Current Events > Proof that 0.999...=1

(+) Yeah! (+)

http://imgur.com/a/0craz

:)

EDIT: Proof of the statement in the beginning: http://imgur.com/a/R0OTE
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Good proof, although someone skeptical would probably also want proof of that infinite series formula.
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.999...=1 is true under the rules of algebra, but not actually true
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Damn_Underscore posted...

.999...=1 is true under the rules of algebra, but not actually true

Third post.
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Damn_Underscore posted...

.999...=1 is true under the rules of algebra, but not actually true

It's true in any context that involves normal, everyday numbers. There are number systems where it becomes untrue, but those number systems are far more esoteric than the level of understanding needed to grasp the idea in the first place.
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tiornys posted...

Good proof, although someone skeptical would probably also want proof of that infinite series formula.

the proof for that is published and public but if you're the type of person who can't grasp the much simpler 1/3 + 1/3 + 1/3 = 0.333... + 0.333... + 0.333... proof, you definitely won't understand a mathematical proof.
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Damn_Underscore posted...

.999...=1 is true under the rules of algebra, but not actually true

lazy trolling
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tiornys posted...

Good proof, although someone skeptical would probably also want proof of that infinite series formula.

Good point. Edited.

http://imgur.com/a/R0OTE
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There are number systems where it becomes untrue, but

I tend to hear this a lot. Seems like no matter what ypu say, it's not true in some number systems. But nobody ever names these number systems.

So question, which sytems are you talking about?
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Damn_Underscore posted...

.999...=1 is true under the rules of algebra, but not actually true

Also infinite series is not algebra.
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DevsBro posted...

There are number systems where it becomes untrue, but

I tend to hear this a lot. Seems like no matter what ypu say, it's not true in some number systems. But nobody ever names these number systems.

So question, which sytems are you talking about?

In this case, one of the most straightforward examples is the surreal numbers: https://en.wikipedia.org/wiki/Surreal_number
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DevsBro posted...

There are number systems where it becomes untrue, but

I tend to hear this a lot. Seems like no matter what ypu say, it's not true in some number systems. But nobody ever names these number systems.

So question, which sytems are you talking about?

There isn't one.

As has been explained multiple times over the years, there is a number that is infinitely close to 1 but isn't 1, however that number is not 0.999... That number is objectively equal to one in all systems of math which are based on the standard definitions and axioms that we do all math with today on planet earth.

If a system exists where 0.999... doesn't equal one, it doesn't exist on planet earth, and may not exist anywhere except inside the head of a few dingbats.

I suppose it technically doesn't equal 1 if you're writing a hexadecimal number. You'd want 0.FFF... for that.
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tiornys posted...

DevsBro posted...

There are number systems where it becomes untrue, but

I tend to hear this a lot. Seems like no matter what ypu say, it's not true in some number systems. But nobody ever names these number systems.

So question, which sytems are you talking about?

In this case, one of the most straightforward examples is the surreal numbers: https://en.wikipedia.org/wiki/Surreal_number

0.999... is not a surreal number. It is a real, rational number.
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ChromaticAngel posted...

As has been explained multiple times over the years, there is a number that is infinitely close to 1 but isn't 1, however that number is not 0.999...

Good point; I was oversimplifying things.
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ChromaticAngel posted...

DevsBro posted...

There are number systems where it becomes untrue, but

I tend to hear this a lot. Seems like no matter what ypu say, it's not true in some number systems. But nobody ever names these number systems.

So question, which sytems are you talking about?

There isn't one.

As has been explained multiple times over the years, there is a number that is infinitely close to 1 but isn't 1, however that number is not 0.999... That number is objectively equal to one in all systems of math which are based on the standard definitions and axioms that we do all math with today on planet earth.

If a system exists where 0.999... doesn't equal one, it doesn't exist on planet earth, and may not exist anywhere except inside the head of a few dingbats.

I suppose it technically doesn't equal 1 if you're writing a hexadecimal number. You'd want 0.FFF... for that.

tiornys was speaking of analysis-type number systems that are not used by people outside the field of mathematics and theoretical physics
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Darkman124 posted...

tiornys was speaking of analysis-type number systems that are not used by people outside the field of mathematics and theoretical physics

I already said (and he already agreed) The number he's talking about exists, but it isn't 0.999...
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ChromaticAngel posted...

Darkman124 posted...

tiornys was speaking of analysis-type number systems that are not used by people outside the field of mathematics and theoretical physics

I already said (and he already agreed) The number he's talking about exists, but it isn't 0.999...

yeah, you ninja'd me
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DevsBro posted...

There are number systems where it becomes untrue, but

I tend to hear this a lot. Seems like no matter what ypu say, it's not true in some number systems. But nobody ever names these number systems.

So question, which sytems are you talking about?

P-adic numbers are one example, I think (definitely not my area of expertise though).

Basically, the real numbers are constructed as a kind of "completion" of the rational numbers.

Take the rational numbers, and then pick your favourite irrational number, like sqrt(2). You can find a sequence of rational numbers which converges (becomes arbitrarily close to) to this chosen irrational number, and so you'd say that sqrt(2) is a limit of a sequence of rational numbers; it can be approximated as closely as you'd like by rationals (which is why you can find a good value for it using modern calculators).

Nevertheless, sqrt(2) is not itself rational, and this means that the rationals are "incomplete" in the sense that not all convergent sequences of rational numbers have rational limits. We can complete them by creating a new set of numbers which contains the rationals, and also contains every limit of convergent rational sequences. This is the real numbers.

However, if you look at the definition of the limit of a convergent sequence (which admittedly is kind of frightening), you'll see that the absolute value function makes a prominent appearance. This function is known as a "metric" - basically, it's a good way of measuring distances between two rational numbers x and y (the distance between them being |x-y|), but it's not the only metric that can be imposed on the rational numbers.

Number theorists like to use the 2-adic metric, which says that two rational numbers x and y are close in the 2-adic metric if x-y is divisible by a high power of 2. As an example, 3-1 is divisible by 2, but 9-1 is divisible by 8=2^3, so 9 and 1 are closer than 3 and 1 in the 2-adic metric. By the way, you can replace 2 in this paragraph by any prime number p to obtain the general p-adic metric.

Once we've established the p-adic metric, we can write down the definition of convergent sequences of rationals with respect to this metric - that is, we consider sequences which get "closer" to a particular limit, where "closer" means in the p-adic sense. Then we'd notice the same phenomenon - there are p-adic-convergent rational sequences whose limits aren't rational. We can create a new set of numbers which includes the rationals, and all p-adic limits of p-adic-convergent rational sequences. The resulting set is called the P-adic Numbers.


The constructions seem similar, maybe, but the p-adics are very strange compared to the reals.
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ChromaticAngel posted...

If a system exists where 0.999... doesn't equal one, it doesn't exist on planet earth, and may not exist anywhere except inside the head of a few mathematicians.

Fixed ;)

That's the beauty of math. We can come up with things that have no chance of existing on planet Earth and still make conclusions about them.
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I think maybe a better way of explaining the topic scenario is this:

Saying that 1 = 0.999... is similar to saying 1 = sin(pi/2) or 1 = the integral of (x*dx/1122) evaluated from -16 to 50. It's an unnecessarily complex way of writing "1" that draws on mathematical concepts outside the scope of everyday life.

It's just that 0.999... looks less complex than it actually is.
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tiornys posted...

I think maybe a better way of explaining the topic scenario is this:

Saying that 1 = 0.999... is similar to saying 1 = sin(pi/2) or 1 = the integral of (x*dx/1122) evaluated from -16 to 50. It's an unnecessarily complex way of writing "1" that draws on mathematical concepts outside the scope of everyday life.

It's just that 0.999... looks less complex than it actually is.

That's a good way of explaining it. 0.999... is a simple notation for an awfully complication gadget that needs a theory of infinite series to have a value.
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Cheese_Crackers posted...

ChromaticAngel posted...

If a system exists where 0.999... doesn't equal one, it doesn't exist on planet earth, and may not exist anywhere except inside the head of a few mathematicians.

Fixed ;)

That's the beauty of math. We can come up with things that have no chance of existing on planet Earth and still make conclusions about them.

Ok, sure.

Euclid proved his postulates 1, 2, 3, and 4, but was never able to prove his 5th postulate.

Despite him not being able to prove it, people took it as true anyway, but a while back people decided "Well what if it isn't true?" and came up with non-euclidean Geometry as a result.

But redefining basic algebra concepts spirals upwards. If you change it so that 0.999... doesn't equal 1, then that's a definition change that impacts every other fundamental aspect of math.

which is fine for a thought exercise, but not a good subject to have a serious discussion about, especially on a forum limited to 500 posts per topic as people who come up with these ideas almost invariably didn't think them all the way through to the end.

A good example of this is the constant c. It's a constant for the speed of light in a vaccuum used in the popular Energy/Mass Equivalence equation, E=mc^2.

If c were to change, even by a little, you fuck up literally everything everywhere and now all scientific math ever done is now wrong.
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Cheese_Crackers posted...

tiornys posted...

I think maybe a better way of explaining the topic scenario is this:

Saying that 1 = 0.999... is similar to saying 1 = sin(pi/2) or 1 = the integral of (x*dx/1122) evaluated from -16 to 50. It's an unnecessarily complex way of writing "1" that draws on mathematical concepts outside the scope of everyday life.

It's just that 0.999... looks less complex than it actually is.

That's a good way of explaining it. 0.999... is a simple notation for an awfully complication gadget that needs a theory of infinite series to have a value.

i think a good approach next time someone says .999...=/=1 is to play dumb and ask them to define it explicitly
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actually if you multiply 0.9999... by 0.9999... you'll get a number smaller than 0.9999... hence 0.9999.. does not equal 1
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Turtlebread posted...

actually if you multiply 0.9999... by 0.9999... you'll get a number smaller than 0.9999... hence 0.9999.. does not equal 1

Would you like to explicitly define the result of 0.9999... times 0.9999...?

*evil grin*
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yes it's obviously 0.999....8
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i like the cut of your jib, tiornys.
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ChromaticAngel posted...

Cheese_Crackers posted...

ChromaticAngel posted...

If a system exists where 0.999... doesn't equal one, it doesn't exist on planet earth, and may not exist anywhere except inside the head of a few mathematicians.

Fixed ;)

That's the beauty of math. We can come up with things that have no chance of existing on planet Earth and still make conclusions about them.

Ok, sure.

Euclid proved his postulates 1, 2, 3, and 4, but was never able to prove his 5th postulate.

Despite him not being able to prove it, people took it as true anyway, but a while back people decided "Well what if it isn't true?" and came up with non-euclidean Geometry as a result.

But redefining basic algebra concepts spirals upwards. If you change it so that 0.999... doesn't equal 1, then that's a definition change that impacts every other fundamental aspect of math.

which is fine for a thought exercise, but not a good subject to have a serious discussion about, especially on a forum limited to 500 posts per topic as people who come up with these ideas almost invariably didn't think them all the way through to the end.

A good example of this is the constant c. It's a constant for the speed of light in a vaccuum used in the popular Energy/Mass Equivalence equation, E=mc^2.

If c were to change, even by a little, you fuck up literally everything everywhere and now all scientific math ever done is now wrong.

0.999...=1 relies on some fairly sophisticated mathematical machinery, though, and some of which is chosen arbitrarily. Read post 18 for a hopefully straightforward example, but making just one choice differently changes that fact.

I'm not saying that those choices should be made by physicists, for example - I don't know enough physics to judge that - but it's certainly a worthwhile discussion and yields some fantastic results in number theory.

Besides, no amount of choices can change the fact that in the standard context in which 0.999... is defined, it's equal to 1.
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If something's missing it isn't whole.
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If I couldn't be bothered to understand those images does it make me stupid at math? I've literally never had to use algebra since high school and I dropped out a year early.
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Turtlebread posted...

yes it's obviously 0.999....8

What makes it have an 8 at the end? What makes it end in the first place?
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Virus731 posted...

If I couldn't be bothered to understand those images does it make me stupid at math? I've literally never had to use algebra since high school and I dropped out a year early.

That makes you ignorant at math. It says nothing about how stupid or smart you might be at it.
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Virus731 posted...

If I couldn't be bothered to understand those images does it make me stupid at math? I've literally never had to use algebra since high school and I dropped out a year early.

No one is stupid at math. Don't bother learning it if you don't want to. I don't know lick about Shakespeare and it's never hurt me in life.
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Cheese_Crackers posted...

0.999...=1 relies on some fairly sophisticated mathematical machinery, though, and some of which is chosen arbitrarily. Read post 18 for a hopefully straightforward example, but making just one choice differently changes that fact.

nah. the 0.999... = 1 having any kind of complexity behind it is nothing but smoke and mirrors that are a consequence of a base 10 number system.

It's fairly obvious to see when you explain it under another number system.

For example, going back to the 0.333... proof, why is that number 1/3rd written that way in base 10? Because it's actually impossible to accurately depict that number as a decimal in base 10. You'd have to write 3s to infinity. But if you were to to write it as a decimal in a base 3 number system, it's actually just 0.1

And so what else is true about a base 3 system? Well, 1+1 = 2, but 1+1+1 = 10.

so if you add 0.1 + 0.1 + 0.1 in base 3, you get 1.

But in base 10, you'd get 0.999...

The number has the same mathematical value no matter how you picture it with symbols.

And by the same token, in base 3, 0.222... = 1.

It's just a property of the base number system. For any Base N, 0.(N-1) repeating = 1.
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ITT, delivery happened. I learned a lot today.
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ChromaticAngel posted...

nah. the 0.999... = 1 having any kind of complexity behind it is nothing but smoke and mirrors that are a consequence of a base 10 number system.

It's fairly obvious to see when you explain it under another number system.

...

It's just a property of the base number system. For any Base N, 0.(N-1) repeating = 1.

So you either have the sophisticated mathematics that go into understanding numerical bases and base changes, or the the sophisticated mathematics that go into understanding infinity and infinite sums. Either way, we've moved beyond the scope of everyday life.
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I never got how both this and asymptotes exist at the same time. They both get infinitely close but are never actually at the "real" value.
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FrndNhbrHdCEman posted...

If something's missing it isn't whole.

this
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Yes, but what does 999... equal?
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Hey posted...

I never got how both this and asymptotes exist at the same time. They both get infinitely close but are never actually at the "real" value.

Asymptotes are a different kind of object. They occur when you look at the relationship between a continuous set of values and certain computations on that continuous set of values.

With 0.999... you're not "getting close" to anything. You simply are at the value 1, and you're writing it in an unusual way.

Now, if you want, you could look at the set {0.9, 0.99, 0.999, 0.9999, 0.99999, etc.}, and in that case you do have a series of numbers that eventually gets "infinitely" close to 1 but never reaches 1. However, that set of numbers doesn't actually contain the number 0.999..., because every number in that set eventually stops having 9's and 0.999... does not.
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x = 0.999...
10x = 9.999...

10x - x = 9.999... - 0.999...
9x = 9
x = 1

0.999... = 1
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Hey posted...

I never got how both this and asymptotes exist at the same time. They both get infinitely close but are never actually at the "real" value.

An Asymptote is a function of a dependent variable. the number 0.999... is a constant. It's a flat value on a line and it doesn't change. Try graphing 1/x and then also graph 1/5 and see how different it looks.
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DevsBro posted...

ITT, delivery happened. I learned a lot today.

Good to hear :) I'm always interested in making math more accessible to people.
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Is this topic about category theory yet?
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MutantJohn posted...

Is this topic about category theory yet?

I'm sure that 0.999... is a limit object in some category which is also complete, co-complete, and autobotically isomorphic to the very large category of cats
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Except the sum of (1/10)^n does not equal 1, it has a limit of 1 but does not equal it.
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0.999... does not equal 1 because Wildberger told me the reals don't exist.
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That image went to a lot of trouble but I don't see how it's any more definitive than saying

1/3 = 0.3333...

2/3 = 0.6666...

0.3333... + 0.6666... = 1/3 + 2/3

1/3 + 2/3 = 1

0.3333... + 0.6666... = 0.9999...

Therefore 0.9999... = 1

I mean this seems like it's just a much more convoluted version of the same thing. If the transitive property proof doesn't convince you than how is this any better?

Really this all gets down to a semantics argument. Repeating numbers aren't real numbers so much as imperfections of the decimal system in showing certain values. So whether it equals one or not depends on how you look at it. I wouldn't say 0.9999... equalsone so much as it is a redundant and pointless because there is no actual imperfection of the decimal system it applies to.

I just say 0.9999... isn't a number.

Or a better way to put it is that, I don't see these "proofs" as being proof that 0.9999... = 1. I see them as proof that repeating decimals are imperfect representations of the values they represent. They are just proving the flaws of the decimal system.
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SpoiltTrouser posted...

Except the sum of (1/10)^n does not equal 1, it has a limit of 1 but does not equal it.

Actually that part was equal to 1/9. Any sum of the form (1/10)^n with finitely many terms doesn't equal 1/9, but if you take the sum with infinitely many terms, then it converges to 1/9, and convergence is the only way to assign a meaningful value to infinite sums.

The reason it feels so weird is that 0.999... is really simple to write down, but to make sense of it in a rigorous way (i.e. a way that allows you to do proofs), you need to talk about convergent infinite sums. It's a complicated gadget with a deceptively simple representation.

action52 posted...

That image went to a lot of trouble but I don't see how it's any more definitive than saying

1/3 = 0.3333...

2/3 = 0.6666...

0.3333... + 0.6666... = 1/3 + 2/3

1/3 + 2/3 = 1

0.3333... + 0.6666... = 0.9999...

Therefore 0.9999... = 1

I mean this seems like it's just a much more convoluted version of the same thing. If the transitive property proof doesn't convince you than how is this any better?

It depends on how pedantic you're feeling. Writing down 0.333... = 1/3 isn't a given statement, it requires its own proof. There's no obvious reason a priori that 1/3, the multiplicative inverse of 3, should be equal to an infinite representation of some kind. The way to prove that is essentially identical to the way I proved that the sum of (1/10)^n is 1/9.

Your proof is probably convincing to people that aren't extreme skeptics, but there's still a something in it that needs proving beforehand.

As for the rest of your post, I understand the hesitation to accept it, but the fact is that 0.999... is an infinite sum, which is much more complicated than writing it down would seem to imply. Infinite sums lead to unintuitive results all the time.
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Nice to read about p-adic numbers. You don't see that very often. I guess one could say that hyperreals contain numbers that "kind of" lie in-between 1 and 0.999..., but hyperreals aren't so common either.
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