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0.999...= 1

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Re: 0.999...= 1

#211

Post by dubcanada »

undecisive wrote: post #115

There is no proof that will ever satisfy a person dead-set against this. Ever since I brought this home from school as a child, my whole family ribbed me mercilessly for it. If you tell a person that 3/6 = 1/2, they'll believe you - because they have been taught from an early age that fractions can have multiple "representations" for the same underlying amount. People mistakenly believe that decimal numbers don't hav…

I feel like most people are not answering this, they are giving the proof in a different way.

Answering how to teach a child not to be bound to 100%'s is really hard. I personally would say just let them explore on their own, teaching a person that 0.99999 = 1 results in the same as if you taught them that 0.99999 != 1. You need to teach them that all sciences and maths are changing constantly, what might be a fact today could change tomorrow. You need to teach them that anything can become wrong or right as we progress and to be open about accepting new information, while being hesitant enough not to not succumb to false/fake information.

That's a very hard lesson to learn and an even harder one to practice. But one that I think a lot of people need to learn.

Re: 0.999...= 1

#212

Post by syockit »

iso1631 wrote: post #12
ginko wrote: post #9

I remember being doubtful when being presented with this in middle school, but after being shown this as fractions makes it obvious: 1/3 = 0.333.. 3 * 1/3 = 3 * 0.333.. 3/3 = 0.999.. 1 = 0.999..

Another secondary school 'proof' x = 0.9999..... 10x = 9.9999..... (10x -x) = 9x = (9.9999.... - 0.9999....) = 9 x = 9/9 = 1

The difficult part is to convince that (9.9999.... - 0.9999....) = 9. Someone else mentioned about Zeno's paradox involving Achilles and the tortoise. Since you multiplied x by 10 to get 9.9999...., you know that it's always going to be 1 decimal place slower in pace to reach convergence compared to 0.9999.... What you might get instead is (9.9999.... - 0.9999....) = 8.9999....

Re: 0.999...= 1

#213

Post by OscarCunningham »

hilbertseries wrote: post #177
bognition wrote: post #168

Earlier quoted context omitted.

No .666666 is not equal to .6700000 0.666... is equal to 0.666...7

Not sure if you’re joking, but 0.666...7 is not a real number. Can you define it?

It's unfair to ask 'Can you define it?' to someone not educated in real analysis because they don't know which definitions you'll accept and which you won't. They already think that '0.666...7' is a valid definition.

Re: 0.999...= 1

#214

Post by parski »

ping_pong wrote: post #160

My 5 year old stumped me with this, and I had to look it up. He asked me why 1/3 + 1/3 + 1/3 = 1, since it's equal to 0.333... + 0.333... + 0.333... which is 0.999... How can that possibly equal 1.000...? And is 0.66... equal to 0.67000...? I didn't have a good enough answer for him, so I had to look it up and found this page. I tried to explain it to him but since I'm a terrible teacher and he's only 5, it was hard…

I asked my math teacher this when I was a kid. He told me to accept that's the way it is so I did.

Re: 0.999...= 1

#215

Post by jonsen »

_bxg1 wrote: post #204
Okx wrote: post #182

Earlier quoted context omitted.

I usually say "If and only if two numbers are different, then you can find a number between them". People often accept this axiom. Then, I offer them to find a number between 0.999... and 1.

Why not 0.00...1?

This is not an infinite decimal. The digit 1 is somewhere out there.

Re: 0.999...= 1

#216

Post by caspper69 »

solotronics wrote: post #202
Okx wrote: post #182

Earlier quoted context omitted.

I usually say "If and only if two numbers are different, then you can find a number between them". People often accept this axiom. Then, I offer them to find a number between 0.999... and 1.

is there any difference between a black hole and nothing? (somewhat joking but I was thinking of a physical analogy of the limit approaching zero)

A black hole has mass. Nothing has no mass.

(also a somewhat joking answer)

Re: 0.999...= 1

#217

Post by amelius »

danharaj wrote: post #179
amelius wrote: post #163

Earlier quoted context omitted.

I think you should look more closely at what I did with the order of operations, and the fact that "<" now is part of the expression acted over by the limit.

It is not reasonable to interpret the notation that way. For one, variable binding becomes incoherent.

I think it is perfectly reasonable for two reasons. One is that (when it works) the notation becomes more intuitive, as in

    0.9999...
And the second reason is that mixing implicit limits (or series or infinity) and numbers is a mathematical hack, and when the notation doesn't work it is a clear indication that something is wrong.

Re: 0.999...= 1

#218

Post by CogitoCogito »

dahart wrote: post #210
CogitoCogito wrote: post #193

Earlier quoted context omitted.

> You can. The proof still works if you do that. > What you’re refusing to accept here is the definition of infinity. I'm refusing to accept the definition of infinity? I have no idea what you mean by that. Would you make the same statement were you aware that I do in fact have a PhD in mathematics in the field of analysis? That I have in fact studied logic? Just as a hypothetical scenario.

The proof hinges on the definition of infinity. 0.9bar7 is a completely nonsensical number precisely because of the definition of infinity. Yes, I think you’re failing to accept the definition of infinity. You’re rejecting a proof that many other PhDs in math accept along with some notable mathematicians like Euler. I reject your appeal to authority; having an advanced degree doesn’t mean you’re somehow automatically…

> The proof hinges on the definition of infinity. 0.9bar7 is a completely nonsensical number precisely because of the definition of infinity. Yes, I think you’re failing to accept the definition of infinity. You’re rejecting a proof that many other PhDs in math accept along with some notable mathematicians like Euler. I reject your appeal to authority; having an advanced degree doesn’t mean you’re somehow automatically right, lots of PhDs in math have failed the Monty Hall problem, lots of people with advanced degrees are wrong all the time. I myself am a walking example on occasion.

I wasn't appealing to authority. I never said that implied that I was correct. (Had I wanted to do that, I would have brought that up much earlier in this comment thread.) I was really just curious if you still believed that I was "failing to accept the definition of infinity" even if you knew that about me. Apparently you do.

> I haven’t heard a reason yet why the logic of the proof doesn’t work, I’ve only heard that you don’t accept it. What is the reason it doesn’t work?

https://news.ycombinator.com/item?id=23007600

Re: 0.999...= 1

#219

Post by klodolph »

hilbertseries wrote: post #177
bognition wrote: post #168

Earlier quoted context omitted.

No .666666 is not equal to .6700000 0.666... is equal to 0.666...7

Not sure if you’re joking, but 0.666...7 is not a real number. Can you define it?

You would define it as a digit sequence, using ω + 1 as the indexing set. I would consider it to be most naturally an element of the hyperreal numbers, although it is also contained in smaller extensions of the real numbers.

Re: 0.999...= 1

#220

Post by 2OEH8eoCRo0 »

Yes it does but it seems like a less correct way of writing it. Like you could represent the number 10 as 10/1 (ten over one) but why would you? Why would you represent 1 as .9 repeated?
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