is about 100 posts on the topic in another thread... I'll try to find it again.

EDIT: And I can't: it seems to be gone past the point of no return. The problem is dealing with infinity. We tend to use it like a "formal/normal" number and then it sends our intuition packing: infinity is no number... one nice way to explain it is to define real number as equivalence classes of Cauchy sequences. Then 1 and 0.99999999......... are the same real number as they live in the same classes.

 

[Hundegesicht]

Multiplication and devision by infinity is not defined. The reason 0.999999... = 1 is because 0.9999999... isn't a real number. In real life, there'll always be a smallest part, even if it's down to the last quark. therefore there's always a "last 9" in reality. Mathematics (which is only a way of trying to explain reality, not reality itself) doesn't have any way to deal with infinitely small numbers, mainly because they don't exist. Since the number in question isn't real, it cannot be treated as a real number and therefore there is either a place were it stops at 8 or it is actually 1. If you were to try to make it stop at 8 somewhere, how would you judge where without being arbitrary? 1, on the other hand, is always static.

It needs no explaination beyond that simply because it isn't a real number and unreal numbers, which are inheriently alogical, cannot be defined logically.

 

[pboily]

The reason 0.999999... = 1 is because 0.9999999... isn't a real number.

How could two things be equal if one of them is real and the other is not?

 

[Hundegesicht]

Since 0.9999... isn't real, in math you'll just have to substitute for the closest real number. In this case, 1. You'll never, ever experience 0.99999... in real life so there's no point in trying to define it or how it'd interact with other numbers. You simply have to take it for what was it is, as there's no way of understanding it within the confines of mathematics.

 

[microbe]

x=1

well i was just wondering if someone can explain to me why something that seems so wrong can be proven right.

This isn't wrong. You just proved it.

 

[The Last Conformist]

It should be fairly easy to convince oneself that every decimal of y=1-0.999... is zero. The only number which has zero for all digits is zero. Any two numbers with a difference of zero are the same.

 

[Duke of Marlbrough]

There is about 100 posts on the topic in another thread... I'll try to find it again.

EDIT: And I can't: it seems to be gone past the point of no return. The problem is dealing with infinity. We tend to use it like a "formal/normal" number and then it sends our intuition packing: infinity is no number... one nice way to explain it is to define real number as equivalence classes of Cauchy sequences. Then 1 and 0.99999999......... are the same real number as they live in the same classes.

http://forums.civfanatics.com/showthread.php?t=82933

 

[The Last Conformist]

Thanks, Duke!