.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.

Comments

[jeremiah johnson]

speaking in numbers, and not how they're represented in the decimal system, they are different. they differ by an infinitely small amount, but they're different.

0.9999... != 1

[Wesner Moise]

This always confuses people.

Each number has identical mathematical properties and are, therefore, the same.

When performing long division, and dividing a number by itself, the result can either by 1.000... or 0.999... depending on whether chooses 0 or 1 as the first digit.

[Wesner Moise]

This by the way is generally accepted by mathematicians, so, for example, 5 = 5.000.. or 4.999...

[Eric TF Bat]

But it's not true! And I can prove it! First, let's redefine "equals" to mean "flibbly wibble spoo"! And now you see your error! Shame on you!

[John Stewien]

It's always so tricky trying to explain mathematical things like this to people because visually .999 looks different to 1.

What is the difference between an infinately small amount and 0? Answer = nothing, they are the same.

1 - 0.999... = 1/infinity = 0

You learn this stuff in high school when you learn about differential equations and integration. Integrals are derived from the fact that 1/infinity = 0.