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# A question of rationality !!

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How do find out if a decimal fraction such as 0.18439716312056737588652482269504........can be expressed as rational fraction, or to find out that, indeed,it is a rational fraction in the first place? Any help would be great. I thank you and "Happy Holidays" to all.

Dec 23, 2017

#1
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You just have a decimal.....not a "decimal fraction"

If  the  digits repeat the same pattern   [  a cycle of 32 digits], the fraction  expressing this decimal is just  the digits in this pattern the numerator / the same number of 9's in the denominator

For instance.....

.167   (repeating)  is  just   167 / 999

.1477 (repeating)  is just  1477 / 9999  .....etc.

If the digits don't repeat.....then the decimal is irrational and can't be expressed as a fraction

Dec 23, 2017
#2
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Actually, that is a very good question! It is not very easy to determine if it is rational or irrational. I have written a computer code that attempts to convert a "decimal" of up to 50 digits to a rational fraction, IF it is a rational fraction! And when I plugged this decimal into the program, it instantly gave this rational fraction:0.18439716312056737588652482269504 = 26 / 141.

What the program does is repeated reciprocals minus the whole integer until it converges to a whole number, if it does converge! It does about 50 such reciprocals and if it doesn't converge, it comes out with the phrase " it is irrational" !!. In this case, it took only 5 reciprocals to converge to 3 and then goes backwards and reconstructs the rational fraction.

I forgot to add that it has a period of 46 digits, and then it repeats every 46 digits.

Dec 23, 2017
edited by Guest  Dec 23, 2017
edited by Guest  Dec 23, 2017
edited by Guest  Dec 24, 2017
#3
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Your computer program is a classic example of GIGO. Even if your program worked properly, it will never determine if a number is irrational. At best, it will give the fractional equivalent of a decimal proving it rational. It should always do that, because you are entering a finite number of decimals. You coded it to say otherwise, but that doesn’t make it true.

Computers are not magical machines Mr. BB. Computer coding isn’t the same as an incantation you’d say to a crystal ball.  Computers do what you tell them to do; that’s not the same as what you want them to do.

I’m not the only one who knows you as an incompetent mathamation and coder.   I’m not your first Troll, either. That honor goes to Dragonlance. He’s one brilliant kid, for sure.

Here are two of his comments:

... One more thing computers and calculators can’t cope because they are machines. They don’t think or love or hate or give a rat’sass about anything. That is a good thing because it would never do anything for you. Only humans cope. Well some humans do. You do not know much about computers or calculators do you?—Source

He has several troll posts directed toward you. He addresses and refers to you as “Anol Retentive” sometimes adding the “constipated hog” description, and as the “banker man.” One time he suggested, “You should go lock you self in a bank vault until you feel better. Make sure the time clock is set for next year.” Source

Sounds like good advice! While you are in there, you should study up on irrational numbers and computer programming.

GA

Dec 24, 2017