Does 0.9999999 = 1?

Akae47 said:

I am shocked at some of these answer here. 0.99999.... is not 1 and it never will be. In terms of money I'm a accountant I can make .9999 = 1,000,000

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As an accountant, it'd be more profitable to make it 0,9

Pan1cMode said:

If it has infinite zeros, it can't end in a 1. In fact, it literally can't end. There will never be a point at which you can say, "oh now comes the 1".

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Same with 0,9. You'll never say "Oh, here it equals 1". So it'll never equal 1.

Pan1cMode said:

No, this 'problem' has long since been solved. The fact that you can't wrap your head around the answer is another thing entirely.

To @Prinz I don't think you've understood the squeeze theorem explanation so I'll go over it again. Basically it states that two numbers are the same number if there are no numbers in between them. 1 and 1.000000000... Do not have any numbers in between them so they are the same number. Likewise, there are no numbers in between 0.9999999... and 1; hence they are the same number.


Yes it is 1. There are plenty of explanations as to why in this thread. An infinite number of 9s following a decimal point is the same as 1.

This is the simplest algebraic proof;

1/3 + 1/3 + 1/3 = 0.33333... + 0.33333... +0.33333... = 0.99999... = 1

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Well maybe we need to leave fractions out of this, because fractions are using whole numbers and decimals are never complete. For example 100 divided by three with no decimals is 33. So maybe we just have to except there is two ways to look at it. Infinetly recurring numbers were probably created to have it make more sense, because there is a noticeable difference between 9 and 10 and 99 and 100. Is 99.9999999999 the same as 100? If it is why not just call it 100... If I said I am 99.9999999999999 sure I'm the greatest injustice player (lol) I didn't say 100 because there is a shred of doubt... Or maybe

Illmatix said:

Well maybe we need to leave fractions out of this, because fractions are using whole numbers and decimals are never complete. For example 100 divided by three with no decimals is 33. So maybe we just have to except there is two ways to look at it. Infinetly recurring numbers were probably created to have it make more sense, because there is a noticeable difference between 9 and 10 and 99 and 100. Is 99.9999999999 the same as 100? If it is why not just call it 100... If I said I am 99.9999999999999 sure I'm the greatest injustice player (lol) I didn't say 100 because there is a shred of doubt... Or maybe

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But in your gut you know there's that 0,0000.....1 chance

Iyou're ix said:

Well maybe we need to leave fractions out of this, because fractions are using whole numbers and decimals are never complete. For example 100 divided by three with no decimals is 33. So maybe we just have to except there is two ways to look at it. Infinetly recurring numbers were probably created to have it make more sense, because there is a noticeable difference between 9 and 10 and 99 and 100. Is 99.9999999999 the same as 100? If it is why not just call it 100... If I said I am 99.9999999999999 sure I'm the greatest injustice player (lol) I didn't say 100 because there is a shred of doubt... Or maybe

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Or maybe youre right and it is the same and this is the only way to rationalize it
I dunno this is super interesting tho
If I asked you this: is .9 equal to one? No. Is .99 equal to one? No. If I ask you forever adding a nine to it every time will it be 1? Yeah ain't nobody got time for that! @Pan1cMode

Illmatix said:

Well maybe we need to leave fractions out of this, because fractions are using whole numbers and decimals are never complete. For example 100 divided by three with no decimals is 33. So maybe we just have to except there is two ways to look at it. Infinetly recurring numbers were probably created to have it make more sense, because there is a noticeable difference between 9 and 10 and 99 and 100. Is 99.9999999999 the same as 100? If it is why not just call it 100... If I said I am 99.9999999999999 sure I'm the greatest injustice player (lol) I didn't say 100 because there is a shred of doubt... Or maybe

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But we're not talking about a lot of 9s, we're talking about an infinite number of 9s. If you said I'm 99.999...% it is exactly the same a saying you are 100% sure.

We can't 'leave fractions out of this', since fractions are just another representation of decimals. And I'm not talking about rounding to equal 1. I'm saying it is exactly equal to 1.

Anyways, if you guys don't wanna understand that's fine by me. If you're so sure about your answer, you can publish it in a peer reviewed journal and overturn hundreds of years of mathematical understanding because even though you've no proof, "in your gut you know there's that 0,000.....1 chance "

Pan1cMode said:

But we're not talking about a lot of 9s, we're talking about an infinite number of 9s. If you said I'm 99.999...% it is exactly the same a saying you are 100% sure.

We can't 'leave fractions out of this', since fractions are just another representation of decimals. And I'm not talking about rounding to equal 1. I'm saying it is exactly equal to 1.

Anyways, if you guys don't wanna understand that's fine by me. If you're so sure about your answer, you can publish it in a peer reviewed journal and overturn hundreds of years of mathematical understanding because even though you've no proof, "in your gut you know there's that 0,000.....1 chance "

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Same to you.

Prinz said:

Same to you.

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My argument is already documented mathematical fact, published in many peer reviewed journals and is supported by a mountain of proofs so don't know what you mean by that =D. Unless you wanna talk about some highly weird alternative number systems (such as the hyperreals) then your statement is just false.

My only qualm with this is it just spreads ignorance and the idea that 'something is too difficult to understand therefore it's wrong'.

Prinz said:

I'm not dismissing anything. I'm just saying this particular problem is not yet entirely solved.

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Mathematicians seem to consider is solved.

Hey do you guys know if .9 is equal to 1 ?

Illmatix said:

Hey do you guys know if .9 is equal to 1 ?

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0.9 recurring.

Here is the Wikipedia article on the subject. We've posted several proofs here along with explanations. It hasn't really been up for debate in the mathematical community.

In the number system we use everyday (consisting of the real numbers), there are no non-zero infinitesimals and 0.999... is just another way of writing 1 same as 0.4999... is another way of writing 0.5 or 12.356999... is another way of writing 12.357. They are exactly equivalent.

Prinz said:

0,(0)1 also has infinite 0's, but it doesn't mean it equals 0.

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LOLOL that's fucking godlike. You have a number with infinite 0's and then a 1 attached as an endpoint xD

I'm sorry, but as an amateur numberphile and college student, that was just pure gold.

Also, I don't quite recall all the retorts you made to our posts Prinz, and I'm not particularly interested in explaining everything again in multiple different manners, but to reply to one thing you said specifically regarding my mile example:

first, it's not Xenon's paradox, that's an element on the periodic table.
second, that doesn't apply here because that problem takes halves and completely ignores the concept of limits (Xeno's paradox has gotten absolutely demolished by mathematicians.)
third, 0 does have a finite value and does have meaning past the decimal point. If I told you I measured the radius of the Earth to be 6000 kilometers or 6000.0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 kilometers, you should be absolutely fucking awestruck at my accuracy.
fourth, since when can mathematicians not define their own rules within math as long as they're logically consistent? You're basically saying we can't have negative numbers because negative amounts do not exist other than in fictitious systems like... oh I don't know... withdrawing money from the bank for a summer course in introductory calculus.

If there's anything I've learned in the incredibly short time that I've been on this planet, it's that when something is too difficult to understand, people will fall back on their original convictions instead of going deeper with the subject.

To make things worse, people here are speaking on a subject of which they're not immediately familiar and only helping to spread ignorance.

Cyber Sub's parry and bombs need to be nerfed. Sub-Zero's resets need to be patched. J360 is a tournament viable lethal killing machine. Laughable.

Might Tested said:

LOLOL that's fucking godlike. You have a number with infinite 0's and then a 1 attached as an endpoint xD

I'm sorry, but as an amateur numberphile and college student, that was just pure gold.

Also, I don't quite recall all the retorts you made to our posts Prinz, and I'm not particularly interested in explaining everything again in multiple different manners, but to reply to one thing you said specifically regarding my mile example:

first, it's not Xenon's paradox, that's an element on the periodic table.
second, that doesn't apply here because that problem takes halves and completely ignores the concept of limits (Xeno's paradox has gotten absolutely demolished by mathematicians.)
third, 0 does have a finite value and does have meaning past the decimal point. If I told you I measured the radius of the Earth to be 6000 kilometers or 6000.0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 kilometers, you should be absolutely fucking awestruck at my accuracy.
fourth, since when can mathematicians not define their own rules within math as long as they're logically consistent? You're basically saying we can't have negative numbers because negative amounts do not exist other than in fictitious systems like... oh I don't know... withdrawing money from the bank for a summer course in introductory calculus.

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Yes I have that number. How is it not possible when 0,(9) is possible? Maybe someone should explore this.
Yeah, sorry, it's Zenon, not Xenon. I like how you say something doesn't apply here because bla, bla, but what you're saying applies because bla, bla. Isn't this implying that you're using specific situations to make a point?
About that 6000.000 example. Nope. If you told me that, I'd ask you what's the point of all those 0's after the dot since they have absolutely no value. Isn't this a number rule that no matter how many 0's you put after a dot it doesn't change the value of the number?
And that bank example, you're right. It depicts deficit. But if you had put two accounts, one being your current amount and an other, represented also in positive values, representing your debt, a negative representation wouldn't be needed. Example: you have 1000$ on you current account that you can withdraw, and 0$ on your debt account. You withdraw 1010$, your current account becomes 0$ and your debt account becomes 10$. That's just a way to represent without negatives.

Might Tested said:

If there's anything I've learned in the incredibly short time that I've been on this planet, it's that when something is too difficult to understand, people will fall back on their original convictions instead of going deeper with the subject.

To make things worse, people here are speaking on a subject of which they're not immediately familiar and only helping to spread ignorance.

Cyber Sub's parry and bombs need to be nerfed. Sub-Zero's resets need to be patched. J360 is a tournament viable lethal killing machine. Laughable.

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Why try to understand something that is obvious? Grass is green, water is wet and 1=/=0,(9). If something, the problem is some people try to explain obvious things and that's when deviations can occur.

Prinz said:

Yes I have that number. How is it not possible when 0,(9) is possible? Maybe someone should explore this.
Yeah, sorry, it's Zenon, not Xenon. I like how you say something doesn't apply here because bla, bla, but what you're saying applies because bla, bla. Isn't this implying that you're using specific situations to make a point?
About that 6000.000 example. Nope. If you told me that, I'd ask you what's the point of all those 0's after the dot since they have absolutely no value. Isn't this a number rule that no matter how many 0's you put after a dot it doesn't change the value of the number?
And that bank example, you're right. It depicts deficit. But if you had put two accounts, one being your current amount and an other, represented also in positive values, representing your debt, a negative representation wouldn't be needed. Example: you have 1000$ on you current account that you can withdraw, and 0$ on your debt account. You withdraw 1010$, your current account becomes 0$ and your debt account becomes 10$. That's just a way to represent without negatives.



Why try to understand something that is obvious? Grass is green, water is wet and 1=/=0,(9). If something, the problem is some people try to explain obvious things and that's when deviations can occur.

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Your number is not possible because in the number system we use, non-zero infitesimals are not possible. Basically if you have an infinite amount of zeros, there will never ever be a point at which you can place a one at the end. There is no end. It's an infinite amount of zeros. You can't say okay one can come now. It literally cannot have an end point. That's kinda the definition of this type of infinity.

Have you read the article I've linked twice? It explains it with several proofs of varying complexity and also discusses how many students have a difficult time grasping it because of several reasons (they lack understanding of infinity being one).


For the extra zeros at the end, they indicate significant digits. They are important insofar as they indicate the accuracy of your measurement.

The problem with 0.999... Is it seems sorta counterintuitive to some people (hence it's not obvious) but that doesn't make it any less equal to 1.

Also, just for your information there are some alternative number systems (such as the hyperreals) with different logical rules that do allow non-zero infinitesimals, but from the way you are speaking I am certain you are not talking about them and they would be beyond your comprehension.

Pan1cMode said:

Your number is not possible because in the number system we use, non-zero infitesimals are not possible. Basically if you have an infinite amount of zeros, there will never ever be a point at which you can place a one at the end. There is no end. It's an infinite amount of zeros. You can't say okay one can come now. It literally cannot have an end point. That's kinda the definition of this type of infinity.

Have you read the article I've linked twice? It explains it with several proofs of varying complexity and also discusses how many students have a difficult time grasping it because of several reasons (they lack understanding of infinity being one).


For the extra zeros at the end, they indicate significant digits. They are important insofar as they indicate the accuracy of your measurement.

The problem with 0.999... Is it seems sorta counterintuitive to some people (hence it's not obvious) but that doesn't make it any less equal to 1.

Also, just for your information there are some alternative number systems (such as the hyperreals) with different logical rules that do allow non-zero infinitesimals, but from the way you are speaking I am certain you are not talking about them and they would be beyond your comprehension.

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In that certain numeral system, where other rules apply, 0,(9) may replace 1. But in the system where 1 is finite, and this specific symbol "1" does depict a finite number, it doesn't.
About the infinite 0's. I'm not gonna talk about it, but here you go, a problem to be solved.

Pan1cMode said:

Your number is not possible because in the number system we use, non-zero infitesimals are not possible. Basically if you have an infinite amount of zeros, there will never ever be a point at which you can place a one at the end. There is no end. It's an infinite amount of zeros. You can't say okay one can come now. It literally cannot have an end point. That's kinda the definition of this type of infinity.

Have you read the article I've linked twice? It explains it with several proofs of varying complexity and also discusses how many students have a difficult time grasping it because of several reasons (they lack understanding of infinity being one).


For the extra zeros at the end, they indicate significant digits. They are important insofar as they indicate the accuracy of your measurement.

The problem with 0.999... Is it seems sorta counterintuitive to some people (hence it's not obvious) but that doesn't make it any less equal to 1.

Also, just for your information there are some alternative number systems (such as the hyperreals) with different logical rules that do allow non-zero infinitesimals, but from the way you are speaking I am certain you are not talking about them and they would be beyond your comprehension.

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Panic, you've just saved me what would've been an incredibly douchey/condescending reply to Prinz.

So yeah, to wrap that up really quickly, logically inconsistent choice of number, Zeno's Paradox has been shut-down, significant digits are a big deal, blabla

If you'd like another example, take differential equations that use the concept of infinity, infinitesimals, complex eigen values (imaginary numbers), oscillating functions in the imaginary and real plane... does that sound like a bunch of fancy mathematical bullshit? Well, there are proofs and theorems of why all that bullshit is valid in differential equations, and you can use that to solve second order differential equations which is a fancy way of saying you can be a total badass engineer (the study of making tons of money off of applying all that fancy mathematical bullshit).

Why try to understand something that is obvious? Are you kidding? Your teachers must've been awful to instill such a terrible scientific mind. I'll take your example.

Let's say some total fucking knob wants to know why the grass is green. He spends like ten years in a basement fucking around with prisms and light, and eventually he realizes, that it's not the grass that is green, it's that certain atoms absorb and reflect various wavelengths of light. Then he goes on to discover that these wavelengths that get absorbed are intrinsically characteristic of the molecules that make up that plant, a unique chemical composition. And some other knob says "So fucking what?" Well now this guy realizes you can analyze the chemical composition of a planet or far off planet's atmosphere by its color, actually enabling you to determine if that planet's atmosphere is suitable for life, or if its carbon-14 compositioin suggests life is active. And at this point, you could still say "Well so fucking what?" Well all of a sudden, some other guy by the name of Maxwell starts to figure out that electromagnetic waves are light. And at this point, if you still suck, you could still not give a shit. Well a couple of anti-aircraft gunners during Operation Desert Storm gave a shit, because F-117s bombed the living shit out of everything based on the simple concept that chemical composition determines what wavelengths are absorbed (like radio waves...), among other concepts to improve its stealth.

Prinz said:

In that certain numeral system, where other rules apply, 0,(9) may replace 1. But in the system where 1 is finite, and this specific symbol "1" does depict a finite number, it doesn't.
About the infinite 0's. I'm not gonna talk about it, but here you go, a problem to be solved.

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Prinz, me and Panic really want to get through to you on this, but you have to read some of the links Panic's posted. There's an absurd amount of proofs explaining why .9 repeating is 1, and there's a bunch of explanations that refute some of the common misconceptions of infinity (a few of which you've accepted and why it's holding you back from understanding infinity).

The only realistic number that would involve some quasi-form of . (0) 1 is some recursive function say...

(n + 1) = n/10, where n can be any function or constant, say a = 1,
so you might get something like a/10, a/100, a/1000, .00001, .000001, ... , .(0)1

However, that would be an infinitely long sequence and it's not the same as an actual number, in reality you're just dividing by ten a large finite number of times, so you can never truly have infinite numbers with some sudden endpoint. Infinity does not end. It's like saying I'm going to travel in a circle, and I'll let you know when I get to the endpoint of that circle and put a flag down.

Prinz said:

In that certain numeral system, where other rules apply, 0,(9) may replace 1. But in the system where 1 is finite, and this specific symbol "1" does depict a finite number, it doesn't.
About the infinite 0's. I'm not gonna talk about it, but here you go, a problem to be solved.

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The number system where 0.999...=1 is the number system you use. They are merely different representations of the same number. Just how 3.14159... and π both represent the number obtained when the circumference of a circle is divided by its radius.

And I don't understand what you mean by "another problem to be solved". What you've said is basically akin to saying an infinite is finite (i.e. has an end) which is not only counterintuitive by logically inconsistent.

Prinz said:

About that 6000.000 example. Nope. If you told me that, I'd ask you what's the point of all those 0's after the dot since they have absolutely no value. Isn't this a number rule that no matter how many 0's you put after a dot it doesn't change the value of the number?

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Ugh. Did you even take a basic high school science class? Or rather, did you pass? The zeros after the decimal doesn't change the quantity, but it absolutely has value in that it indicates how precise the calculation was.

aj1701 said:

Ugh. Did you even take a basic high school science class? Or rather, did you pass? The zeros after the decimal doesn't change the quantity, but it absolutely has value in that it indicates how precise the calculation was.

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So you're saying I should question your calculation precision if put less than infinite 0's after the dot? If I showed you 2-1=1, you'd say "Yeah, but is it precise? What if you put a dot after the 1, will there follow a zero? Or 2 zeros? Or an infinite amount of zeros?" and I'd say "Fucking yeah, because there's absolutely no point at showing what's not there". The only reason you could show zeros after the dot is if you wrote it this way: 2,00-1,00=1,00. And that, is just for convenience. For example, percentages are written with at least 1 zero after the dot, currencies with at least 2 zeros, even if the amount is without decimals. But, that doesn't mean 10,0% is more or less than 10%, it's the same amount and the zero doesn't change anything.

Prinz said:

So you're saying I should question your calculation precision if put less than infinite 0's after the dot? If I showed you 2-1=1, you'd say "Yeah, but is it precise? What if you put a dot after the 1, will there follow a zero? Or 2 zeros? Or an infinite amount of zeros?" and I'd say "Fucking yeah, because there's absolutely no point at showing what's not there". The only reason you could show zeros after the dot is if you wrote it this way: 2,00-1,00=1,00. And that, is just for convenience. For example, percentages are written with at least 1 zero after the dot, currencies with at least 2 zeros, even if the amount is without decimals. But, that doesn't mean 10,0% is more or less than 10%, it's the same amount and the zero doesn't change anything.

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Omfg you just don't get it lol. He's not talking about an infinite amount of zeros hahahahaha.

If I measure a distance and say it's 2m. Someone comes along with a more accurate ruler and says, well actually it's 2.01 meters. Then someone comes along with an even more accurate ruler and measures 2.0101 meters. Than it happens again and someone measures 2.01010000 meters. See how the zeros indicate how accurate the measurement is?

Pan1cMode said:

Omfg you just don't get it lol. He's not talking about an infinite amount of zeros hahahahaha.

If I measure a distance and say it's 2m. Someone comes along with a more accurate ruler and says, well actually it's 2.01 meters. Then someone comes along with an even more accurate ruler and measures 2.0101 meters. Than it happens again and someone measures 2.01010000 meters. See how the zeros indicate how accurate the measurement is?

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Yes I do. But, isn't this actually a measurement error of the method or the ruler used ?

Prinz said:

Yes I do. But, isn't this actually a measurement error of the method or the ruler used ?

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No, if I say its 2.000 I'm saying I am certain to the thousandths place which is more accurate than if I just said 2. The latter would imply the measuring instrument isn't accurate to even the tenths place.

aj1701 said:

No, if I say its 2.000 I'm saying I am certain to the thousandths place which is more accurate than if I just said 2. The latter would imply the measuring instrument isn't accurate to even the tenths place.

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I agree about the inaccuracy of the instrument. But you'd say it's equal to 2 since the zeros don't add any real value. Unlike the initial problem.
Btw, is there a way to prove 3-2=0,(9)?