Finite vs Infinite Math Example 3

Follow the full solution, then compare it with the other examples linked below.

Example 3

easy
For each, say whether the process terminates (finite) or goes on forever (infinite): (a) counting to 1,000,0001{,}000{,}000, (b) listing all prime numbers, (c) writing the decimal expansion of 13\frac{1}{3}.

Solution

  1. 1
    (a) Counting to 1,000,0001{,}000{,}000: reaches a definite end. Finite (though time-consuming).
  2. 2
    (b) Listing all primes: by Euclid's theorem there are infinitely many primes, so the list never ends. Infinite.
  3. 3
    (c) 13=0.333…\frac{1}{3} = 0.333\ldots: the 33 repeats forever. Infinite (but with a pattern).

Answer

(a) Finite; (b) Infinite; (c) Infinite.
The distinction between finite and infinite processes is practical as well as theoretical. A task that takes a billion steps is finite (just long); a task with no end is truly infinite. Repeating decimals are infinite processes described by a finite pattern.

About Finite vs Infinite

Finite describes a quantity or set with a definite end; infinite describes something that goes on forever without bound.

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