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feijiduan

[1 paradox] Why 0.999... is not equal to 1?

12 posts in this topic

[1 paradox] Why 0.999... is not equal to 1?

Written in 2012

The current mathematic theory tells us, 1>0.9, 1>0.99, 1>0.999, ..., but at last it says 1=0.999..., a negation of itself (Proof 0.999... =1: 1/9=0.111..., 1/9x9=1, 0.111...x9=0.999..., so 1=0.999...). So it is totally a paradox, name it as 【1 paradox】. You see this is a mathematic problem at first, actually it is a philosophic problem. Then we can resolve it. Because math is a incomplete theory, only philosophy could be a complete one. The answer is that 0.999... is not equal to 1. Because of these reasons:

1. The infinite world and finite world.

We live in one world but made up of two parts: the infinite part and the finite part. But we develop our mathematic system based on the finite part, because we never entered into the infinite part. Your attention, God is in it.

0.999... is a number in the infinite world, but 1 is a number in the finite world. For example, 1 represents an apple. But then 0.999...? We don't know. That is to say, we can't use a number in the infinite world to plus a number in the finite world. For example, an apple plus an apple, we say it is 1+1=2, we get two apples, but if it is an apple plus a banana, we only can say we get two fruits. The key problem is we don't know what is 0.999..., we can get nothing. So we can't say 9+0.999...=9.999... or 10, etc.

We can use "infinite world" and "finite world" to resolve some of zeno's paradox, too.

2. lim0.999...=1, not 0.999...=1.

3.The indeterminate principle.

Because of the indeterminate principle, 1/9 is not equal to 0.111....

For example, cut an apple into nine equal parts, then every part of it is 1/9. But if you use different measure tools to measure the volume of every part, it is indeterminate. That is to say, you may find the volume could not exactly be 0.111..., but it would be 0.123, 0.1142, or 0.11425, etc.

Edited by Tripredacus
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[1 paradox] Why 0.999... is not equal to 1?

The current mathematic theory tells us, 1>0.9, 1>0.99, 1>0.999, ..., but at last it says 1=0.999..., a negation of itself (Proof 0.999... =1: 1/9=0.111..., 1/9x9=1, 0.111...x9=0.999..., so 1=0.999...).

As I see it, 0.999... = 1.

Proof:

1/3 x 3 = 1

.333...x 3 = .999... =1 yes.gif

Hope this helps to clear any misunderstanding.

Edit:

I see a similar proof to my proof is in the original also.blushing.gif

Still I learned in physics (long ago, not sure about current mathematical theory) that it is exactly equal to one (due to above proof) and not less than.

Wikipedia agrees: http://en.wikipedia.org/wiki/0.999...

In mathematics, the repeating decimal 0.999... (sometimes written with more or fewer 9s before the final ellipsis, or as 0.9, 7a8e7bcc450b073330c6e473493504d1.png, 0.(9)) denotes a real number that can be shown to be the number one. In other words, the symbols "0.999..." and "1" represent the same number. Proofs of this equality have been formulated with varying degrees of mathematical rigor, taking into account preferred development of the real numbers, background assumptions, historical context, and target audience.

Every nonzero, terminating decimal has an equal twin representation with trailing 9s, such as 8.32 and 8.31999... The terminating decimal representation is almost always preferred, contributing to the misconception that it is the only representation. The same phenomenon occurs in all other bases or in any similar representation of the real numbers.

The equality of 0.999... and 1 is closely related to the absence of nonzero infinitesimals in the real number system, the most commonly used system in mathematical analysis. Some alternative number systems, such as the hyperreals, do contain nonzero infinitesimals. In most such number systems, the standard interpretation of the expression 0.999... makes it equal to 1, but in some of these number systems, the symbol "0.999..." admits other interpretations that contain infinitely many 9s while falling infinitesimally short of 1.

The equality 0.999... = 1 has long been accepted by mathematicians and is part of general mathematical education. Nonetheless, some students find it sufficiently counterintuitive that they question or reject it, commonly enough that the difficulty of convincing them of the validity of this identity has been the subject of several studies in mathematics education.

Edited by BlouBul
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Won't be drawn into a philosophical debate that can be easily explained through modern calculus. ;)

[...]cut an apple into nine equal parts[...]

Number 3 falls apart because you're not cutting it right. :P

Edited by 5eraph
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Still I learned in physics (long ago, not sure about current mathematical theory) that it is exactly equal to one (due to above proof) and not less than.

Another thing I learned in high school physics was the concept of significant digits. When taking measurements, available precision depends upon the tool or method used to measure.

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Still I learned in physics (long ago, not sure about current mathematical theory) that it is exactly equal to one (due to above proof) and not less than.

Another thing I learned in high school physics was the concept of significant digits. When taking measurements, available precision depends upon the tool or method used to measure.

Exactly. The question is how close do you want to get to the answer? Even if you say one billion significant digits (which in most real world applications application will be closer to one than 1 itself newwink.gif), you can just add one billion and one 9's and you are in spec again.

Edited by BlouBul
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Thanks, Tripredacus. Didn't want to acknowledge the spam links or the added theological nonsense. :)

In theory there is no difference between theory and practice, but in practice there is.

LOL. Very true.

Edited by 5eraph
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A digitally derived value of an analog number is inherently inaccurate.

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I prefer fractions to decimal values; or for irrational numbers, the original algebraic expression or constant names (pi, Euler's number, etc.). Then I can choose how precise I want/need to be.

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Nonetheless, even PI is inaccurate and never WILL be totally "digitally" accurate. It's already been proven. ;)

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I could image the usage of this question in many machines, and blueprints. I think that question is better left off to those operators of those machines.

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