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off topic: 0.999 repeating Equals 1, proof inside!

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Z28Fistergod

Mon Dec 07 2009, 02:07AM

Registered Member #2040 Joined: Fri Mar 20 2009, 10:13PM
Location: Fairfax VA
Posts: 180

I definately like that second proof much better.

Per wikipedia the first proof relies on 1/3 being equal to 0.333...
They arrive at this conclusion based on division

At one significant figures: 1/3 = 0.3
At two significant figures: 1/3 = 0.33
At three significant figures: 1/3 = 0.333

They reason that this will go on forever. I don't disagree, but it doesn't seem very rigorous to me. It seems there should be some type of mathematical induction to prove that it does go on infinitely.

But the second one looks pretty airtight. I wouldn't know how to cause trouble with that one.

Mattski

Mon Dec 07 2009, 04:25AM

Registered Member #1792 Joined: Fri Oct 31 2008, 08:12PM
Location: University of California
Posts: 527

Engineer's "Proof" #1: It is clear from inspection that 0.9999... will be incredibly close to 1, therefore they are identical for most intents and purposes.

Engineer's "Proof" #2: The mathematicians who care about these things seem to be satisfied thet the above relation is proven, therefore it probably is.

Both are joking, but it raises the question of how certain you want to be in your understanding.

I think it is also helpful to look at it in another base, where 1/3 is not a repeating decimal, base 3. There you have 1/3 (decimal) = 1/10 (ternary) = 0.1 (ternary). Then 0.1 * 10 (3 in ternary) = 1.0 (ternary and decimal) just as we expect.

Now if we say that 0.333... (the decimal expansion of 1/3) is not exactly equal to 1/3, then we are saying one of two things: (1) There is another, exact, way of representing 1/3 in decimal, or (2) there exist real numbers which cannot be represented in the decimal system. It is pretty easy to accept that the first is false. If we also accept that the second point is false (it's a bit more general but seems convincing to me), then 1/3 must be exactly 0.33333... and 1 must be exactly 0.99999...

Disclaimer: Not a proof but it convinced me.

Turkey9

Mon Dec 07 2009, 05:07AM

Registered Member #1451 Joined: Wed Apr 23 2008, 03:48AM
Location: Boulder, Co
Posts: 661

It may not be completely accurate, but I think that the reason the 1/3 proof is good is that it demonstrates a math concept that is counter-intuitive. You can argue that 1/3 doesn't equal .333 repeating, but for most people they see 1/3 as equalling .333 repeating because we were taught that in school. So by the same logic .999 repeating does equal 1 and that makes you scratch your head and go, "Huh, that's cool." I like it more as an interesting thought, not a proven fact. But the second proof is great as well.

HV Enthusiast

Mon Dec 07 2009, 01:56PM

Registered Member #15 Joined: Thu Feb 02 2006, 01:11PM
Location:
Posts: 3068

So in otherwords . . .

1Million * 0.999... still equals 1Million

As an engineer, I'm only concerned about 3 decimal places period!

Avalanche

Mon Dec 07 2009, 06:05PM

Registered Member #103 Joined: Thu Feb 09 2006, 08:16PM
Location: Derby, UK
Posts: 845

This is nuts,

it's no more strange than saying one third is equal to .333, except the difference between 1 and .999 is more likely to grab your attention!

Both are equally un-true

Having said that, 3 phases are never spaced exactly 120* apart in software I've written

Dr. Slack

Mon Dec 07 2009, 07:55PM

Registered Member #72 Joined: Thu Feb 09 2006, 08:29AM
Location: UK St. Albans
Posts: 1659

The main problem here is that this is about *real* numbers. When mathematicians talk about real numbers, they include the fact that they are defined in a very specific way, involving infinities and Cauchy series and all sorts of stuff that you don't need if you are dealing with integers, rational fractions, or the sorts of numbers that we use every day as engineers, or check-out persons for that matter.

The nicest demonstration of 0.999... == 1 is DaJ's second one, because when the mathemagican pulls the rabbit from the hat, you saw him putting it in there moments earlier, thus ...

let b=0.999... (shows hat)

then c=10*a=9.999... (quickly shoves rabbit into hat)

... because in left-shifting the digits to multiply by 10, we still have exactly the same number of 9s after the decimal point, because there are an infinite number. Infinity + 1 = infinity. To use the most fundamental method of counting, each 9 after the decimal point of b can be put into a 1:1 correspondance with each in c.

now c-b =9 exactly, as all the 9s to the right cancel (grasps ears and starts to withdraw rabbit)

and c-b = 10*b - b = 9b = 9, so b=1. (viola, regard le lapin!)

For proofs which involve 1/3rd, there is no technical difference between whether 0.333... and 0.999... are equal to 1/3rd and 1, however, it is common that most non-mathematicians accept the first and reject the second. I think the reason is that we tend to want to round a finite length approximation. So if we want to write 1/3rd to 10, or 100 decimal places, we stop at the last 3, and so implicitly round down. However, if we want to do the same with 0.999..., then there is that nagging feeling that we ought to be rounding up at the end, and that's going to ripple back through all the digits, which doesn't feel quite right.

Steve Conner

Mon Dec 07 2009, 08:00PM

Registered Member #30 Joined: Fri Feb 03 2006, 10:52AM
Location: Glasgow, Scotland
Posts: 6706

wrote ...
I state this difference after the first few months of Calculus class in which limits were shoved into our heads and to never think of being infinitely close and there as the same thing

Not "never". Only in certain cases, when you're taking limits of some functions using calculus.

One classic example is lim (x>0) of sin(x)/x. That's undefined (0/0) at the origin, but everyone takes it to be 1, because that's the limit as x tends to 0. Proof: For small angles in radians, sin(x) is roughly equal to x, and x/x is of course 1.

A lecturer on a recent DSP course I attended put this in a very down-to-earth way: "0/0 equals whatever you want." He might have added: Only if you're a math professor. If you're a math student, it equals whatever your professor wants.

But in the case of 0.999... just as with the paradox of Achilles and the tortoise, infinitely close *is* the same as there.

HV Enthusiast

Tue Dec 08 2009, 01:40AM

Registered Member #15 Joined: Thu Feb 02 2006, 01:11PM
Location:
Posts: 3068

Now i'm truly reminded to why i hated math so much at the university.

To be honest, as an EE, the only math i've ever used is algebra, and a derivative and integral every once in awhile.

Bored Chemist

Tue Dec 08 2009, 06:28PM

Registered Member #193 Joined: Fri Feb 17 2006, 07:04AM
Location: sheffield
Posts: 1022

Z28Fistergod wrote ...

I definately like that second proof much better.

Per wikipedia the first proof relies on 1/3 being equal to 0.333...
They arrive at this conclusion based on division

At one significant figures: 1/3 = 0.3
At two significant figures: 1/3 = 0.33
At three significant figures: 1/3 = 0.333

They reason that this will go on forever. I don't disagree, but it doesn't seem very rigorous to me. It seems there should be some type of mathematical induction to prove that it does go on infinitely.

But the second one looks pretty airtight. I wouldn't know how to cause trouble with that one.

The truth of the statement neither knows nor cares what your opinion of the proof might be.
The first proof weas valid. Any further proof, no matter how elegant, is redundant.

Z28Fistergod

Wed Dec 09 2009, 03:02AM

Registered Member #2040 Joined: Fri Mar 20 2009, 10:13PM
Location: Fairfax VA
Posts: 180

Bored Chemist wrote ...

The truth of the statement neither knows nor cares what your opinion of the proof might be.
The first proof weas valid. Any further proof, no matter how elegant, is redundant.

Well I sure hope so. God forbid an attribute be swayed by my opinion. Although I heard those characteristics and properties aren't so rigid; they're pretty impressionable.

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