What you will read below is a brief exercise I completed in 2017 to resolve a point of curiosity in my mind.
As you are aware, in whichever number system you find yourself working, some fractions have finitely many decimals, while other fractions have infinitely many decimals.
But what are the determinants of this? And how does one’s choice of number system affect this? What rules are there to follow to determine if the decimal expansion terminates?
The proof below was my effort to answer these questions. Having had the proof below independently verified as correct, I now present it here for your reading pleasure. Enjoy!
Hello there. And welcome. I trust this finds you well.
Now that you are here, we are ready to begin.
Below is a proof of the following statement: in a base-N number system, the fraction Y / K will have infinitely many decimals unless any of the following four conditions holds:
- K equals 1;
- K is a factor of Y;
- K is a power of a prime factor of N;
- K is a product of powers of prime factors of N.
Note the following key. Roman numerals indicate footnotes, which are available below the main text.
- N is a natural number larger than 1 (natural numbers are positive integers) [i,ii].
- K is a natural number at least equal to 1.
- Y is a natural number at least equal to 1.
We start with the necessary exclusions of the four conditions above.
- If K equals 1, then Y / K is simply Y / 1, which equals Y itself and hence has finitely many decimals.
- If K is a factor of Y, then Y / K is some positive integer Q, so Y / K has finitely many decimals.
- If K equals PX (some power of a prime factor of N, with X being a natural number), then Y / K = Y * N-X * (N / P)X and hence Y / K has finitely many decimals [iii].
- If K equals P1X1 * P2X2 * … * PW-1XW-1 * PWXW (some product of powers of some prime factors of N, with each exponent being a natural number), then Y / K = Y * N-W * (NW / [P1X1 * P2X2 * … * PW-1XW-1 * PWXW]) and hence Y / K has finitely many decimals [iv,v].
With the four exclusions above having been lain out, let K now not be as defined in the excluded cases. (Note: T is a natural number large enough to have NT be greater than or equal to K.)
Y / K
= Y / K * N-T * NT
= Y / K * N-T * [K * Q + R]
= Y * N-T * [Q + R / K]
= Y * N-T * Q + Y * N-T * R / K
= Y * N-T * Q + N-T * R * Y / K
The presence of Y / K in the formula on the right-hand side of the workings above indicates that Y / K has infinitely many decimals and hence does not have a terminating decimal expansion, which proves the statement. (If one continues the workings above, one will find that the Y / K is persistent and remains on the right-hand side throughout.)
QED
[i] An integer, simply put, is a whole number – one without any meaningful decimals attached, e.g. -4, 3, 87, 234, 908787, -3467, etc. I say “without any meaningful decimals” because of the technicality that any whole number’s decimal expansion consists solely of zeros.
[ii] A natural number is a positive integer.
[iii] A prime number is a natural number with only two factors (divisors): 1 and itself. Examples of prime numbers include: 2, 3, 5, 7, 11, etc. There are infinitely many prime numbers, 2 is the only even prime number and any natural number other than 1 that has more than two factors is a composite number.
[iv] The proof below makes use of the Division Algorithm as well as the Fundamental Theorem of Arithmetic. The Division Algorithm states that given any two integers N and D, one can always write N as N = D * Q + R, where Q and R are also integers and R ranges between 0 (zero) and D (zero included; D excluded). The Fundamental Theorem of Arithmetic states that every integer larger than 1 can be expressed as the product of powers of its prime factors and that said expression is unique, up to the order of the factors.
[v] The forward slashes indicate division. The asterisks indicate multiplication. Plus signs, minus signs, equals signs and exponentiation are all to be understood as per normal.