May 29th, 2017, 07:08 PM  #61  
Senior Member Joined: Aug 2012 Posts: 2,356 Thanks: 739  Quote:
Traditionally, the Pythagoreans believe in the whole numbers and the ratios of whole numbers. So they did believe in the existence of all rational lengths like $\frac{2}{3}$ and $\frac{47}{112}$. They believed that all naturally occurring lengths in geometry were ratios of whole numbers. [Of course they put this in the language of proportions. They did not have our modern concept of a rational number]. However they also knew the Pythagorean theorem  they were the Pythagoreans, after all! And anyone would agree that a square of side $1$ is a very natural geometric object. But from the Pythagorean theorem we can determine that the diagonal of a unit square is $\sqrt 2$; and the Pythagoreans knew the theorem written down by Euclid that the square root of $2$ can not possibly be the ratio of two whole numbers. So the Pythagoreans had to accept the existence of irrational numbers. It's said that they threw overboard and drowned the guy who made this discovery. Whether that's true we can never know, it's lost in history. But it's clear that this was a very deep and counterintuitive discovery. Not every naturally occurring number is the ratio of whole numbers. Now you are going a step further. You are saying you don't even believe in the existence of rational lengths! And the reason for your doubt is that some rational numbers, like $\frac{1}{3} = .333 \dots$, have infinitely long decimal expressions. But forget decimal expressions. Pretend you're an ancient Greek. It is entirely straightforward that we could take a unit length and divide it into three equal pieces. There is nothing mysterious about that. Now I'm going to let you in on a mathematical secret. This is not in books and you will never go to any math class at any level where they will tell you this explicitly. It's just something you pick up along the way. And that is: The decimal representation of real numbers is very flawed! It's broken! We tell schoolkids that $\frac{1}{3} = .333 \dots$ and deep down they don't believe it and frankly their skepticism is perfectly justified. The fact that $\frac{1}{3} = .333 \dots$ is a very deep mathematical fact that requires material not taught to anyone until you take an upperdivision math major course in college. That's the third year of a fouryear college degree in the US. Mathematicians themselves only worked all this out 130 or so years ago. So the bullet points are: * Of course $\frac{1}{3}$ exists, we just break a unit length into three pieces. The real problem is with those pesky irrationals! THEY might not exist!. [Of course they do, but it took mathematicians a long time to figure out a framework in which it made sense and could be proven]. * It's a fact, proven to thirdyear math majors in college, that $\frac{1}{3} = .333 \dots$ * But even if you don't quite believe that, it's ok ... because the decimal representation of a number is not the same thing as the number. Just as both $4$ and $2 + 2$ are different expressions that "point to" the same abstract number; the decimal representation of a number is not the same thing as the number. * So $\frac{1}{3}$ is a very simple rational number, that has unfortunately a very mysterious decimal representation. But that does not reflect badly on $\frac{1}{3}$; rather, it reflects badly on the nature of the decimal representation. I hope some of this helps. They do not explain this to anyone in school unless they become a math major. And even then they don't ever make this explicit. But this is what is going on. $\frac{1}{3}$ is a very natural geometric object with a messy decimal representation. Last edited by Maschke; May 29th, 2017 at 07:38 PM.  
May 29th, 2017, 07:36 PM  #62  
Math Team Joined: Dec 2013 From: Colombia Posts: 7,675 Thanks: 2654 Math Focus: Mainly analysis and algebra  Quote:
It so happens, that many other rationals become problematic in this base 3 representation, but for every rational there is a base under which the representation is nice and simple. There isn't a base for which they all are nice though.  

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