How is .999... = 1?

[matt grime]

It isn't necessarily a question of competency, more that this is such an infamous and pointless debating point on this and other math forums that it's best to nip it in the bud.

 

If you don't ask questions you won't learn. And sometimes it is better to be rash and find out quickly. Unfortunately you picked 'the bad question' to ask.

 

One reason why this question keeps coming up is that you aren't taught what the Real numbers really are until university, and only if you do maths (it is a delicate question), yet you have to use them at school.

[Royston]

Thanks Matt, I'm starting undergrad maths near the end of the year...so what you said actually helps a lot, and something I'll look into as preperation.

 

My current course uses maths for scientific applications...so the principles are not clearly defined from a maths perspective, they're more a convenience.

 

Sorry everyone for going off topic.

[JustStuit]

Snail' date=' JustStuit, stop it. There is no grey area, there are no competing thoeries. 0.99... and 1 are different representations of the same number, just like 1/2, 2/4, 3/6, and 1234/2468 are all different representations of the same number.

 

The Real Numbers, the place where this argument takes place, are a complete metric space, 0.99... is the limit of the sum

 

S(n)=9/10+9/100+...+9/10^n

 

and lim S(n) as n tends to infinity is 1. That is easy to prove since the distance between S(n) and 1 is 1/10^n and this converges to zero. Since limits are unique this shows that these two symbols represent the same number.[/quote']

I was talking about how people aren't agreeing on whether they are the same. I believe they are the same since the evidense is pretty thorough and logical and there is no evidense against it (that I have seen.)

[matt grime]

How can you argue against a valid proof. This is maths, not an episode of CSI. It is a consequence of definitions, like all maths. Whilst certain conjectures in maths do have evidential support (ie statements like: For all *something* we believe the following.... do lead to evidential support, ie many cases where it is verified) but this is not one of them, it is a straightfoward consequence of the definitions, you do not need to rely on 'evidence' at all. The 'multiplying 1/3 by 3 argument' is not evidence for this result since no one who cites it ever proves that decimal representations are a model for the real numbers, since such a proof would require you to declare 0.9... and 1 to be the same representations. It is certainly a persuasive argument for why we need to declare them to be the same (otherwise the system is inconsisitent: i can do the same thing two supposedly equivalent ways but get different answers), but the reason they are is because of the properties (that we declare them to have) of the real numbers. In short a proof exists, it is easy to state, possibly hard to understand. But lacj of rigour shouldn't be tolerated like this if it lets these myths continue.

 

Moral of the Story: numbers are not decimal expansions, decimal expansions are representations of numbers.

[the tree]

numbers are not decimal expansions, decimal expansions are representations of numbers.

Confusious say: listen to Matt, for Matt is clever.

[aj47]

Having just started reading about relativity, which I'm finding really interesting but painful to understand I had this random thought.

 

If an object with mass can never reach the speed of light, but can reach infinetly close to it, would that not prove that 0.99999999 is not equal to 1 as if it where then it would violate the laws of relativity.

 

Am I missing something?

[matt grime]

I'm sorry, but that makes no sense.

 

To start with 0.999999 is not 1. You're missing the recurring thing.

 

What has relativity got to do with structure of the real number system and decimals as representations as real numbers? Instead of trying to figure out relativity why don't you find out what the real numbers really are? It'd do you good, perhaps, to see the structure of them.

[aj47]

I'm sorry' date=' but that makes no sense.

 

To start with 0.999999 is not 1. You're missing the recurring thing.

 

What has relativity got to do with structure of the real number system and decimals as representations as real numbers? Instead of trying to figure out relativity why don't you find out what the real numbers really are? It'd do you good, perhaps, to see the structure of them.[/quote']

 

Yea I guess so, I just assumed that if the number exist in the laws of nature, then they must exist mathematically

 

Also I did mean 0.9 reoccuring, just could find a suitable recurring dot.

[JustStuit]

Yea I guess so' date=' I just assumed that if the number exist in the laws of nature, then they must exist mathematically

 

Also I did mean 0.9 reoccuring, just could find a suitable recurring dot.[/quote']

To get a photon to .999 recurring [math]c[/math] would take infinite energy as well since you need to keep adding - it's basically the same as trying to get to the speed of light. This may be another proof type thing (ish) for .99999 recur = 1

[starbug1]

using the assumption that .99999999999 = 1...In any practical circumstances wouldn't this rounding be an error. The methods make sense, but all it is really doing is rounding; would this be an errant miscalculation in anything that needed to be precise, say, in engineering or watchmaking?

[JustStuit]

using the assumption that .99999999999 = 1...In any practical circumstances wouldn't this rounding be an error. The methods make sense, but all it is really doing is rounding; would this be an errant miscalculation in anything that needed to be precise, say, in engineering or watchmaking?

If it truely in .9999 rucurring and not just a lot of 9's

[starbug1]

If it truely in .9999 rucurring and not just a lot of 9's

 

????

[JustStuit]

.999999999 repeating (infinite 9's) would be different than even a large finite amount of nines. I was unsure as to which you were refering.

[starbug1]

.999999999 repeating (infinite 9's) would be different than even a large finite amount of nines. I was unsure as to which you were refering.

 

True; but in practical measurement, infinite 9's cannot exist,..so I guess the question I asked isn't pertinent to the original proof. my mistake.

[matt grime]

Look, people, decimal representations are just representations of Real numbers. Stop thinking of physical things, it is unnecessary and apparently misleading. Whether or not there are 'an infinite number of nines' from measurements is irrelevant, this is maths, it is an abstraction, not real life. You're deducing things about apples from looking at sheep, never mind oranges.

 

Why is 0.9..... equal to 1? Because these are decimal representations of elements of the field of real numbers, and the former is the limit of a cauchy sequence that is equivalent to 1 in the completion of the rationals.

 

 

For the layman who didn't understand that: if we were not to make these identifications then these would not represent real numbers: you would have two distinct elements satisfying |x-y|<1/n for all n in the natural numbers, and that is not possible because of the restrictions we place upon the real numbers.

 

Just to really hammer home the point. If these were base 11 representations of real numbers they would not be equal.

 

This is about the decimal representations of Real numbers, as I keep saying, so why is no one even bothering to think what decimal, representation, or Real means?

[Ragib]

To get a photon to .999 recurring [math]c[/math'] would take infinite energy as well since you need to keep adding - it's basically the same as trying to get to the speed of light. This may be another proof type thing (ish) for .99999 recur = 1

 

It is impossible to have an impossible to have 0.9999 recurring [math]c[/math], as it is impossible to have any irrational quantity in a physical system. This is due to the fact that everything is quantised, that is, in comes in chunks and is not continuous. If you think about it, its impossible to have a near ending number in reality. Velocity is quantised, as is space, time, and everything else.

[the tree]

it is impossible to have any irrational quantity in a physical system.

So? Whoever said the real number system is physical?

Velocity is quantised

wtf?

[matt grime]

It is impossible to have an impossible to have 0.9999 recurring [math']c[/math]

 

 

a photon, by definition, in a vacuum travels at 0.99recurring c because 0.99recurring is the same real number as 1. (irrespective of whether you understand why they are the same number 0.9recurring is clearly a rational number since it has a recurrent decimal expansion).

 

as it is impossible to have any irrational quantity in a physical system.

 

what has this to do with anything (eve if it were true)?

 

This is due to the fact that everything is quantised, that is, in comes in chunks and is not continuous.

 

that is false. certainly there is planck length, and energy in bound systems is quantized, but free energy is not quantized. no discrete quantized model for space-time has yet been proposed that works.

 

 

If you think about it, its impossible to have a near ending number in reality

 

 

what does that even mean and what does it have to do with mathematics? the real numbers are a totally ordered complete field, in such a place 0.99 recurring as a decimal expansion is the same number as 1 and is no more mysterious than why 1/2 and 2/4 are different representations of the same rational number.

[bascule]

matt grime, you have the patience of a saint... if I were you I'd have gone mad long, long ago

[Ragib]

The Tree - "So? Whoever said the real number system is physical?" I was merely talking bout JustStuits comments about getting a photon at "0.9 recurring" of c, which is in a physical system, no pure mathematics. And, velocity is quantised because space is quantised. I believe it is C/the Plank Time.

 

Matt Grime- Read closer, i never actually said 0.999 recurring is irrational, i merely stated its impossible to have an irrantional quantity in a physical system. And what does it have to do with anything? It was just an extra sentence to give some extra information...and don't u try to say free-energy isnt quantised. You find me a photon thats got less energy than h.

"what does that even mean and what does it have to do with mathematics?", once again, i wasnt talkin about mathematics, i was talking about physics. I wonder how old you are and how much fun you get out of harrassing 14 year olds. I bet when you were my age you didnt know half of what i do.

[matt grime]

You find me a photon thats got less energy than h.

 

and photons are the only things with energy? Allowed quanta in some Hamiltonian system are the spectra of certain differential operators, or something, aren't they? And high temperature physical systems have continuous spectra or something. Not that I'm at all a physicist, and will happily admit I'm wrong if indeed I am, but this is what physicists I know and trust say. Besides, planck length, and planck time, are the smallest meaningful measurements of distance and time, we cannot 'measure' beyond these scales, which is strictly different from saying that time is quantized. Exactly what the true nature of any model of 'everything' will be is still unclear, from my limited understanding of mathematical physics, and heavily limited by the unobservability of the phenomena physicists are trying to explain.

 

 

 

It is impossible to have an impossible to have 0.9999 recurring , as it is impossible to have any irrational quantity in a physical system.

 

The direct implication of that sentence is that you think 0.9recurring is irrational. You cite the reason for 0.9recurring not being allowed is because irrational quantities are not allowed (in physical systems).

 

Anyway, the statement 0.9 recurring equals one is a statement about complete metric spaces, not anything to do with physics.

 

You picked an unfortunate topic to post on; the number of cranks out there with their own personal pet theories about this particular subject means that it is necessary (if a website such as this is to have any mathematical integrity) to stop such threads as this going off topic. Indeed it would be better if all 0.9recurring is/is not 1 threads were just locked instantly with a standard explanation of analysis.

 

Further, in the question of mine you object to, you avoid explaining what you meant by:

 

If you think about it, its impossible to have a near ending number in reality

 

what is a 'near ending number'?

[insane_alien]

Ragib: a photon with frequency 0.9Hz has an energy of 0.9h J is that good enought for you

[Ragib]

Sorry Matt Grime, typo, Never, not Near. Never ending number.

 

insane_alien- Good idea lol. I obviously worded that wrong.. I just ment E=hf , it comes in multiples of h.

[matt grime]

But 1/9 is, I suspect, a never ending number (in fact almost all rationals have 'never ending' decimal expansions) in your language; it all depends what base you pick, but then that is immaterial to what 0.9recurring represents, and thes symbols are all just representations of things.

 

Now, if I have 9 atoms, then one of them constitutes a ninth of the total, so that never ending number (1/9) must occur 'in reality'.

 

And what mutliples of 'h' are you allowing? In high energy physics electrons are emitted from atoms at 'all' frequencies. But the notion of measurement is getting even further away from what the mathematics of the real numbers are. The continuum is the 'right' thing to use when modelling high energy stuff.

[Ragib]

Unless I Specifiy, i talk about base 10 numbers, like most other people...

And about the number of atoms thingo, fine, Your right, its only irrational quantitys that are impossible in physical systems. All rational numbers can exist. I made a mistake and im sorry.

And, the reason high energy particle physics uses a continuum of energy is due to the fact the measuring tools used have a measuring error more than the value of h, and its pointless to have that much accuracy for high energy transfers, im 99% of cases. Also, the value of h is so small, and high energy physics is quite, well, high.., making it unfeasable to use such small quanta. Just because humans cant be bothered, and really shouldn't be cause it makes almost no difference, it is still there.