The 15 Most Famous Transcendental Numbers
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 finite series of arithmetical or algebraic operations. Using a fixed-size font, it can't be written on a piece of paper as big as the universe.
I also talk about all the mysteries of pi in my book Keys to Infinity.
Many of you have probably heard of pi and e. But are there other famous transcendental numbers? After conducting a brief survey of readers, I made a list of the fifteen most famous transcendental numbers. Can you list these in order of relative fame and/or usage?
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