I have been taught that 13 is 0.333...
. However, I believe that this is not true, as 1/3 cannot actually be represented in base ten; even if you had infinite threes, as 0.333...
is supposed to represent, it would not be exactly equal to 1/3, as 10 is not divisible by 3.
0.333... = 3/10 + 3/100 + 3/1000...
This occured to me while I discussion on one of Zeno's Paradoxes. We were talking about one potential solution to the race between Achilles and the Tortoise, one of Zeno's Paradoxes. The solution stated that it would take Achilles 1113 seconds to pass the tortoise, as 0.111... = 1/9
. However, this isn't that case, as, no matter how many ones you add, 0.111...
will never equal precisely 19.
Could you tell me if this is valid, and if not, why not? Thanks!
I'm not arguing that 0.333... isn't the closest that we can get in base 10; rather, I am arguing that, in base 10, we cannot accurately represent 13
Answer
Here is a simple reasoning that 1/3=0.3333....
Lets denote 0.333333...... by x. Then
x=0.33333.....
10x=3.33333...
Subtracting we get 9x=3. Thus x=39=13.
Since x was chosen as 0.3333.... it means that 0.3333...=13.
No comments:
Post a Comment