0.999...

From MysteryWiki

0.999... is a repeating decimal.

Fred Richman provided three proofs of the equality of 0.999 and 1 in his 1999 paper in Mathematics Magazine as follows. The first he calls "a valid inference if x is cancellable":

<math>x = 0.999...</math>

<math>(10)x = (10)0.999...</math>

<math>10x = 9.999...</math>

<math>10x - x = 9.999... - x</math>

<math>9x = 9.999... - x</math>

<math>9x = 9.999... - 0.999...</math>

<math>9x = 9</math>

<math>\frac{9x}{9} = \frac{9}{9}</math>

<math>x = 1</math>

The second begins with the fraction <math>\frac{1}{3}</math>:

<math>\frac{1}{3} = 0.333...</math>

<math>(3)\frac{1}{3} = (3)0.333...</math>

<math>\frac{3}{3} = 0.999...</math>

<math>1 = 0.999...</math>

The third involves the "sum of geometric series," i.e. that:

<math>0.999... = \frac{9}{10} + \frac{9}{100} + \frac{9}{1000} + \cdots</math>

<math>\frac{9}{10} + \frac{9}{100} + \frac{9}{1000} + \cdots = \frac{9/10}{1 - 1/10}</math>

<math>\frac{9/10}{1 - 1/10} = \frac{9/10}{9/10}</math>

<math>\frac{9/10}{9/10} = 1</math>

<math>0.999... = 1</math>

However, Richman also provided criticisms of two of these approaches, saying that the first proof actually insinuates that 9x equals 8.999..., instead of 9 and that the third is "equal to 1 in limit, but is not equal to 1".

References

  • Courant, R., and H. Robbins. "Alternative Methods of Defining Irrational Numbers. Dedekind Cuts." What Is Mathematics?: An Elementary Approach to Ideas and Methods. 2nd ed. Oxford: Oxford UP, 1996. 71-72. Print.
  • Jeffreys, H., and B. S. Jeffreys. "Nests of Intervals: Dedekind Section." Methods of Mathematical Physics. 3rd ed. Cambridge: Cambridge UP, 1988. 6-8. Print.
  • Richman, Fred. "Is 0.999 ... = 1?" Mathematics Magazine 72.5 (1999): 396-400. Print.