Assuming that it is given that π is irrational number, show that sum of decimal digits of π?

Question

is infinitely large

Answer 1

I'm not sure I follow.

Rational numbers have a repeating decimal expansion and irrational numbers don't so they have an infinite number of digits which do not repeat. An infinite subset of these digits must then be non-zero. If there was only a finite subset of non-zero digits, the number would end with an infinite string of 0s following the last nonzero digit, which cannot be for an irrational number. So the sum of the infinite subset of non-zero digits will be infinitely large.

Is this the sort of argument you are looking for?

Answer 2

Since it is given that π is irrational, we have that there are infinitely many nonzero digits in the decimal expansion of π. Otherwise, the decimal would terminate, and π*10^N would be in integer for some N, which would make π rational. Thus, for any M>0, there are at least M non-zero digits in the decimal expansion of π. The sum of M+1 of those digits, since all are at least one, would be at least M+1>M. This holds for any M. Letting M become arbitrarily large, we have the result.

Answer 3

Proof by contradiction:

Assume that the sum is not infinite. Then there is a finite number of digits in the expansion of pi. Which means it can be expressed as m/n where m is the finite digits and n is a power of 10. This is a rational number. But we were given that pi is irrational. Contradiction.

Answer 4

by definition, Pi has an infinite number of non repeating positive numbers, call the sum of the first N digits S. Now, add the N+1st term to S and you will produce a sum larger than S, call this S'. Add the N+2nd term to S' and produce a sum larger than S'...we can do this for infinity showing the sum will always increase without bound

Answer 5

I'm not really sure what the question is, but Pi is irrational. It is a fraction. Irrational numbers' decimal places are forever-going. They don't stop. Obviously, the sum of the decimals of pi will keep going and going and going, so the sum of those digits will also keep going and going and going. So, the sum is infinetely large.