DrMatrix,
The number .999… is not defined as a limit. The conversion to a limit is done so that useful calculations can be performed, and it is NOT an exact conversion. .999… approaches 1, so the limit is defined as 1. However, .999… never actually reaches 1. Because you asked, the error that is disregarded is infinitely small, say equal to 1/infinity. Just because it is infinitely small does not mean it does not exist.
Zeno’s paradox only illustrates that a finite can be infinitely sub-divided. It is a nonsensical word-game.
Orbifold,
The moment that you attempt to use .999… in an equation it must be converted into a limit. Otherwise it would be impossible to actually finish said equation. The necessity of such a conversion leads to the introduction of an infinitely small margin of error.
You do not seem to understand that there does not need to be a number between .999… and 1. The fact is .999… is the number that is closest to 1 yet still less than 1. It IS THE CLOSEST! There is no number closer yet still less than 1! Is that so hard to understand?
.999… is not, by your definition, a real number. Why do you insist that it is?
kabbes,
You are so very close to the truth, yet just one step away. 0.333… does not equal 0.4 in base-12. The 3 keeps repeating so that it can get closer to the number intended, yet it must repeat an infinite number of times to reach such a point. However many times it repeats it will always be less than infinity, so it will never exactly equal 0.4 in base-12.
All
Think of .999… as an abstract concept. This number approaches 1 infinitely closely, yet does not ever reach it. In practice it can be considered to be equal to 1 only because the error introduced is infinitely small. However, it is still not equal to 1.
Achernar, I don’t understand how you can determine a limit except as the result of the addition of an infinite number of converging terms.
Zeno, despite what some think, perfectly well understood in his paradox that what he was doing was essentially the equivalent of cutting a finite length into an infinite number of pieces. His only problem was that it was not yet understood how an infinite addition could converge.
And it’s a good leap to go from 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + 1/7 + 1/8 … is infinitely large
especially when you cannot express the fractions in so easily manipulable a form.
I guess I have to pass the question back to you. How do you understand the concept of a limit without involving infinity?
And as a side point, I understand the formalist objection to the algebraic proof. But it’s so incredibly useful and such a wonderful aha! experience for schoolkids that I’d hate to eliminate it on a technicality.
Phage, your first point is wrong. Your second point is wrong. Your third point is wrong. Your fourth point is wrong.
It all seems to be the same error. You do not “convert” .999… to a limit. The infinite expansion is a limit. And therefore it is not infinitely close to 1, but exactly equal to 1.
By definition, .333… (notice the ellipsis) is 1/3. Same for .999…, which is equal to 1. They are both real numbers, by the definition of “real numbers.”
By “by definition”, I mean, using the standard mathematical notation and standard interpretation of that notation.
By the definition of real numbers, there cannot be another number (which you call .999…) that is the next number less than 1.0. So, .999… has to equal 1.0 in order for all our definitions to be consistent.
No, numbers in general and 0.999… in particular do not approach. They have fixed values. The sequence (0.9, 0.99, 0.999, …) approaches 1, the number 0.999… equals 1.0. There is no real number that is infinitely small.
Zeno’s paradox is the same as the OP.
If 0.999… is the closest number to 1 with nothing in between, then what about (1 + 0.999…) / 2? Surely it is a number. How would you represent it in decimal?
Really? Consider the sequence { 1, 1/2, 1/3, 1/4, … }. Do you think of the limit of this sequence in terms of an infinite sum? If so, what would that be? All convergent sequences are fundamentally like this one, even if they’re stated as series. The limit of a series should be thought of as the limit of partial sums, not as the sum of an infinite number of things.
Whatever the heck .999… is, it’s just one number. And it’s the same number today that it was yesterday. It ain’t going anywhere. So how is it approaching anything?
The sequence [.9 .99 .999 .9999 .99999 …] is a bunch of numbers. It is approaching something. In fact, it’s approaching 1.
But here’s the key point. The number .999… is not one of the numbers in that sequence. .999… is defined to be the number approached by the sequence [.9 .99 .999 .9999 .99999 …]. The number approached by that sequence is 1. Therefore, .999… is defined to be 1.
Phage, what do you think it means when a mathematician says that “something exists”? He means exactly that the thing in question does conform to all of the properties of the set in question (such as real numbers, plane figures, vectors, whatever). If the thing in question (i.e. 0.999…!=1) can be shown to conflict with the properties of the set in question (i.e. the real numbers) then you have to conclude that this thing (i.e. 0.999…) is not a real number . Mathematicians have a shorthand saying for this, “it does not exist”.
Now, if you still want to argue that it has some kind of properties of some other kind of number, well, have at it. I can even suggest a starting point, Surreal numbers are discussed at this link, and seem to have some kind of the properties that you’re talking about.
But, if you claim that 0.999… exists and is a real number not equal to 1, you are not going to get along with all of the mathematical richness associated with calculus, algebra, differential equations, and so forth.
Whether you like it or not, what I did is the standard method for turning a repeating decimal into a Fraction. It works for any decimal that can be expressed in a fraction. If you can find anything wrong with the math, I’d love to hear it (and your talk about convergence is irrelevant. I’m not doing calculus or limits; I’m subtracting numbers.)
x = 0.3333…
10x = 3.3333…
10x - x = 3.3333… - 0.3333…
9x = 3.0000
x = 3/9 = 1/3 thus 0.3333… = 1/3
Now no one is arguing that .33333… <> 1/3, are they? That was, IIRC, one of the first premises of this thread.
Or there’s this:
x = 0.142857142857142857142857142857…
1000000x = 142857.142857142857142857142857142857…
999999x = 142857.00000
x = 142857/999999 = 1/7
or to simplify:
x = 0.5
10x = 5.0000
(The decimal element has been removed from the equation, so there’s no need to subtract.) Thus:
10x = 5 x = 5/10 = 1/2
Now, are you willing to argue that this is false?
The fact is, this proof always turns a decimal into it’s equivalent fraction. You’re welcome to try to find a counterexample.
I’m not doing anything to trick anyone. I’m not arguing anything like convergeance. I’m not talking about infinity. I’m talking about numbers. It works for any repeating decimal and is accepted throughout mathematics. Unless you can actually find an error in the formula I’ve shown, you have no basis of arguing it is wrong. And if the method is correct, then 0.9999… = 1.
Whether you like it or not, what I did is the standard method for turning a repeating decimal into a Fraction. It works for any decimal that can be expressed in a fraction. If you can find anything wrong with the math, I’d love to hear it (and your talk about convergence is irrelevant. I’m not doing calculus or limits; I’m subtracting numbers.)
x = 0.3333…
10x = 3.3333…
10x - x = 3.3333… - 0.3333…
9x = 3.0000
x = 3/9 = 1/3 thus 0.3333… = 1/3
Now no one is arguing that .33333… <> 1/3, are they? That was, IIRC, one of the first premises of this thread.
Or there’s this:
x = 0.142857142857142857142857142857…
1000000x = 142857.142857142857142857142857142857…
999999x = 142857.00000
x = 142857/999999 = 1/7
or to simplify:
x = 0.5
10x = 5.0000
(The decimal element has been removed from the equation, so there’s no need to subtract.) Thus:
10x = 5 x = 5/10 = 1/2
Now, are you willing to argue that this is false?
The fact is, this proof always turns a decimal into its equivalent fraction. You’re welcome to try to find a counterexample.
I’m not doing anything to trick anyone. I’m not arguing anything like convergeance. I’m not talking about infinity. I’m talking about numbers. It works for any repeating decimal and is accepted throughout mathematics. Unless you can actually find an error in the formula I’ve shown, you have no basis of arguing it is wrong. And if the method is correct, then 0.9999… = 1.
Achernar, now I’m confused. {1, 1/2, 1/3, 1/4 … } is divergent, not convergent. It “sums” to infinity. How is this an example of a limit? All it does is lead to impossible results as lucwarm shows because he is using regular arithmetic on an infinite expansion. Infinity has its own arithmetic. Seems to me that this is just another reason for requiring infinity in one’s thinking about these matters.
And I stress in thinking about it. This may not be technically correct in the formalist sense, but it’s the way to get non-mathematicians to make the mental leap to comprehension.
For those who still don’t think that a infinite convergent summation is exactly equal to its limit, go back to Zeno’s paradox. (The one in which an arrow goes halfway to the target and then half the remaining distance etc.)
As DrMatrix says, this is the equivalent to the statement that .999… = 1. Instead of expressing it in halves, you are merely expressing it in tenths instead. Obviously, the arrow does not get infinitely close to the target – it reaches the target.
Your method hinges on the assumption that .3… or .9… actually represents a real number. It does, but once you’ve got that, you’ve already got the desired result, and you don’t need the rest of the argument.
IOW, what you’re saying is correct, but redundant.
Yes, I do and IANA mathematician but everytime this subject comes up in GQ or GD (or anywere) I open the thread and read it top to bottom, love it, dig it, but can’t really contribute anything. You guys are doing a wonderfull job. Keep fighting, every lurker who graps this is another soul saved from ignorance.
RealityChuck,
The problem with your formula is the first operation you perform.
x = 0.999…
10x = 9.999…
How exactly did you do the math for a number that has no end? You would never be able to complete that first operation, so your formula is useless.
DrMatrix, Chronos,
When I refer to 0.999… approaching 1, I am referring to a common way of thinking of infinity; very few people can picture infinity at once, so they take it in chunks. As you consider more and more of the infinite 9s you get closer to the real standing of the number.
DrMatrix,
(1 + 0.999…) / 2
The above cannot be completed. You would be attempting to divide an infinite; it cannot be solved in a finite time.
Race Bannon,
There are plenty of numbers like 0.999… that still exist. How about pi? The ending point of pi has not been found, and is projected to be unending. Pi does not have a limit such as 1!
All,
The problem with the number 0.999… and others such as 0.333… is the inability of such numbers to be expressed absolutely accurately in decimal form. (1/3) is an exact number, and can be expressed exactly in base-6 as (.2). However, base-10’s best approximation is (0.333…) which is slightly less than (1/3).
It seems to me that anyone who believes “.9999… = 1” does not understand the pure meaning of infinity.
Note that, according to the link provided, infinity holds a quality of “having no boundaries or limits”. Equalling “1” would be a limit, correct? I smell an obvious contradiction of reason here.
.99999…, having nines that list on and on forever, infinitely, is a non-terminating decimal. The difference between this value and “1” will always decrease tenfold as long as another nine is revealed (which it is, or course). The difference gets smaller and smaller, endlessly, trivial and infinitesimal. But it is always there.
It’s easier to say that .99999… = 1 than it is to believe that something can get smaller and smaller without becoming non-existent. This is the rub that so many of you deny.
I don’t claim to be a mathematician, or that I have extensive education on the subject. But I have realized that mathematics is our most useful tool for understanding the world we live in. Unfortunately, a very nasty side effect is the way a mathematic mind tends to limit itself to reason built from systems that are too rigid, formulaic, and limited by “boundaries” (ah? the post recirculates itself!) which reflect an unopen mind.
Really? Consider the sequence { 1, 1/2, 1/3, 1/4, … }. Do you think of the limit of this sequence in terms of an infinite sum? If so, what would that be? All convergent sequences are fundamentally like this one, even if they’re stated as series. The limit of a series should be thought of as the limit of partial sums, not as the sum of an infinite number of things.