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Definite proof that most of Digg readers are idiots
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CptObvious
Platinum Member
i dunno about the 1/3 = .3333 part. that requires the same premise as 2/3 = .6666 and 3/3 = .9999 = 1. each 3, 6, or 9 you add brings it to closer approximation, but it's an imperfect expression of a fraction. but then again i'm no math major so take it fwiw
How about Zeno's dichotomy paradox? If phrased slightly differently, it's pretty much what is being argued about on this thread. To wit, imagine you are shooting an arrow at a target exactly 1 kilometer away. Zeno would argue that before that arrow would hit its target, it would have to travel 9/10th of the way there (or .9 km). Once at that point, it would have to travel 9/10th of the remain distance (for a total travel distance of .99 km). Once there, it would again have to travel 9/10ths of the new remaining distance (for a new total travel distance of .999 km). And so and so forth ad infinitum.
How far does the arrow travel? Zeno would say .9999999.... kilometers, and it would therefore never reach the target. Common sense tells us, of course reaches it target and travels 1 km.
Born2bwire
Diamond Member
The problem is that these decimal equivalents have an infinite number of decimal places. So the idea that each extra 3 or whatever digit brings you closer to an approximation is incorrect. We already dictated that we have an infinite number of decimal places. By trying to think about adding an extra 3 to .3333333333 requires you to inherently truncate the decimal to a finite number of decimal places. 1/9=.11111...=the infinite summation of 1/10^x of x=1 to infinity. There is no approximation in the equality between those three. But as soon as you start thinking of how the summation converges towards 1/9, then you get into the trap of thinking of it as a finite problem, where we have a finite summation. But .11111... != summation of 1/10^x of x=1 to 'some incredibly large number'. The equality only holds for infinity. Conceptually it's one way to show the equivalence of the three, but it also becomes a point which people try to use to incorrectly discount the equality (ie: It never equals 1/9, it approaches and approximates it). In the physical sense of real life, infinity is something that we can never comprehend or obtain. But in the sense of mathematics, infinity is tangible value that we can use.
This is not a proof.
Going back to fundamentals.
10x = 0.9999.... + 0.9999.... + 0.9999.... + 0.9999.... + 0.9999.... + 0.9999.... + 0.9999.... + 0.9999.... + 0.9999.... + 0.9999....
10x - x = 0.9999.... + 0.9999.... + 0.9999.... + 0.9999.... + 0.9999.... + 0.9999.... + 0.9999.... + 0.9999.... + 0.9999.... + 0.9999.... - 0.9999....
cancel the last two terms out and do the addition
9x = 0.9999.... +
0.9999.... +
0.9999.... +
0.9999.... +
0.9999.... +
0.9999.... +
0.9999.... +
0.9999.... +
0.9999....
= 8.99999...
!= 9
So the basis for the proof is based upon an algebraic trick.
Not saying that that it cannot be proved by another method, but this is not the proof.
HA! I can do that 🙂
Whoozyerdaddy has made the assumption 1/3 = .33333...... exactly, let's test this assumption.
back to basic division. Sorry for the bad (or total lack of) formatting.
1/3 = 0.3 with 0.1 remainder.
= 0.333 with 0.001 remainder
= 0.3333 with 0.0001 remainder
= 0.33333 with 0.00001 remainder
= 0.333333 with 0.000001 remainder
rinse and repeat an infinite number of times
1/3 = 0.3333.... with an infinitesimally small remainder divided by 3.
Therefore 1/3 = .33333...... is an approximation.
The assumption does not hold up, this is not the proof.
Running throught the calculation again (I am calling the infinitesimally small addition to the number (to make it exact) "remainder", I am not sure of the mathematical nomenclature for such a concept);
1/3 = .333... + remainder/3
+
2/3 = .666... + 2*remainder/3
-------------------
3/3 = .999... + remainder
3/3 = 1
footnote. If you say that the infinitesimally small remainder can be ignored then you are making the assumption that an infinitesimally small amount can be ignored to prove that an infinitesimally small difference = 0. This is circular reasoning.
EDIT: clarified statement.
glenn1
Lifer
You should both try using their other brain hemisphere to re-examine the question. For example, most folks would inherently understand that "gorgeous" and "beautiful" are different ways of expressing the same concept yet are undeniably different words. Likewise, 1 and .99999 still express the same concept but are not the same number. While Unheard and his like grasp the second concept, they incorrectly understand the = sign to mean "exactly alike" instead of two mathematically same values. Unfortunately for Mwtgg and the others supporting his position in the thread, they seem incapable of explaining this to the non-math folks (or not understand it themselves) so can only impotently rage that "of course .999999 = 1" without resorting to algebra or calculus to do so, which is like using the very word you're trying to define in your definition.
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