### The Number 0.999...

October 14, 2010 The recent content of this blog has been somewhat heavy reading, Nobel Laureates, general relativity, and all that, so it's time for some light diversion. Mathematics has its lighter and more accessible moments, although a perusal of recent papers on arXiv might convince you otherwise. Maurice Allais, an able practitioner of applied mathematics whom I eulogized in yesterday's article, had the following opinion of mathematics:"Any author who uses mathematics should always express in ordinary language the meaning of the assumptions he admits, as well as the significance of the results obtained. The more abstract his theory, the more imperative this obligation. In fact, mathematics are and can only be a tool to explore reality. In this exploration, mathematics do not constitute an end in itself, they are and can only be a means."[1]Now, the promised diversion. So, you really think that the number "0.999..." is smaller than one? "0.999..." appears to be a real number, not a rational number, like 1/3, so how could it possibly be equal to the rational number 1? There's a 28 page paper in the The Montana Mathematics Enthusiast that reviews proofs that "0.999..." is identical to one.[2] I discussed this number in a previous article (A Not So Limiting Case, November 8, 2007), where I presented the following proof:

• Let a = 0.999...Here's another proof.

• then 10a = 9.999...

• subtracting 10a - a = 9.999... - 0.999

• 9a = 9

• a = 1

• 1/9 = 0.111...Another proof, presented by Leonhard Euler in his Elements of Algebra, appears below. Euler's equation is for 9.999..., which resolves to 10, but the idea is the same.

• 9 x (1/9) = 0.999...

• 1 = 0.999...

*Section 468 of Euler's Elements of Algebra (John Farrar, trans, Hilliard, Gray & Co., Boston, 1836).*

The principle behind Euler's proof is that any convergent infinite series of the form

a(r)has the sum (ar)/(1-r), provided that |a| < 1. In our case, using a = 9 and r = (1/10) gives the decimal representation for 0.999..., and the sum 0.9/0.9 = 1. Many students encounter this identity with disbelief. After all, even after writing many nines after the decimal point, the number still seems to fall short of one by a known amount that can be expressed exactly. That's why mathematics is both wonderful and intimidating at the same time.^{1}+ a(r)^{2}+ a(r)^{3}+ a(r)^{4}+ ...

### References:

- Maurice Allais, "La formation scientifique, Une communication du Prix Nobel d'économie," address to the Académie des Sciences Morales et Politiques (1997), as quoted on Wikiquote.
- Karin Usadi Katz and Mikhail G. Katz, "When is .999... less than 1?," The Montana Mathematics Enthusiast, vol. 7, no. 1 (January, 2010), pp. 3-30; available also at http://arxiv.org/pdf/1007.3018.
- 0.999... Page on Wikipedia.

*Permanent Link to this article*

Linked Keywords: Nobel Laureates; general relativity; mathematics; arXiv; Maurice Allais; The Montana Mathematics Enthusiast; Leonhard Euler; convergent infinite series.

RSS Feed

### Google Search

Latest Books by Dev Gualtieri

**Previews Availableat Tikalon Press**

*STEM-themed novel for middle-school students*

*Mathematics-themed novel for middle-school students*

*Complete texts of LGM, Mother Wode, and The Alchemists of Mars*

Other Books

- US Independence Day, July 4, 2024

- Copper and Green Energy, July 1, 2024

- Baseball Mud, June 24, 2024

- Wiedemann-Franz Law, June 17, 2024

- A.I., Wine, and Beer, June 10, 2024

- Bergmann's rule, June 3, 2024

- 60 Hertz Standard, May 27, 2024

- Droplets, May 20, 2024

- Levitation, May 13, 2024

- 100 Years of IBM, May 6, 2024

- Cicada Apocalypse, April 29, 2024

- Critical Materials, April 22, 2024

- Eclipse 2024, April 15, 2024

- Reverse Sprinkler, April 8, 2024

- T Coronae Borealis, April 1, 2024

- Coffee and Tea, March 25, 2024

- Arno Penzias (1933-2024), March 18, 2024

- Wave-Particle Duality, March 11, 2024

- Offshore Wind, March 4, 2024

- High Entropy Ceramics, February 26, 2024

- Methane Emissions, February 19, 2024

- Thales, Heron, and Experimental Mathematics, February 12, 2024

- Mica, February 5, 2024

- Disk Packing, January 29, 2024

- Is e
^{π}> π^{e}?, January 22, 2024

- Brownian Motion Energy Harvesting, January 15, 2024

- Coin Flip, January 8, 2024

### Deep Archive

Deep Archive 2006-2008

**Blog Article Directory on a Single Page**