Poll: Does 0.999.. equal 1 ?
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I originally picked no, despite seeing all the various proofs to the contrary (which are infact correct). I, like a lot of people, was incapable of understanding how 0.999... = 1
Human perception makes it hard to accept the idea. We think to ourselves, "but the 9s go on forever! They will always get closer to 1, but never reach it." Unfortunately, the nature of infinity and infinity small numbers works in a highly counter-intuitive fashion: it is entirely because the 9s go on forever that you can't say it is any less than 1.
The straight dope article gives a practicle example of this problem, known as Zeno's paradox: If a man is racing against a tortoise, and the tortoise has a ten meter head start, you would expect the man to reach the tortoise very quickly. But by the time the man has run the ten metres, the tortoise has moved foward 1 meter. By the time the man has run another meter, the tortoise will have gone on another .10 meter, and so on and so forth. If this goes on for infinity, how can the man ever overtake the tortoise?
In the real world, we know that the man can obviously overtake the tortoise in barely any time, so what does this say about the nature of infinity? Just because it goes on forever, doesn't make it an unachievable task.
Human perception makes it hard to accept the idea. We think to ourselves, "but the 9s go on forever! They will always get closer to 1, but never reach it." Unfortunately, the nature of infinity and infinity small numbers works in a highly counter-intuitive fashion: it is entirely because the 9s go on forever that you can't say it is any less than 1.
The straight dope article gives a practicle example of this problem, known as Zeno's paradox: If a man is racing against a tortoise, and the tortoise has a ten meter head start, you would expect the man to reach the tortoise very quickly. But by the time the man has run the ten metres, the tortoise has moved foward 1 meter. By the time the man has run another meter, the tortoise will have gone on another .10 meter, and so on and so forth. If this goes on for infinity, how can the man ever overtake the tortoise?
In the real world, we know that the man can obviously overtake the tortoise in barely any time, so what does this say about the nature of infinity? Just because it goes on forever, doesn't make it an unachievable task.
Most analytical proofs rely on the fact that you can make small perturbations to an object without it altering it (these are usually denoted epsilon or delta). So by the same logic any infinitessimal difference between 0.999... and 1 is mathematically insignificant. Although if you study analysis you should already know decimals are a really bad way of trying to express irrational numbers anyway.
0.999... = 1.000...
This is universally agreed upon by mathematicians and scientists - basically, people who are experts in the nitty gritty of mathematics.
As everyone else has said, Wikipedia is your go-to place for proofs of this fact.
Common arguments against it are things like 'Oh, but there's an infinitesimally small amount between them' - NO THERE'S NOT, that's the whole point.
Whether you like it or not, mathematics allows us to prove things that our little fleshy brains might not like. Did you know you can draw a shape with finite area and infinite perimeter?
Infinity is not for the faint of heart. 0.999... is as equal to 1 as 4/8 is equal to a half. They are numerically identical, two difference ways of writing the same scalar value.
This is universally agreed upon by mathematicians and scientists - basically, people who are experts in the nitty gritty of mathematics.
As everyone else has said, Wikipedia is your go-to place for proofs of this fact.
Common arguments against it are things like 'Oh, but there's an infinitesimally small amount between them' - NO THERE'S NOT, that's the whole point.
Whether you like it or not, mathematics allows us to prove things that our little fleshy brains might not like. Did you know you can draw a shape with finite area and infinite perimeter?
Infinity is not for the faint of heart. 0.999... is as equal to 1 as 4/8 is equal to a half. They are numerically identical, two difference ways of writing the same scalar value.
Incorrect. Vast majority of engineers know math past 5th grade, and so will correctly answer that the two numbers are equal. Again, this is not an opinion, an open discussion, or a debate.Thaliur said:
"Pragmatic/realistic" mind will also realize, that a *number* and a *usable in practice* number are two different things, and in this question *usable in practice* was never raised. Therefore, *numbers* are being considered.
Zeno's paradox confuses by talking about distance, but measuring with time. Given the Paradox as it stands, the Man can never pass the tortoise because the time at which he does is never reached.maninahat said:
This is why the equivalency was brought in, but also why 0.9 recurring cannot equal 1. Because .9 recurring cannot finitely exist; it is, in itself, an irrational number - therefore it has a rational equivalency of 1. It can't equal 1, because 1 is rational.
If people want to argue and throw wiki's at me, then that's fair enough. But I'd also challenge you to find a definition of a solid material that's over 99% space - which is what it's like in real physics (Atomic Theory).
http://en.wikipedia.org/wiki/Equivalence_relationVolkov said:
*rolls eyes*
These are several incorrect statements.The_root_of_all_evil said:
1. 0.(9) can finitely exist. It exists as 1.0.
2. It is NOT an irrational number.
3. There is no such thing as "rational equivalency" of an irrational number.
This is not an argument. This is a precise mathematical statement based on direct, unambiguous conclusions from founding axioms of real number arithmetic, versus numerous incorrect statements using incorrect terms. Only one side is correct.
Solid matter is matter which can sustain shear, tensile stress, and therefore can hold shape without boundary forces acting on it. It has nothing to do with atom density, mass density, or anything similar. Therefore asking for "a definition of solid material that's over 99% space" is like "asking for definition of the term 'video game' that necessarily mentions erythrocytes." The two are entirely unrelated, and therefore the latter will not be in the definition of the former.
This shouldn't be a poll. Maths has already proven that .999 recurring is indeed 1. No other opinions needed.