Transcendental Numbers

The number pi, the ratio of circumference to diameter of a circle, is a rather special and unique number. Approximately, it is given as 3.1459265358….It is an irrational number, which means a real number that is not rational. Where a rational number is any number that can be expressed as the quotient or fraction (p/q) of two integers. That is pi cannot be expressed as a common fraction, (except as an approximation 22/7) and so its decimal identity is a continuing sequence. Another interesting thing about pi is that it is not the root (solution) of any non-zero polynomial equation having rational coefficients. This makes it a transcendental number, which is any real or complex number that is not algebraic.

An interesting observation is that all real transcendental numbers are irrational numbers, since all rational numbers are also algebraic. However, it is not the case that all irrational numbers are transcendental numbers. An example of this is the case of the square root of 2. This is an irrational number but it is also the root of the polynomial equation x^2 - 2 = 0. This means that the square root of 2 is not a transcendental number. Another example is the so called golden ratio, which is equal to 1.6180339887….and is also known as the golden mean. It is an irrational number and is the root of the polynomial equation x^2 - x - 1 = 0, but which also means it is not a transcendental number. Whilst a polynomial with rational coefficients is countable since each polynomial has a finite number of zeros, a set of transcendental numbers is uncountably infinite.

This article was originally posted on a previous web site for the Asterism project on 15th Jul 2020 and it has been copied here since that site was closed down.