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Any mathematicians willing to do a little math?

#1
Ok, so I formulated this paradox based on how i see fractions, although it may have a flawed premise, please let me know if I made a mistake!

Liviticus's Paradox of Fractions
Flawed Theory wrote:1/3 = 0.333... Therefore, 1/3 = 0.333... + ...333, meaning you can add a 3 after every last decimal, from this you can occlude that if you map out 1/3 you will find that it is ever expanding based off of the numbers although we know this is not the case as dividing a candy bar (for example) into 3 equal parts would mean each individual part is ever-expanding, therefor the whole candy bar is ever expanding, this is not possible because we know that 1 < 1 is not logically possible, although this means that there should be an end to the decimal form of 1/3, although this will not mathematically make sense as it is reoccurring.

After further calculation I saw that the 0.333... Will never reach 0.4 therefore it isn't ever-expanding, although the 2nd paragraph is still relevant.

Also, with this particular example, 0.333... Is a 3rd, meaning that 0.333... * 3 = 1, although we know that this is not the case because 0.333... * 3 = 0.999... Therefor, 1/3 *3 = 0.999... We also know that this is not the case because a 3rd * 3 = 1.

The paradox comes in when you conclude from this that 1/3 * 3 is either >1 or <1 but never 1, although 1 being the only answer we can give to 1/3 * 3, neither of these are possible. From this we can conclude that 1/3 does not equal 0.333... And that the true solution is either an imaginary number or an undiscovered irrational digit.

I don't know if this made any sense to you, or if my premise is flawed.
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Re: Any mathematicians willing to do a little math?

#2
Your logic makes sense, but I think you're forgetting a key principle of recurring decimals.

1/3 is NOT equal to 0.333 nor 0.3333 or anything like that. It's not even equal to 0.3333333333333 (with a billion 3s). The fact with recurring decimals is that they are infinitely long. The more 3s you add, the closer you are getting to the TRUE value of 1/3, but never quite reaching it. This is called TENDING to 1/3. So, 0.333 * 3 will be CLOSE to 1, and 0.33333 * 3 even closer. However, because the nature of writing decimals, you cannot ever reach infinity 3s, so you will TEND TO 1, but never quite reach it. So, how do we get around this? This is why we use fractions here. 1/3 is equal to 0.3333... with infinity 3s. 1/3 * 3 IS one, because it is a way of writing the decimal with infinite threes.

If you divide the candy bar, you're dividing it into 1/3 pieces. You're dividing it into 0.333333333333333333333333333 (to infinity) but not into 0.3333. The fact you use the ellipse (...) goes to show that it tends to infinity, where your reasoning seems like you are confusing 0.333 to equal 0.3333 to equal 0.33333 and so on, which it doesn't. 1/3 equals only one number and that's 0.3333 with infinity 3s.

Hope this makes sense! :)
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Re: Any mathematicians willing to do a little math?

#3
1/3 * 3 = 1 is true

1/3 = 0.3333 Is false since 0.3333 Is only a decimal approximation of the fraction.

The candybar analogy is flawed since you cant divide it into 3 equal parts, it will have 0.33, 0.33 and 0.34 pieces. Its not an ever expanding entity. Just like a dollar divided into 3 people, one will get 34 cents.
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Re: Any mathematicians willing to do a little math?

#10
@Aileron

So, deviding by 3 and getting an exact result is technically impossible? Or am i missing something? Is this only if it is a devisible of 10 or a similar number?

What if i take a strand of 3 atoms of the same diameter, now i break the strand into the 3 equal pieces each consisting of no more than or less than an atom, will this technically be devided into 3 equal pieces?

Sorry if i'm asking obvious quastions, i'm more of a mechanical/ science guy, although i want to know why i am writing a number before i write them, same with x^0=1.
I am a Moderator Need any help? Private message me here or send me a mail in-game, Liviticus on every world.

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