Why 1/3 Never Ends: Fractions, Decimals and Common Denominators

A fraction is in lowest terms when the numerator and denominator share no factor other than 1. To get there, divide both by their greatest common divisor (GCD). 18 and 12 have a GCD of 6, so 18/12 reduces to 3/2.

Listing factors works for small numbers and becomes hopeless for large ones. The method this calculator uses, along with practically every computer algebra system, is the one in Book VII of Euclid's Elements, written around 300 BC. Replace the larger number with the remainder of dividing it by the smaller, and repeat until the remainder is zero. The last non-zero remainder is the GCD.

For 1071 and 462: 1071 = 2 × 462 + 147, then 462 = 3 × 147 + 21, then 147 = 7 × 21 + 0. The GCD is 21, found in three divisions. Gabriel Lamé proved in 1844 that the number of steps is never more than five times the number of digits in the smaller number, which is why a 15-digit fraction reduces as fast as a 2-digit one.

To add 3/4 and 5/6, both fractions need the same denominator. Multiplying the two denominators always works: 4 × 6 = 24 gives 18/24 + 20/24 = 38/24, which then has to be reduced to 19/12. The least common denominator, LCM(4, 6) = 12, gives 9/12 + 10/12 = 19/12 directly.

Both routes reach the same answer. What changes is the size of the numbers along the way, and that gap grows fast. Adding 1/12 + 1/18 + 1/30 over the product of the denominators means working in 6,480ths. The least common multiple is 180.

The LCM comes straight from the GCD: LCM(a, b) = a × b ÷ GCD(a, b). When the denominators share no factor, as 4 and 7 don't, the GCD is 1 and the least common denominator is simply their product, 28. When one denominator divides the other, as 4 divides 8, the larger one is already the answer.

Write the fraction in lowest terms and factor the denominator. If the only primes in it are 2 and 5, the decimal terminates. If any other prime appears, it repeats forever.

The reason is that a terminating decimal is a fraction over a power of ten, and 10 = 2 × 5. 1/8 can be rewritten as 125/1000 because 8 = 2³ divides 1000 = 2³ × 5³. No power of ten is divisible by 3, so 1/3 can never be written over one, and the long division never comes out even.

The number of decimal places is predictable too. It equals the larger of the two exponents. 40 = 2³ × 5, so 1/40 ends after three places at 0.025. 16 = 2⁴, so 1/16 needs four: 0.0625.

The rule only applies after reducing. 3/6 seems to have a 3 in its denominator, but it reduces to 1/2, which is 0.5.

Long division can only leave as many different remainders as the denominator minus one. Dividing by 7, every step leaves a remainder from 1 to 6. As soon as a remainder comes back, everything after it repeats, because the division from that point on is identical. That is how the calculator finds the repeat: it records each remainder and stops when one reappears.

So 1/p repeats with a period of at most p − 1 digits, and by Fermat's little theorem the period always divides p − 1 exactly. Primes that reach the maximum, like 7, 17 and 19, are called full reptend primes. For them every multiple uses the same digits, rotated: 1/7 = 0.142857…, 2/7 = 0.285714…, 3/7 = 0.428571….

Gauss worked through these periods at length in Disquisitiones Arithmeticae (1801). A further pattern was published by E. Midy in 1836 and now carries his name: when the period of 1/p is even, its two halves add up to a row of nines. For 1/7, 142 + 857 = 999. For 1/17, 05882352 + 94117647 = 99999999.

The rule holds in every number base, with the base's own prime factors in place of 2 and 5. Most software stores numbers in binary floating point as defined by IEEE 754, where the base is 2. In binary, a fraction terminates only if its denominator is a power of 2.

1/10 has a 5 in its denominator, so in binary it repeats forever as 0.000110011001100… and has to be rounded to fit 53 significant bits. The same happens to 2/10. Add the two rounded values and the error surfaces in the 17th significant digit: 0.1 + 0.2 gives 0.30000000000000004.

This is why accounting software stores money as whole cents or uses a decimal type, and why this calculator takes whole-number numerators and denominators instead of decimals. Integer arithmetic is exact, so 1/10 + 2/10 is 3/10 with nothing lost.