Pi is an irrational number. As such, it has an infinite number of digits.
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A set which containing $and pi are the end blocks are the finite and without these are infinite
Some finite numbers in a set: the number of digits on your hand, the number of seats on a bus, and the number of people on earth. Some infinite numbers in a set: the number of positive integers and the number of digits in pi.
The square root of pi is a finite number. However, it is an irrational number. That implies that if you try to write it out as a decimal number, it will stretch out to an infinite number of decimal places to the right of the decimal point.
Pi is an irrational number so it will go on to infinity. That also means it will take to infinity to calculate the infinite point. By definition a finite number will never be found.
(Pi) is not an 'infinite number'. It just can't be written exactly with any finite number of digits. But there are plenty of digits available, to make your division as accurate as you want it. You can never get an exact answer, just like you can never write the exact value of (pi), but you can get as close as anyone will ever need to get.