For quite a long time i did not know how to untie the definition of rational and irrational numbers from base 10.

There was the obvious "p/q is rational for p and q integral q!= 0" definition - which also said numbers otherwise are not - did not illuminate the issues too deeply.

It was then (well a little late) that i came to know about continued fractions and the use they can be put to. They almost magically (beautifully) do the untying work.

However, and this is my question, base 10 still looks like a good way to differentiate rationality and irrationality of numbers. In fact, it looks superior to other bases i have chanced upon in this regard because in base 10 a number is rational if its dotted (decimal i.e something like 2.34) expression either terminates or shows periodicity as it continues indefinitely. A number failing this is irrational.

This definition makes base 10 sound superior because in base 2 we can have a dotted (becimal ???) representation for rational numbers which continues indefinitely and does not show any obvious pattern. 1/3 is a case in point (if i remember correctly)...

So, my question is - is there something really interesting about base 10 or is it just a property many other numbers can show...

(that is, is 10 really a privileged base???)

Einstein - You know, earlier my preference mathematics. Later on i changed it to physics

Pincare: And why is that?

Einstein: I could not tell important facts from non-important ones.

Poincare: Starne that you say so. But now that you mention it, earlier i cherished physics..Now i cherish mathematics.

Einstein: And why is that

If B is a base with prime factors p, q, r, ... then any fraction whose denominator is a product of powers of p, q, r, ... has a finite expansion in base B. All others are periodic. Those that do not have either finite or periodic expansion are irrational. Being irrational is an intrinsic property of a number independent of a base used in its representation.

For B = 10, the factors are 2 and 5 so that all fractions with denominators in the form 2^m5ⁿ have finite expansions. No other fraction does. If B is a prime number, then only fraction k/Bⁿ have finite expansions.

This is the reason for B = 12 to be considered superior to B = 10. For B = 10, only denominatros 2 and 5 (below 10) lead to finite expansions. But for B = 12, you have denominators 2, 3, 4, 6, 8. For example,

base 10		base 12
1/2		.6
1/3		.4
1/4		.3
1/6		.2
1/8		.16
1/12		.1