1 = .999?

[LukaP]

Actually I don't think that's a thing either, an infinite number cannot have a "last digit" if you know what I mean. 0.0*1 is not something you can write. 0.9* is simply 1 with no approximation, for the reasons we showed earlier - there is nothing missing in 1/9*9=1, it's 100% true. You'd need to use lim(1*10^infinity), which is quite simply 0.

nono, im not saying it has a last digit at all, thats my point, there is no first or last digit in the decimal notation. hence in decimal one cant write anything accurately without some standards.

 

we all know that 1 in decimal is just how we agreed to write numbers such as 0*01.0* and 0*0.9*.

 

/\

this

 

 

 

@Deniers

Take Calculus, learn math

then come back here and speak.

[Speedyv]

One is already an integer. Why would you turn it into an irrational number?

 

Also when you think about it, there an infinite number of numbers between each integer. 

 

Infinity is an amazing mathematical wonder....

[Sauron]

10 divided by three equals an irrational number

 

The DEFINITION of a rational number is that it can be written as n/m where n and m are integers.

 

Both 10 and 3 are integers. 10/3 is rational.

[Elehat]

it's provable by math, but it doesn't make logical sense like 1/3 recurring does so it's an argument of discussion. what's there to deny.

 

 

the point of limit is that you can never reach it, it never gets there, that's why you don't type = 0 it can't equal that at infinity because there is no such thing as infinity in the world we live in. 

 

Like I said, it makes sense in math, it doesn't make nearly as much sense in life like 1/3 or 2/3 does.

Maths does not simply imagine a place where the limit has been obtained and then tell everyone to believe it. When the proof is done more formally than mentioned by others, a monotonic sequence is defined and then a limit is hypothesised. At this point it's just a hypothetical number, there's nothing intrinsic stopping it from existing. The definition of a limit is a supremum of a set (not its maximum), and doesn't have to 'equal' this hypothetical 0.9999... (which you don't formally define by the way, but it's ok because maths does ) In fact, in topology, a space which has this property is considered very nice and said to be 'compact' (or 'complete', i forget terminology). An alternative proof would be to show that the real topology has this nice feature. The existance of this limit is then formally proven, by showing an existance of a supremum for the set, and then finally we can show 0.99999... actually is this limit. We then show that the limit is also the same thing as 1 and we're done. This is how it is done properly; there's no hand waving and saying 'it's really close so it must be the same', the equality is actually proven. (The whole of calculus also gets this rigorous seeing to, but most scientists don't bother with it and just use the results, because it's a bit boring and mathematicians have already told them the answer).

You say that 0.999999... only equals 1 in this abstract world, but it's only defined in maths, so I don't see the problem. Similarly, 0.333...... is only defined in maths, because nowhere else has arbitrary accuracy. If you say that you can imagine what 1/3 is, you're right, but you can also imagine what 1 is. And to the same extent that 0.999999... doesn't exist outside of maths, neither do 1/3 or 1, because you can only have arbitrary accuracy in maths. 

[Stefanopolis]

Does it look like .99 and 1 are the same thing? No, it's just a cool math trick for 7th grade Pre-Algebra teachers to tell their students. If you give a cashier 99 Cents for something that costs a dollar, you won't be able to buy it. Yes, since there are an infinite amount of Nines you are very very close to 1 but always infinitely far away due to the nature of infinity.

[Kalle Hänninen]

Nor, do I.  My point was simply that it would never equal 1, but how you could do some math that doesn't work to try and get another number.  I do know the math as I am in higher college Math classes, and there are actual theories of how it could though I don't support them as the math doesn't make sense.

I tried to show you without any proper math that no other number than 1 fulfill given definition, so even if it's not perfect, it's sufficient precept.

 

And FYI famous mathematicians have tried it since the dawn of math without any kind of resemblance to solution. Functions themselves give us ways to handle these situations without any contradiction. Math is not pure, it's just consistent. Like getting same meth every time. Who cares if it's cut. At least you get the same results every time.

[TheRandomness]

n = .5

10n = 9.5

 

9n = 9

n = 1

 

1 = .5 ???

 

whyh is it wrong

Because 10*0.5555 isn't 9.5555

same problem as with 0.99999999, you can't write 0.55555555555 as a fraction.

I just saw you corrected yourself nvm.

[TheRandomness]

I hope they fixed that last part because 0.55555555555 as a fraction is 11111111111/20000000000.

Actually;

n = 0.5555

10n = 5.5555

9n = 5

n = 5/9

 

simple.

[Kalle Hänninen]

My answers, yesterday, were more sarcasm/jokes than anything, mostly.  The problem with 1 = 0.999 is that it depends on the applied math.  0.999 is not "auto-equal" to it.  It can be equal to it with applied math, but it can also be rounded or estimated to be 1.  The problem with the last two is that those do not "equal 1", but rather are "approximately 1."  Yes, it's not perfect, but math has to be perfect is the problem there.  So, for the sake of math, making it exactly equal is the only true way.

Maths not perfect. Its been proven long time ago and even has a measure called ohm.

[Essemm]

I feel like half the confusion in this thread is in the notation. The infinite form which is equivalent to 1 I would write as " ". For example here it has been interpreted that 0.99 is proposed to be the same at 1 and has a suitable counter argument as it is obviously incorrect:

Does it look like .99 and 1 are the same thing? No, it's just a cool math trick for 7th grade Pre-Algebra teachers to tell their students. If you give a cashier 99 Cents for something that costs a dollar, you won't be able to buy it. Yes, since there are an infinite amount of Nines you are very very close to 1 but always infinitely far away due to the nature of infinity.

However the recurring form, as correctly stated by many people form is in every way the same as unity in all but how we choose to write it. It's like writing 1/2 instead of 0.5 which is much more acceptable.

[Kalle Hänninen]

Well, it depends on the type of math then I guess.  Or, I might just be a Nazi about it.

We are talking about rationals. They are given. (well not exactly, but are given by the system of numbers) Numbers are indeterminants of your equation. You don't even have to consider them to do in most cases. On other cases you need to be explicit about those possible contradictions, like Riemann hypothesis.

 

[Kalle Hänninen]

If we are just talking about rationals then I can accept that.  But, technically in the sense of math that's not, exactly equal, but rather "can be expressed as."

infinitesimals is a different notion. Useful to have, but is as given as infinite amount of natural numbers.

 

And still, you've done nothing else but objected. Now I've said enough and you need to start producing.

[manikyath]

Nope, it isn't. 1 divided by three equals an irrational number the entirety of which no device can express.

 

.99999999999999 recurring is not 1. However, if you add enough to make it an even one, then you would be adding

0.00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000..........

 

You'll never reach 00000000000000001. Never. And the difference is so insignificant that for all intents and purposes it doesn't exist and has never existed. it may as well be zero. If at some point the number ends, then you can solve for it. There's no problem. It's a quantifiable amount. If it's an irrational number made up solely of zeroes, then the one at the end doesn't exist. But it must! But it doesn't.

 

This is where mathematics and philosophy collide, this is a problem that human nature will never get over because our brains are wired to complete stuff. we're completionists.

But it will never be solved. Deal with it. And be happy about dealing with it, because at the end of the day...

It doesn't really matter that much!

you're entirely 100% right, congrats.

 

the reason math states 1 = 0.999... is because the row of zeroes before the 1 is infinite. the 1 will never come. its there, but it actually isnt.

numberphile has a lot of interesting videos on infinity, that explain why everything containing infinity is so weird.

its like you say "i will go in this direction forever, and then turn right" which.. well.. the turning right is something on the plan, but you will never come to the point of turning right, hence turning right wont happen, and thus 1 = 0.999... because you will never come to a point where the row of zeroes ends, if that made sense

PS: i edited out a small flaw on your end ^^

[Kalle Hänninen]

Here Simon practically proves that glitch primes aren't interesting.

And that's interesting.

[SpaceNugget]

Maths not perfect. Its been proven long time ago and even has a measure called ohm.

wtf are you talking about. an ohm is a measure of resistance. which is the subject of physics and you are just talking out of your butt. there is nothing wrong with math.

 

This thread is full of incredibly stupid replies. 1 does exactly equal 0.99 repeating. end of story.

[KemoKa]

you're entirely 100% right, congrats.

 

the reason math states 1 = 0.999... is because the row of zeroes before the 1 is infinite. the 1 will never come. its there, but it actually isnt.

I feel like this is what they're talking about when the subject of quantum mechanics and theory come up in discussion... It's mind-bending.

 

It is, but it isn't at the same time... but still is.

[Kalle Hänninen]

wtf are you talking about. an ohm is a measure of resistance. which is the subject of physics and you are just talking out of your butt. there is nothing wrong with math.

 

This thread is full of incredibly stupid replies. 1 does exactly equal 0.99 repeating. end of story.

Man you're stupid. Ohm is a fucking symbol, numbfuck.

[manikyath]

I feel like this is what they're talking about when the subject of quantum mechanics and theory come up in discussion... It's mind-bending.

 

It is, but it isn't at the same time... but still is.

infinity is not a concept that exists in classic math, we need to twist on the same roads as quantum mechanics while talking about infinity.

 

one of my favourites is the hilbert hotel.

[Vivilacqua1]

Are you wanting stupid math terms, or good engineering terms?

Engineering is so much better with numbers imo.

[Stefanopolis]

I feel like half the confusion in this thread is in the notation. The infinite form which is equivalent to 1 I would write as " ". For example here it has been interpreted that 0.99 is proposed to be the same at 1 and has a suitable counter argument as it is obviously incorrect:

However the recurring form, as correctly stated by many people form is in every way the same as unity in all but how we choose to write it. It's like writing 1/2 instead of 0.5 which is much more acceptable.

I know that there must be the proper denotation that the number is repeating, I just don't know how to do that on my keyboard.

[.spider.]

1=.9 recurring

Who is it possible to discuss about this so long? The proof is already on page one.

[LukaP]

Maths does not simply imagine a place where the limit has been obtained and then tell everyone to believe it. When the proof is done more formally than mentioned by others, a monotonic sequence is defined and then a limit is hypothesised. At this point it's just a hypothetical number, there's nothing intrinsic stopping it from existing. The definition of a limit is a supremum of a set (not its maximum), and doesn't have to 'equal' this hypothetical 0.9999... (which you don't formally define by the way, but it's ok because maths does ) In fact, in topology, a space which has this property is considered very nice and said to be 'compact' (or 'complete', i forget terminology). An alternative proof would be to show that the real topology has this nice feature. The existance of this limit is then formally proven, by showing an existance of a supremum for the set, and then finally we can show 0.99999... actually is this limit. We then show that the limit is also the same thing as 1 and we're done. This is how it is done properly; there's no hand waving and saying 'it's really close so it must be the same', the equality is actually proven. (The whole of calculus also gets this rigorous seeing to, but most scientists don't bother with it and just use the results, because it's a bit boring and mathematicians have already told them the answer).

You say that 0.999999... only equals 1 in this abstract world, but it's only defined in maths, so I don't see the problem. Similarly, 0.333...... is only defined in maths, because nowhere else has arbitrary accuracy. If you say that you can imagine what 1/3 is, you're right, but you can also imagine what 1 is. And to the same extent that 0.999999... doesn't exist outside of maths, neither do 1/3 or 1, because you can only have arbitrary accuracy in maths. 

All of my other proofs also showed that not only it comes reaally close, but that it equals as well.

[Elehat]

All of my other proofs also showed that not only it comes reaally close, but that it equals as well.

Thread now too long to read but I'll assume you did x=0.9..., 10x=9.9..., 9x =9, x=1. If there're others let me know. 

This is a pretty good proof for most purposes, but if we're asked to prove that 0.9...=1 then we're dealing in a situation where the questioner does not yet accept any features of the algebra in which we work.

 

How do you prove that 10x=9.9.... really thouroughly? 

 

The decimal shift is just something that we take to be true because it seems to work. To actually prove it you need to be more pedantic, and not use any of the algebraic features you're used to using. The longer proof i gave on page 1 or 2 uses things like powers and reciprocals (Because we can give a definition for them and prove that they exist in this case), but never features anything that makes assumptions about recurring decimals as yours did.

 

As an example, consider someone who has never used the decimal system, and represented numbers using continued fractions instead. They would have a hard time working out what this whole recurring thing is, and certainly wouldn't find it obvious that multiplying by 10 'shifts everything along one'. We need to assume this is the kind of person we're making the proof for.    

[LukaP]

Thread now too long to read but I'll assume you did x=0.9..., 10x=9.9..., 9x =9, x=1. If there're others let me know. 

This is a pretty good proof for most purposes, but if we're asked to prove that 0.9...=1 then we're dealing in a situation where the questioner does not yet accept any features of the algebra in which we work.

 

How do you prove that 10x=9.9.... really thouroughly? 

 

The decimal shift is just something that we take to be true because it seems to work. To actually prove it you need to be more pedantic, and not use any of the algebraic features you're used to using. The longer proof i gave on page 1 or 2 uses things like powers and reciprocals (Because we can give a definition for them and prove that they exist in this case), but never and features anything that makes assumptions about recurring decimals as yours did.

 

As an example, consider someone who has never used the decimal system, and represented numbers using continued fractions instead. They would have a hard time working out what this whole recurring thing is, and certainly wouldn't find it obvious that multiplying by 10 'shifts everything along one'. We need to assume this is the kind of person we're making the proof for.    

i did that, i did the philosophical "divide the interval to 10 pieces infinite times" one, and i did one with the limit of 1/10ⁿ when n goes to infinity, which is the one that is actually legit undisputable, unless someone doesnt believe in supremums and series in general. 

 

also if one doesnt believe in algebra, how am i supposed to prove anything in maths to him? unless is show him proofs for everything in algebra, which he can again deny, like he denies the algebraic proof of .9999 = 1....

[Elehat]

i did that, i did the philosophical "divide the interval to 10 pieces infinite times" one, and i did one with the limit of 1/10ⁿ when n goes to infinity, which is the one that is actually legit undisputable, unless someone doesnt believe in supremums and series in general. 

 

also if one doesnt believe in algebra, how am i supposed to prove anything in maths to him? unless is show him proofs for everything in algebra, which he can again deny, like he denies the algebraic proof of .9999 = 1....

The 10 pieces infinite times is a heuristic not a proof, though it's a good way to introduce the idea. 

 

The limit is the correct way to do it.

When doing it we need to be really careful that we define the limit correctly (sup of sequence) and then prove that the limit exists (by using the monotone convergence theorem, and proving this as well if neccessary), and then showing that both terms are equal to it. It needs to be done without asserting anything else along the way, we need to be really careful about this.

 

It's not that the 'questioner' doesn't believe in algebra. It is assumed that they accept the axioms of the algebra in which we're working (in this case peano axioms, or the ZF+C set theoretic axioms if they want to make you do a lot more work in defining what numbers are). However, that's the point at which philosophy stops, and we presume nothing else. Features of decimal representation are not included in these canonical axioms, you need to prove them before you use them in a proof. Now it is possible prove these statements too, it's just that it would take about the same amount of work to do it. In the more formal proofs, each statement is a consequence of a theorem that can easily (this is a notion of convention, but if you use it you should be able to present it) be traced back to axioms, and is a logical argument that is formally accepted within the algebra that it's agreed we're working in.