I need to find a set of two Integers P and Q such that P/Q comes out to be non recurring non terminating decimals . Such as 106951484895/47666297253.
How can we find such two numbers. Is there any formula for this?
I need the result to be a decimal that does not end, nor do the numberals repeat themselves in any recognizable sequence.
Example: 106951484895/47666297253 = 2.24375483430840090053512174785916761672572536877348541885492361677904883097801043287594504105040726394682939649060976328569701751152716078581955550664345621014373276853529464561869479096457226131837381633508104200383965998089941882484488017422971446512579402430136563475351346061895880989881427322957327423437070470786123178209350642720039431463073874604572487035088141630189988694987841412729755839421966691187327329614175517087337289424929857158670121550588652768665265722816433005640…
EDIT: From the comments I got that such numbers don't exist.
So how can I get Fractions which approximates Irrational numbers?
An example is Fraction such as 22/7 which is commonly used to approximate π
Best Answer
In the comments, OP clarifies, "I want a fraction with a particularly long period in its decimal expansion."
The longest the period of $m/n$ can be is $n-1$ (because when you calculate the decimal expansion, the period begins when you see the same remainder twice, and there are only $n-1$ nonzero remainders available when you are dividing by $n$).
The period $n$ can only be achieved when $n$ is prime (because remainders that have a common factor with $n$ can't be achieved, and if $n$ isn't prime there are numbers between 1 and $n-1$ that have a common factor with $n$).
But not every prime number $n$ works. E.g., $13$ is prime, but the period of $m/13$ is $6$, not $12$ (try it!).
The primes $n$ for which you get period $n-1$ are tabulated at https://oeis.org/A001913, and the first 10,000 of them are listed at https://oeis.org/A001913/b001913.txt. It is widely believed, but not yet proved, that there are infinitely many of them.