Mathematically speaking, is 1/3rd of a second a paradox?

Question

Say for example that a task takes 1/3rd of a second to complete.

1/3rd of a second is 0.33333333 etc., it's a number which never ends, no?
So, every time you reach the last 3 in 1/3rd of a second, there is another 3 after it. Theoretically it should be impossible to reach full 1/3rd. And yet simple experience tells us otherwise. Is this a paradox? If not, please tell me why.

Answer 1

In formal terms, you are confusing the abstraction of a mathematical construct with an application of a number system.



In regular English:



The fraction 1/3 is a perfectly legitimate number in every math sense. It is the multiplicative inverse of 3. Three has an inverse because the integers form a ring (an abstract algebra term).



Thus, the fraction 1/3 has a purpose and a value in mathematics.





To measure it outright, on the other hand, is difficult. It is, however, by no means impossible. No more so than measuring 10.9 or 45.63 or even 5.0 What happens is that we realize that we will never have precision that is infinitely small (precision is the possible error in measurement). Another way of saying this is that we will always have limited accuracy.



Thus, we accept that a (wide) range of exact values be represented by a number of a particular precision.



A perfect example of this is when we buy gas. The pumps measure the price of gas in tenths of a cent, but we round to the nearest cent. We accept this "inaccuracy" because our smallest coin is a penny (one cent).



And so it is with other measurements. We accept an approximation for 1/3, not because it is a paradox or doesn't make sense, but because we realize that we can never measure it perfectly anyways.



For that matter, we can never measure ANY NUMBER perfectly. Try measuring 1.0 inches (or centimeters). All you really know is that you have approximated that length to a precision beyond which you cannot discern.



In other words, "it is close enough to your eyes". That is the way we apply these perfect notions we often find in mathematics. But more often than not, these applications work and work well.





Something else that may boggle your mind is to take your decimal representation of 1/3 = 0.333... and multiply it by three:



3*(1/3) = 3* (0.333333.......)



1 = 0.99999999999........





Isn't math fun?

Answer 2

First of all, saying that 1/3 of a second has a decimal representation that does not terminate is actually quite anthrocentric. If you have an alien with 3 fingers, they'll count in base 3, not base 10. In that case, the decimal representation of 1/3 is actually 0.1, since they would count in 'thirds' going from 0.1 (one third), 0.2 (two thirds), and since the digit 3 is outlawed (base 3 only uses 0, 1, and 2), you carry the number to get 1.0 (three thirds).



This paradox is actually quite like a paradox called Xeno's paradox. I'm sure there's a good wikipedia article on it. Basically, you're adding an infinite series of numbers:



3/10 + 3/100 + 3/1000 + 3/10000 + 3/100,000 + ...



Notice that there's a "dot dot dot" meaning that this number goes FOREVER. If it DOES go forever, you reach exactly 1/3. However, if it terminates at ANY point, you will NEVER, EVER get higher than 1/3. You'll be lower, but not higher. This is the basic concept of the 'limit', and is actually quite a good deal of time is spent on problems like this in calculus.

Answer 3

It's only a paradox if you change it to a decimal. But if you leave it as a fraction there is no problem. Theoretically, if a task takes 1/3 of a second, you could do the task 3 times in a second. The second would be split into 3 equal parts.



That's why, when dealing mathematically it is so much more accurate to work with fractions (if they are repeating or non terminating) than with decimals.

Answer 4

A funny thing, I thought of this too.



I was thinking if 1 = .9999 and saw this online too



if 1/3 = .3333 = .3 repeating



2/3 = .6666 = .6 repeating



3/3 = .9999 = .9 repeating



but isn't 3/3 = 1



So what happens to that little 1 who know wheres...



I think its infintely small that we can disregard it because the 9 goes on forever.



For your question if someone reaches 0.33 sec most ppl would call that a 1/3 of a second, but speaking theoretically yes it is impossible.



Edit: yeah I agree with Kristi, I think it's because decimals are approximations whereas fractions are precise



now one can argue well 1/5 = 0.20 its exact and precise, but thats only because their are infinite amount of 0s after 2. When you have something like 1/9 = .1111 it loses its finity, if thats a word.

Answer 5

you can definite wait 1/3 of a second, you just don't have the tool to measure it exactly...or you can wait 1 second then you know you have waited 1/3 of a second three times.

Answer 6

This is just Zeno's paradox. Fortunately, it's not much of a paradox anymore.



http://en.wikipedia.org/wiki/Zeno%27s_paradoxes#Achilles_and_the_tortoise

Answer 7

No. "Theoretically it should be impossible to reach full 1/3rd." has no meaning.

Answer 8

I have seen similiar.