What is the Dottie Number?

The Dottie number (represented by d) is the unique real number whose cosine is equal to itself. In other words, it is the solution to the equation x = cos(x).

It is easy to find an approximation to d by repeatedly pressing the cos button on a calculator until the result stops changing. (Be sure to use radian mode.) Many people discover the Dottie number this way.

The Dottie number serves as a nice example of an irrational number easily understood by beginning trigonometry students. Since d is irrational we can never give a exact representation with the decimal or fraction notations we are all used to. Still, we can approximate it in decimal. The number is approximately equal to:

0.73908 51332 15160 64165 53120 87673 87340 40134 11758 90075

For a longer approximation, see d to 100,000 digits.

The Dottie number marks the coordinates, both x and y, of the point where the cosine and arccosine curves intersect. The line y = x also intersects this point.



The number serves as a non-trivial example of a fixed point and an attractor. It is also a transcendental number.

The name "Dottie number" comes from a 2007 article by Kaplan, who tells a story of a woman who like many others made the discovery on a calculator:

Dottie, a professor of French, noticed that whenever she put a number in the calculator and hit the cos button over and over again, the number on the screen always went to the same value, about 0.739085.... She asked her math-professor husband why the calculator did this no matter what number she started with. He looked. He tried it. He said he had no idea, at least not that day. The next day he realized not only what was happening, but that his wife had found a beautiful example of a global attractor.

It is important to note that the term "Dottie number" has not been universally accepted by the mathematical and scientific community. Indeed, the number receives little attention in math research and as far as I know, no effort has been made among academics to decide on a standard name. Other names that have been used include cosine constant, cosine fixed point, and cosine superposition constant. Dr. Arakelin represents the number with the Armenian letter ա in his work.

While I do use the name "Dottie number", I do not believe that any single person deserves credit for the number's discovery. I consider the name an informal term sufficient for the purposes of this blog. I have no objections if others want to use a different name.

The Dottie number can succinctly be described with a limit formula:


Where cosn(x) is the cosine of x taken n times. For example cos3(x) = cos(cos(cos(x))). Actually there is nothing special about the number zero here. Repeatedly taking the cosine of any real number leads to d. Although the formula above is short, I am seeking a simpler formula that does not require the cosine or any other non-algebraic function.

Kaplan has discovered an infinite series that exactly equals d. Furthermore Newton's or Halley's method can approximate the number to any accuracy, given enough time and computing power. I continue to search for faster ways to approximate d.

The number has received less attention than π or e, but may have its own interesting properties waiting to be discovered.

Further Reading

The Dottie Number on MathWorld

Reference

Kaplan, S. R. "The Dottie Number." Mathematics Magazine Vol. 80, No. 1, February 2007. p 73-74.


Revised 15 Jan. 2011. Added links to 100,000 digits, irrationality, and transcendence of d. Also added a limit formula, a link to Kaplan's sequence, and a discussion of the search for a simple formula.
Revised 5 Feb. 2011. Added a sentence about d being accessible to pre-calculus students.
Revised 21 Mar. 2011. Added two paragraphs about the name "Dottie Number".
Revised 28 Nov. 2011. Added a Wiktionary link for the letter ա.