.999... = 1

6/20/2006 2:21:54 PM

.999... = 1

I looked at this post by a mathematics teacher explaining that 0.999… = 1.000 via Digg. I always assumed that this was common knowledge, but Digg warns that “readers indicate that this story contains information that may not be accurate.”

I don’t recall encountering this relationship in school but only stumbling on it on my own as a child. The poster includes an algebraic proof and the calculation of a geometric series, but he doesn’t include my own childhood observation.

To arrive at the fractional representational of a repeating decimal value, take the repeating digits and divide them by a number consisting only of nines with the same number of digits.

So,
.000… becomes 0/9 = 0.
.111… becomes 1/9.
.222… becomes 2/9.
.333… becomes 3/9 = 1/3.
.999… becomes 9/9 = 1.

Similarly,
.090909… becomes 09/99 = 1/11.
.121212… becomes 12/99 = 4/33.
.333333… becomes 33/99 = 3/9  = 1/3.
.868686… becomes 86/99.

Any repeating decimal is a rational value, expressible as a fraction of two integers; all rational values are repeating decimals. For any arbitrary base b, dividing any integer from 0 to b-1 by b-1, results in that integer turning into a repeating number in base b.

This knowledge can be used to quickly convert a number to its rational form with a numerator and denominator. Another cool algorithm for doing this, but that suffers from rounding errors, is to take the integer portion of a number as the partial calculation of the number. The remaining fraction can then be calculated recursively by using the reciprocal of the calculation of the reciprocal; the reciprocal of the fraction, whose magnitude is less than one, will necessarily have a non-zero integer portion that can later be stripped.