Math paradox
 

Hi all!
Here's my math paradox, which I already posted on a few forums, and now I thought let's post it here!

To prevent that this post goes over the character limit, let's say a sequence of numbers between brackets stands for infinite repetition.
Example: 0.[3] = 0.333333333333333...

I say that 0.[9] = 1

People have been arguing about this forever. Sometimes, decimals *cannot* be as accurate as fractions.

3/3 ~= .999999999...

It just gets you "close enough" to 1.

It's true that sometimes, decimals *cannot* be as accurate as fractions, but in this case I'm not discussion a fraction :) I'm discussing the number 0.[9]. In fact, there is no fraction accurate enough to display 0.[9]. So also the opposite of what you said is true: Sometimes, fractions *cannot* be as accurate as decimals.

If 0.[9] extend to infinity, when you multiply by 10, it now extends to infinity minus one, therefore when you subtract you have an extra decimal place in x that is not in 10x.



Yay for crazy math. :p

"If 0.[9] extend to infinity, when you multiply by 10, it now extends to infinity minus one" is wrong, "If 0.[9] extend to infinity, when you multiply by 10, it now extends to infinity plus one" is correct, but since calculus operations on infite are forbidden(see bottom of my post) we multiply the starting number by 10 resulting in the decimal point moving one position to the right.

1 != 1 - 0.[1]

Every single decimal brings it further away from 1:
1-0.1 = 0.9
1-0.11 = 0.89
1-0.111 = 0.889


Why calculus operations are forbidden on infinite:

When I want to browse 4chan I usually go to the site, not wowinterface.

The weird thing is that some people can accept that

1/3 = 0.3̅
2/3 = 0.6̅

And not that

3/3 = 0.9̅

(The overbar is important.)

Chit-Chat
A place to chit-chat about anything off topic.

Yes, but this has to be the most re-posted subject on 4chan.

Maybe a new rule: Do not repost items commonly found on 4chan :banana:

Incorrect. Someone forgot how to convert fractions to decimals.

3/3 = 1. 100% of the time. Always. Any fraction in which the numerator and denominator are equal non-zero values is equal to 1. Let's go back to the time-honored "pizza analogy", shall we?

You have three pizzas. Your three pizzas (as a "whole entity" of food) are separated into three parts. How big is each of the three parts of three pizzas? One pizza. 1. Uno.

You see, you guys are looking at this "paradox" the wrong way. The paradox lies within the fallacy of attempting "accurate" math using the infinite. You might as well divide by zero. In fact, I'll try that now...

OH SHI-

It's not incorrect because 0.999, repeating = 1

They are exactly the same value, there's no difference between them, they're just different ways of writing the same thing. That is not new information; it is taught in high school math. The fact that it seems weird is an artifact of human thinking. It's not a paradox anymore than "Zeno's paradox" is.

As a thought experiment for those who aren't convinced by anything above: what is 1 minus 0.9999 repeating. It would be 0.000000, followed by an infinite number of 0s, followed by a 1 right? There's a reason that sounds impossible. :p

No. They're not equal. That'd be like saying 99/70 is the same as √2.

They're close, yeah. But definitely not equal. You're rounding infinity, which eliminates the entire point of the concept of infinity (otherwise you might as well say √2 = 1, or π = 3).

Take for example your "thought experiment". The correct way to think of the answer is not 0, but an infinitely small amount of something. Something cannot be nothing.

Also, infinity isn't a number. Take for example 1/∞. What you're attempting to say (in a roundabout fashion) is that 1/∞ = 0. Sorry:



That hyperbola is gonna go on for infinity, but it's never gonna touch an axis, ever. In fact, by looking at a plot one could make the argument that ∞ is nothing more than the opposite of 0.

Oh, and to help explain it here's a bit on it known as the "ghosts of departed quantities", which is a paradox involving infinitesimal calculus. Sorry it's a bit late...took me a while to find this. It's been a loooong time. :p

To consider an example, the function y = x^2 is differentiated in calculus by forming the quotient

Δy/Δx

of the y-increment, usually denoted Δy, over the x-increment, usually denoted Δx. The resulting expression simplifies algebraically to

2x+Δx

To obtain, instead, the familiar expression for the corresponding derivative, namely 2x, one needs to strip away the infinitely small quantity Δx. Thus, the infinitesimal quantity Δx seems to be assumed nonzero at the stage of calculating the quotient, and yet it is assumed zero in the last phase of the calculation when Δx is stripped away. In summary, we have departed quantities (Δx = 0) which are yet present in some ghostly fashion (Δx ≠ 0).

Everyone here is trying to make mathematical arguments, when the real issue everyone has is with the decimal numbering system.

The infinite series 9/10 + 9/100 + 9/1,000 + 9/10,000 + ... isn't a number. It converges to a number, which happens to be 1, but the series itself isn't equal to 1. However, that's not the issue.

The issue is that the part of a decimal number which comes after the dot has two parts—the part that is always there, and the part that repeats. Every rational number, in decimal form, has two parts—some leading digits and then a repeating, periodic part.

For example, 1/2 is 0.5[0]—a leading "5" and then an infinite number of zeroes. The infinite number of zeroes doesn't make 0.5 any less accurate. Or 1/3, which is 0.[3]—no leading digits, and an infinite number of 3s. Or 1/12, which is 0.8[3]—a leading 8, then an infinite number of 3s. Written as a decimal with a finite number of digits, only 0.5 is exactly equal to 1/2—the rest could only be approximations. But the decimal system we use isn't a system of approximations—it's a system that embraces infinite repetition wholeheartedly.

The weird thing about the decimal number system is that some numbers have multiple representations. For example, one can be written as 1.[0] or 0.[9]. If you were to posit that 0.[9] wasn't equal to one, well, what is it? I don't want to define 0.[9] as an infinite series—I'd prefer all my closed-form decimal expansions to come out to rational numbers, thanks all the same.

The real reason I believe all the explanations that say that 0.[9] = 1 is that every decimal number we've ever used has an infinite repeating portion, and that doesn't keep them from being real numbers.

Also: Knuth said it, it must be true.

Uhm...wow. :eek:

*deep breath*

What's 0.[0]?

Think about that.
1. No.

2. Ok, you can say that, yeah. What's the difference here? The repeating portion is zero. You're saying 0.[0] is the same as 0.[1], 0.[2], 0.[3], etc etc...

This isn't that hard, guys. You can try to convince yourself in every way imaginable...but 0.[9] is NOT one, and 0.0000......1 is NOT 0.

You cannot append to an infinitely sized list.

0.[0] is the same general idea as 0.[1] and 0.[2], etc. Without the ability to use infinite repeating sequences after the decimal point, we limit the decimal numbering system to only being able to accurately represent numbers that evenly divide into whatever number we choose as the base (10, in this case). At some point, mathematicians decided that having to deal with infinite repeating sections after the decimal point was worth the hassle—as a result, we have a decimal numbering system that is well defined for all rational numbers. What's even better is that absolutely nothing breaks if we consider 0.[3] exactly equal to 1/3, or 0.[9] exactly equal to 1.

Sure, we could just as easily define the decimal numbering system in such a way as to only be exact when we're talking about nice, neat numbers like 1/2 or 1/5. But then we've have to start throwing around the ≈ symbol around a lot more.

There are multiple potential ways to define the decimal numbering system, and the math world in general happened to pick one that has a weird consequence. We could've defined it differently, but it's still a definitional issue, not an absolute rule of the universe.

Well of course not...just like how an infinitely small list still exists.

(Not really following where you were going with this...just a random declarative statement?)