And now for something completely different. A math puzzle. Or conundrum, if you will.
In the figure to the left, the bar above the number 9 indicates that it is to be repeated forever. For the remainder of this post, we will represent that concept by several nines with an ellipsis (.999...).
Now, here is the conundrum. .9 repeating is EQUAL TO ONE. Not CLOSE to one, mind you, but EQUAL to one.
Nonsense, you reply. It is obviously less than one. Not by much - by an infinitely small amount, in fact. But the simple fact (?) that it is not one is enough to demonstrate that it can't be equal to one. It's as close as you can get to one without being one.
Wrong. It is in fact equal to one, and that fact can be demonstrated mathematically in several ways.
The most easily understood is to revert to other familiar repeating digits. Everyone knows that 1/3 is 0.333... and that 2/3 is 0.666... If you add them together, you get 3/3, which is one.
But now note that the sum of the decimals on the right side of the equation is 0.999...
Therefore, one is equal to (not close to) .999...
You don't agree? Then try this. Subtract .999... from one. What you have is 0.000... An infinitely long string of zeroes, which can only be equal to zero. And if the subtraction of .999... from one leaves zero, then the .999... must be one. But, you say, there's a one at the end that string of zeroes. No, there isn't, because the string of 9s doesn't end.
There are other proofs at the Polymathematics blog, along with a long series of comments, an update refuting the counterarguments, and a final refutation of the most stubborn skeptics.
Link.
In the figure to the left, the bar above the number 9 indicates that it is to be repeated forever. For the remainder of this post, we will represent that concept by several nines with an ellipsis (.999...).
Now, here is the conundrum. .9 repeating is EQUAL TO ONE. Not CLOSE to one, mind you, but EQUAL to one.
Nonsense, you reply. It is obviously less than one. Not by much - by an infinitely small amount, in fact. But the simple fact (?) that it is not one is enough to demonstrate that it can't be equal to one. It's as close as you can get to one without being one.
Wrong. It is in fact equal to one, and that fact can be demonstrated mathematically in several ways.
The most easily understood is to revert to other familiar repeating digits. Everyone knows that 1/3 is 0.333... and that 2/3 is 0.666... If you add them together, you get 3/3, which is one.
But now note that the sum of the decimals on the right side of the equation is 0.999...
Therefore, one is equal to (not close to) .999...
You don't agree? Then try this. Subtract .999... from one. What you have is 0.000... An infinitely long string of zeroes, which can only be equal to zero. And if the subtraction of .999... from one leaves zero, then the .999... must be one. But, you say, there's a one at the end that string of zeroes. No, there isn't, because the string of 9s doesn't end.
There are other proofs at the Polymathematics blog, along with a long series of comments, an update refuting the counterarguments, and a final refutation of the most stubborn skeptics.
Link.
The fact that you can rationalize that it is thru mathematics does not in fact prove the author's point.
I believe the fault here is assuming that mathematics simply reflects some pure objective fact, when it doesn't, or something.
x = 0.999..
10x = 9.999..
10x - x = 9.999.. - 0.999..
9x = 9
x = 1
Since we assumed that x = 0.999.., and have shown that x = 1, it must be the case that 0.999.. = 1
Then the same can be said for 0.000000...1 you never reach the 1 at the end therefore you haven't really done the subtraction properly. To put it your way "the string of 0s [before the one] doesn't end."
http://xkcd.com/169/
This is old, pointless and not funny geeky maths to even the maths geeks.
roll on by, old news is old new - blizzard even did it as a joke one time its that popular - so come on....
It is what it is, a theorem if you will; and a infinitely repeating number can not be applied to the finite things we deal with on a regular basis. If we wanted to talk about infinite things this would apply and perhaps be useful but how many times have you tried to measure something infinite in the past 6 months? and how many true statements can we, as the human race, make to describe things we think to be infinite?
Human beings tend to think what's easy to understand is correct and what is hard to comprehend must be wrong. Just because you can't find a use for this or because it seems like a difficult concept does not mean it is invalid or wrong or not worth talking about it. If it does not interest you.. walk away.
http://en.wikipedia.org/wiki/Zeno%27s_paradoxes
I've asked three people about this (two are my math professors and one is my really smart friend) and they said it wasn't exactly equal to it (my current professor told me she forgot why.).
I thought since repeating decimals can be turned into x/9 as fractions, .99999999999 became 9/9 thus equal to one, but zkeletenz' argument nullifies this, I guess.
a. The expression is a convention. Its generally agreed to be true. The convention is not without its uses, but as conventions go, you will always have nay sayers.
b. 0.000...1 cannot exist as a non-terminating decimal. Essentially, it terminates at the decimal place where the 1 occurs. This means that you cannot subtract 0.999... from 1. If you don't believe me, try solving 0.999... + X = 1 within the constraints of Dedekind cuts.
c. Until someone within the mathematical community has the spine to stand up and say 1/0 = [infinity], one will always have to defend the idea that certain non-terminating decimals are rational while a non-terminating decimal like Pi is irrational.
Those who simply dismiss this age old discussion as a waste of time, fail to recognize the importance in accurately defining basic principals of math.
Keep the discussion going Netorama. I'm glad to see someone in the main stream bringing this discourse back to light.
0.3333... is not 1/3rd and 2/3rd is not 0.66666 - those are just an approximates we use for convenience. There are many fractions that can't be perfectly expressed by the decimal system. The fact that the decimals must go on forever is a symptom of that fact. Whoever originally thought this up is just using this gotcha either on purpose or unwittingly to make the 0.999 == 1.0 point.
There is no such point of contention.
0.3333... is not 1/3rd and 2/3rd is not 0.66666 - those are just an approximates we use for convenience.
No, they're exactly equal to 1/3 & 2/3. That is the entire purpose the bar notation was created, to allow an exact notation for fractions like 1/3 in decimal notation. If bar notation was not used in that way, mathematicians would have had to invent some other way to represent 1/3 in decimal.
Furthermore they are not "approximates we use for convenience". Approximates are decidedly inconvenient for countless engineering and scientific problems. Significant digits are important.
It's true that .33 is not the same as 1/3, but .333... (with the ellipsis) IS the same. [[It's awkward to use the ellipsis because the proper conventional mathematical symbol is a vinculum above the repeating decimals, as is shown in the figure, but that isn't available in the character set of the font I'm using here, so I'll continue to use the ellipsis to designating a repeating string of digits.]]
But back to the argument. .333... IS the same as 1/3 because that's the way 1/3 is expressed in decimals. One-third could be expressed in other ways in other base systems, but in a base10 system the only choice is .333... (or with the vinculum).
Understanding that, just say to yourself .333... is a way to REPRESENT 1/3. And .666... is a way to REPRESENT 2/3. Then .999... is equally a way in the decimal system to REPRESENT the number 1. [You can also represent 1 as 1.0 or 1.00. Other ways to represent it are 7/7 or 10 to the power 0.]
.999... IS the number one.
And it's not correct to say that "an infinitely repeating number can not be applied to the finite things we deal with on a regular basis." To prove it, tonight I will eat .333... of a pizza.
.3333 is as close as we can write 1/3 in numbers without having a fraction, but .999 repeating is not 1,
I think I encountered this idea in 1st grade or something when I told the teacher I'd found the smallest number EVER! and said it was 1 - .9 repeating. She thought it was cute and told me whats up.
In this case it's a function of dividing by 10 and adding to the summation. The function is asymptotic to 1. To me, asymptotic to 1 and equal to 1 are different, but not for any practical purpose.
What people above said about '.3 repeating' being the closest we can come to digitally displaying the value of 1/3, is true.
The concept of infinity is hard for the human mind to grasp; we can really only understand it abstractly. Thats why .3 repeating (and .9 repeating) cause such mental dissonance. Its kinda the same reason some people are so obsessed with finding digits of pi.
There are several proofs for 0.999... = 1, and the some are already here. Math deals with infinity and repeating decimals ALL THE TIME. If 0.333... was just an approximation, all of our upper mathematical theories would just collapse in failure.
And don't even get started with e^(pi*i) + 1 = 0. That might make people's heads explode.
x = 0.999...
10x = 9.999...
10x - x = 9.999... - 0.999...
9x = 9
x = 1
vs.
x = 0.8
10x = 8.0
10x - x = 8 - 0.8
9x = 7.2
x = 0.8
Therefore, infinite numbers are introducing calculation problems. Infinity minus infinity does not equal zero.
/takes his bow
//knows he is a wise-a&&
To explain it on its own terms, with no need for analogy. Let's say that you have a rifle with 10 horizontal aiming positions. Now, you have a target that lands at position 4 (8/2nds from your example) position. Can you hit it? Yes.
But can you hit a target at the 1/3rd horizontal position? Unfortunately, no. You need a better scope. You can change to a 100-point scope, or 1000 or 1 google plex. But you still have the same problem.
This is not a paradox - it is perfectly consistent. It is just another way of saying that you have a digital scale (10-point, 100, 1000, or whatever digital limit you are using). An infinitely-repeating digital expression will never perfectly express a fraction that does not divide evenly. That's what infinite repetition means - it is NOT FINITE.
Thanks for the support, Hoax.
Mr. Lombard: I actually thought of the concept when I was a fetus. So, as they say in Intellectual Property parlance, I "swear behind your conception date." :)