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0.999...= 1

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Re: 0.999...= 1

#541

Post by naikrovek »

2OEH8eoCRo0 wrote: post #221
gspr wrote: post #162

Earlier quoted context omitted.

> There is no proof that will ever satisfy a person dead-set against this. Indeed. I've torn my hair out trying to convince smart people with PhDs in hard sciences and had to give up in frustration. I usually find that the most success can be had by kicking the ball to them immediately and having them define what they actually mean when they say "0.999…". If we're going to debate whether that thing equals another thi…

Ask for a number between .9 repeated and 1

I don't understand how they're the same number.

I will never accept that they are the same. The difference between 0.9 repeating infinitely and 1 is infinitely small, but it isn't zero.

Re: 0.999...= 1

#542

Post by ramshorns »

tom-thistime wrote: post #248
ramshorns wrote: post #222

Earlier quoted context omitted.

Well, a limit of a series is just a number and doesn't approach anything either. If the series approaches something, we say the limit exists and is equal to that thing. Anyway, in the surreal numbers you could probably make up a notation where 0.999... actually denotes 1 - ε or something. But I daresay it might not be very useful because then how do you denote 1 - ε/2 or anything else.

We've defined the series we're talking about, and we've defined 0.99... as the limit of that series. I don't know whether repeating decimals are useful for testing equality of surreal numbers. All I'm trying to say is we're all talking about anything and everything except the definition "when are two real numbers equal?" And then we're saying people who don't understand the consequences of that definition are kind of…

Fair. Maybe the definition is that two real numbers are equal iff there is no real number between them. Which is nonintuitive if you're used to things like natural numbers which have successors. But if you teach people this way of thinking about the real numbers, then arguments like "it's the last number riiight before 1" stop working.

Re: 0.999...= 1

#543

Post by brlewis »

hansvm wrote: post #530
brlewis wrote: post #408

Earlier quoted context omitted.

Whatever base you use, it will only be useful for rationals. You can't use any base to significantly improve the representation of pi. In base 7, 3.1 would be a nice approximation, but you can't go beyond approximations.

It doesn't have a terrible representation in base pi ;)

;) As someone fond of integers, I can't advocate base pi.

Re: 0.999...= 1

#544

Post by fluganator »

Can someone help me out here with least upper bounds?

Generally the proofs of .9...=1 rely on the fact there is no number that exists that can be between .9.. and 1 and therefore .9... is equal to 1.

.9... is the least upper bounds of the set. My question is if .9... was removed from the set what would be the new least upper bounds. Another way of asking the question is if we define it in this context doesn't any set bounded by a real number have a least upper bounds and aren't all real numbers equal to each other?

Thanks!

Re: 0.999...= 1

#545

Post by austincheney »

ISO-morphism wrote: post #527
austincheney wrote: post #519

Earlier quoted context omitted.

Proof: bring up removing time zones, such that there is only one time zone. The responses are comical, even here on HN, such as not being able to wake up or not knowing when morning is or when meals are. Let’s not forget that time zones are a human invention less than 200 years old.

200 years ago we didn't need timezones because we couldn't communicate or travel fast enough or often enough for them to matter. With railroads and telegraphs the level of granularity required for "some time 300 miles away" went from days to hours and minutes. No one in the U.S. west wanted to be told "The sun is at its zenith around 8 a.m. for you." For most people talking about time in their day to day lives it's f…

> 200 years ago we didn't need timezones because...

We still do not. China and India are examples of large geographies spanning across a vast amount of longitude and yet each are a single time zone. Time zones are a political entity only, an unnecessary complexity. The absence of time zones will not halt business or communication over long distances.

> For most people talking about time in their day to day lives it's far more useful to communicate a relative time of day

People have done this for thousands of years without modern chronometers. Examples: dusk, dawn, morning, midday, afternoon, evening, twilight.

Re: 0.999...= 1

#546

Post by brlewis »

SkyBelow wrote: post #512
gspr wrote: post #345

Earlier quoted context omitted.

It isn't.

Or is it? Say I'm a layman and I decide that in the system of math as I understand it, 0.000... is larger than 0. Yes, if I was going to be completely form with my own system of math I would eventually have to face the problems this introduces and resolve it, but until then I can generally adopt a self contradictory system and continue to live my life unaffected. Much like many people live their whole lives using nai…

Then in your system of math 0.999... is also less than 1.

However, basic arithmetic taught to children requires that adding trailing zeros does not change the value of a number. You'll have a hard time doing arithmetic once you change that assumption.

Re: 0.999...= 1

#547

Post by austincheney »

bmn__ wrote: post #523
austincheney wrote: post #519

Earlier quoted context omitted.

Proof: bring up removing time zones, such that there is only one time zone. The responses are comical, even here on HN, such as not being able to wake up or not knowing when morning is or when meals are. Let’s not forget that time zones are a human invention less than 200 years old.

> Let’s not forget that time zones are a human invention less than 200 years old. What's the point of that statement? Before the introduction of formal time zones, we had thousands of informal ones, one for each settlement, calibrating noon to the zenith of the sun.

The point is to indicate that humans knew how to wake up, eat food, communicate, and conduct business (even over long distances) before they had clocks much less the relatively recently invented time zones.

Re: 0.999...= 1

#548

Post by gowld »

naikrovek wrote: post #541
2OEH8eoCRo0 wrote: post #221

Earlier quoted context omitted.

Ask for a number between .9 repeated and 1

I don't understand how they're the same number. I will never accept that they are the same. The difference between 0.9 repeating infinitely and 1 is infinitely small, but it isn't zero.

What is an "infinitely small" number?

Is 9999..... the same as infinity?

What is 1.0 - 0.99999.... = ?

What does it mean to say X is a number, if you can't subtract it from another number and get a number as an answer?

Re: 0.999...= 1

#549

Post by gowld »

burmer wrote: post #450
2OEH8eoCRo0 wrote: post #221

Earlier quoted context omitted.

Ask for a number between .9 repeated and 1

How about, ask for an integer between 1 and 2. Can't think of one? Guess they're the same number then.

http://kasmana.people.cofc.edu/MATHFICT/mfview.php?callnumbe...

http://strangehorizons.com/fiction/the-secret-number/

Bleem

Re: 0.999...= 1

#550

Post by novacole »

cthalupa wrote: post #437
novacole wrote: post #413

So 99.999..% of the speed of light is just the speed of light?

Yes. How much faster than 99.999..% the speed of light would you need to go to get as fast as the speed of light? If your answer is 0.00..1% the speed of light, this answer is nonsensical because infinity never ends, and you never ever have the ability to add that 1 at the end. So, the only answer can be 0.00.., and 0.00.. = 0, so 99.99.. has to = 1.

Hmm, This reminds me a lot like Zenos paradox.

Also, I’d like to understand how if we can say 99.999... is = 100, wouldn’t we also be able to say 99.9999888999... = 99.9999..., and therefore also just 100? And so on?

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