Math paradox

[nightcracker]

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

  x     = 0.[9]
10x     = 9.[9]

10x - x = 9.[9] - 0.[9]
 9x     = 9
  x     = 1

x = 1
x = 0.[9]

x = x
1 = 0.[9]

[Seerah]

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.

[nightcracker]

Originally Posted by Seerah 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.

[Starinnia]

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.

[Slakah]

0.[9] = 1 - 0.00000000....1
so:
1 = 1 - 0.00000000....1
1 = - infinity?

[nightcracker]

Originally Posted by Starinnia 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.

"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.

x = 1.000000x
10x = 10.00000x

Originally Posted by Slakah 0.[9] = 1 - 0.[1] so: 1 = 1 - 0.[1] 1 = - infinity?

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:

I say x + 1 → x

First we prove it wrong:
x + 1 → x -- minus x
1 → x - x
1 → 0 -- FALSE

If we would assume calculus operations on infinite would be ok, look what happens:
x → ∞
x + 1 → x -- divide by x
1 + 1/x → 1
1 + 1/∞ → 1 -- CORRECT!

[Nexuapex]

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.)

Originally Posted by Nexuapex 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.)

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-

[Akryn]

Originally Posted by Amenity Incorrect.

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.

Originally Posted by Akryn 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.

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.

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).