RE: Maths proves 1=0.999.. thus ends in self contradiction
November 21, 2014 at 6:51 pm
(This post was last modified: November 21, 2014 at 7:23 pm by Smaug.)
On the first point. I don't know the historical background that well and can't tell for how long it has been that way (so feel free to correct me) but 0,999... is 1 by definition. The trick is that 0,999... is not just 0,999...9 with a finite number of 9s in between, it's has an infinite number of 9s. Prooving that 0,9... equals exactly 1 had been a sort of a philosophical problem indeed but only before the establishment of Calculus (limits, infinite series and sums in particular). One of the most important concepts of Calculus is a limit. Not going into details, the concept of Limit is a way to understand, to formalize and to deal with certain types of infinity such as the one present above. 0,9... may be represented by an (infinite) power series: 9*10^(-1) + 9*10^(-2) + ...+ 9*10^(-n) + ... . As you can see, it's an infinite sum. Although you obviously can't find the sum by adding all the members one by one Calculus provides a way to calculate it. Now I'm not going to explain all the theorems that allow the following actions, just going to show schematically how that's done. A partial sum is a sum of the first n elements of the series. If you try to calculate several partial sums in this exact case you notice that with an increase of n they get closer and closer to 1. If you make n approach infinity you get a limit of partial sums which is the sum of given infinite series. Formally it looks like this:
lim (sum (9*10^(-n))) = lim (1 - 10^(-n)) = 1 (where n -> infinity)
Let me explain a bit here. The sum in the limit is not an infinite one. It's a partial sum of first n elements of the series so there IS a final n'th term in it. With that in mind, we can rewrite the sum as 1 minus that one 0.00...1 (10^(-n)) that separates the sum from being exactly 1. And after that we proceed to calculate the limit: with an increase of n the term 10^(-n) gets infinitesimaly small and ends up zero when n -> infinity.
To sum it up, 1 = 0,999... was a sort of contradiction untill people got better understanding of the concept of infinity. And no surprise that it's still confusing for those who look at it from an arythmetical point of view.
As for point two, I don't really get what you exactly mean but there are plenty of things that may look paradoxal for people who are not that much into Mathematics. In Abstract Algebra you can come up with sets with some pretty exotic properties. Examples can be also found in Functional Calculus and other highly abstract fields of Mathematics.
As for the rest, I'm not prepared to talk right now.
lim (sum (9*10^(-n))) = lim (1 - 10^(-n)) = 1 (where n -> infinity)
Let me explain a bit here. The sum in the limit is not an infinite one. It's a partial sum of first n elements of the series so there IS a final n'th term in it. With that in mind, we can rewrite the sum as 1 minus that one 0.00...1 (10^(-n)) that separates the sum from being exactly 1. And after that we proceed to calculate the limit: with an increase of n the term 10^(-n) gets infinitesimaly small and ends up zero when n -> infinity.
To sum it up, 1 = 0,999... was a sort of contradiction untill people got better understanding of the concept of infinity. And no surprise that it's still confusing for those who look at it from an arythmetical point of view.
As for point two, I don't really get what you exactly mean but there are plenty of things that may look paradoxal for people who are not that much into Mathematics. In Abstract Algebra you can come up with sets with some pretty exotic properties. Examples can be also found in Functional Calculus and other highly abstract fields of Mathematics.
As for the rest, I'm not prepared to talk right now.
