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May 29th, 2017, 08:08 PM   #61
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Quote:
 Originally Posted by Microlab I mean the number that has exact value. There is two equal sections of 10, 5 and 5. In reality there is 9 equal sections of 10, but we can't measure it. Either they not exists at all, or lack of knowledge. Decimal with three dots does not represent exact value of number.
Oh I understand exactly what you mean now. You have raised a really good point. Let me see if I can put this in context for you.

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 upper-division math major course in college. That's the third year of a four-year 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 third-year 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 08:38 PM.

May 29th, 2017, 08:36 PM   #62
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Quote:
 Originally Posted by Maschke * 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.
This is very well put and extremely important. If you are having trouble with the decimal expansion of $\frac13$ you should bear in mind that we can produce a similar expansion in base 3 (rather than the decimal base 10). In this case $\frac13 = 0.1$ which is a very simple expression that doesn't imply any of the problems you are seeing.

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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