### 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

Thanks to Cory Doctorow of BoingBoing for his favorable review of Secret Codes!

Other Books

- The First Angiosperm - September 18, 2017

- Noise Thermometry and the Boltzmann Constant - September 11, 2017

- Walking in the Rain - September 4, 2017

- Agitated Atoms - August 28, 2017

- Partial Solar Eclipse at New Jersey - August 24, 2017

- Magnetocapacitive Tunnel Junctions - August 21, 2017

- Tardigrades - August 14, 2017

- Roman Concrete - August 7, 2017

- Solar Spicules - July 31, 2017

- Schroeder Diffuser - July 24, 2017

- Rough Microparticles - July 17, 2017

- Robot Musicians - July 10, 2017

- Walter Noll (1925-2017) - July 6, 2017

- cosmogony - July 3, 2017

- Crystal Prototypes - June 29, 2017

- Voice Synthesis - June 26, 2017

- Refining Germanium - June 22, 2017

- Granular Capillarity - June 19, 2017

- Kirchhoff–Plateau Problem - June 15, 2017

- Self-Assembly - June 12, 2017

- Physics, Math, and Sociology - June 8, 2017

- Graphene from Ethylene - June 5, 2017

- Crystal Alignment Forces - June 1, 2017

- Martian Brickwork - May 29, 2017

- Carbon Nanotube Textile - May 25, 2017

- The Scent of Books - May 22, 2017

- Patterns from Randomness - May 18, 2017

- Terpene - May 15, 2017

- The Physics of Inequality - May 11, 2017

- Asteroid 2015 BZ509 - May 8, 2017

- Fuzzy Fibers - May 4, 2017

- The Sofa Problem - May 1, 2017

- The Wisdom of Composite Crowds - April 27, 2017

- J. Robert Oppenheimer and Black Holes - April 24, 2017

- Modeling Leaf Mass - April 20, 2017

- Easter, Chicks and Eggs - April 13, 2017

- You, Robot - April 10, 2017

- Collisions - April 6, 2017

- Eugene Garfield (1925-2017) - April 3, 2017

- Old Fossils - March 30, 2017

- Levitation - March 27, 2017

- Soybean Graphene - March 23, 2017

- Income Inequality and Geometrical Frustration - March 20, 2017

- Wireless Power - March 16, 2017

- Trilobite Sex - March 13, 2017

- Freezing, Outside-In - March 9, 2017

- Ammonia Synthesis - March 6, 2017

- High Altitude Radiation - March 2, 2017

### Deep Archive

Deep Archive 2006-2008

**Blog Article Directory on a Single Page**