Can factors determine where a decimal number ends?

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Does anyone know of any research or math records on determining when a decimal number will terminate based on the inverse of its factors?

Yes I know that I would be working with whole numbers not integers. But for integers what if you took the factors of a quotient, say circumference divided by diameter and factored the numerator and denominator. The equation would be = to itself, but if you multiplied the factors together you already know that those factors terminate at the product.

Someone has probably done a similar technique. But factoring large numbers is recursive and then you add to the process recursion again to do the multiplication and you get a problem you can’t solve.

Also can someone explain the process that was used when they solved Pi to a trillion digits and don’t know where it stops. The stopping point is what I would describe by the multiplication of the factors. The factors multiplied together still equal the original number, but the product determines where the factors stop.

Hope this makes sense.

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Posted (edited)

Decimal is just one of infinite number of numeral systems. It is used in computers just to show result to user. Computers don't compute in decimal system.

Irrational number never ends.

Rational number, if it has no repeating fraction, will end.

(different numeral system have different repeating fraction)

But from computer programming point of view, you should start from making rational number class (i. e. taking p/q as arguments), or download 3rd party from net. Override ToString() (e.g. Java or C#) or << operator (C++) or method used to convert to decimal string to support repeating fraction.

Edited by Sensei

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Here is something about how well irrational numbers (such as Pi) can be approximated by fractions: https://www.quantamagazine.org/new-proof-settles-how-to-approximate-numbers-like-pi-20190814/

As numbers like pi don't "stop", anything expressed using integers (other than an infinite series) can only be an approximation.

7 hours ago, Trurl said:

Also can someone explain the process that was used when they solved Pi to a trillion digits and don’t know where it stops.

I assume by evaluating one of the (many) infinite series to that number of decimal places.

Also, surprisingly, it is simpler to calculate the nth digit of pi than all n digits: https://math.hmc.edu/funfacts/finding-the-n-th-digit-of-pi/

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Seems like it is pretty involved to do this kind of thing. I will use it to strengthen my programming.

I was taught that that something written as say: 28/7 is in exact form. It is when you divide it is an approximation in decimal form.

I read the article and they prove that the fractions are infinite.

Here is what I was trying to do is invert the quotient and factor both the numerator and denominator. Divide and reduce those factors; then decide if there is any common modulus where the factors will reach a whole number. I guess you could say the fraction will terminate where the factors multiplied together will equal. (If they don’t terminate at a smaller modulus).

I know you are thinking how do we know by division that this modulus will be reached. But it would simply be the modulus of the numerator by the denominator would tell how many iterations, until the infinite fraction terminates. We know the numerator and denominator will terminate at their original product. However there is no guarantee that the inverse quotient of those factors will terminate.

I know this is flawed. But in the article they were testing fractions for accuracy. They prove the accuracy. But I am arguing that with infinite iterations the decimal may terminate. Yes, I know they went to Oxford and I went to Point Park, but this hypotenuse is why I was interested in this process in the first place.

Hope this makes sense.

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If I am understanding this correctly, a fraction represents a terminating decimal (so there is an "end") if and only if the denominator has factors of "2" and "5" only.  How many decimal places is a little harder.  1/2= .5, 1/4= 1/2^2= .25, 1/8= 1/2^3= .125, 1/10= 1/(2*5)= 0.1, 1/16= 1/2^4= .0625, 1/20= 1/(2^2*5)= .05, etc.  Do you see a pattern?

Edited by Country Boy

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