Re: Extrapolate divisor & dividend from quotient?

On 6 Jul, 22:25, "G.E. Ivey" <george.i...@xxxxxxxxxxxxx> wrote:
I agree with Julio Di Edigio that you are probably thinking of the method of converting any rational number to a fraction.  (Rational numbers are DEFINED as numbers that can be written as a fraction.)

  If the rational number is terminating, it is easy.

  For example, if x= 0.342346, then multiplying both sides by 1000000, which has 6 zeros, moves the decimal point 6 places: 1000000x= 342346 so x= 342346/1000000.  Now you can factor and cancel like factors to reduce the fraction.

  If a rational number does not terminate, then it must be "eventually repeating.

   For example, if x= 0.231578157815791578...  where the "1578" part repeats forever, multiplying both sides by 100 moves the "non-repeating" 23 part outside the decimal: 100x= 23.157815781578...  Since 1578 has four digits, multiplying both sides by 10000, with 4 zeros, moves ONE "repeat" outside the decimal: 1000000x= 231578.157815781578...  Notice that the right side STILL have an infinite number of the repeating '1578' term.
  Subtracting both sides we have
1000000x- 100x= 231578.1578...- 23.1578... or
999900x= 231578.  

  Dividing both sides of that gives x= 231578/999900 which can be reduced.

  As I said before, any rational number is either a terminating decimal or an eventually repeating decimal.  Any other number is irrational and cannot be written as a fraction.


You are very right: given the OP, I should have not mentioned periodic
vs. non-periodic expansions at all.

I'll take the chance for a couple of questions that where in my mind
anyway:

Can all irrational numbers be written down (at least in principle) as
infinite decimal expansions? (I might ask: can all numbers in R be
expressed or at least thought of as decimal periodic expansions with a
period that can be infinite? Is there anything in the domain of
"numbers" that escapes this definitions?)

For instance, Phi is an irrational number whose decimal expansion can
be easily expressed with a rule for a sequence, or otherwise as a
continuous fraction, or a nested radical, etc. Are all real numbers
like this?

-LV
.

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