Fifteen Most Famous Transcendental Numbers

Summary

The 15 Most Famous Transcendental Numbers Cliff Pickover. Follow me on Twitter I am in love with the mysterious transcendental numbers. Did you know that there are "more" transcendental numbers than the more familiar algebraic ones? Even so, only a few classes of transcendental numbers are widley known to humans, and it's very difficult to prove that a particular number is transcendental. In 1844, math genius Joseph Liouville (1809-1882) was the first to prove the existence of transcendental numbers. (More precisely, he was the first to prove that a specific number was transcendental.) Hermite proved that the number e was transcendental in 1873. Lindeman proved that pi was transcendental in 1882. For more information, see my book Wonders of Numbers from which this is excerpted. The mathematical constant pi represents the ratio of the circumference of a circle to its diameter. It is the most famous ratio in mathematics both on Earth and probably for any advanced civilization in the universe. The number pi, like other fundamental constants of mathematics such as e = 2.718..., is a transcendental number. The digits of pi and e never end, nor has anyone detected an orderly pattern in their arrangement. Humans know the value of pi to over a trillion digits. Transcendental numbers cannot be expressed as the root of any algebraic equation with rational coefficients. This means that pi could not exactly satisfy equations of the type: pi2 = 10, or 9pi4 - 240pi2 + 1492 = 0. These are equations involving simple integers with powers of pi. The numbers pi and e can be expressed as an endless continued fraction or as the limit of an infinite series. The remarkable fraction 355/113 expresses pi accurately to six decimal places. In 1882, German mathematician F. Lindemann proved that pi is transcendental, finally putting an end to 2,500 years of speculation. In effect, he proved that pi transcends the power of algebra to display it in its totality. It can't be expressed in any finit...