I discovered something very surprising and somewhat disturbing recently - that almost all real numbers are transcendental and irrational.

The Real Numbers - A real number is any number that can be written as an infinite string of digits, i.e. 1.30000... or 1.310219029100... The real numbers are uncountably infinite, which means that we cannot assign a whole number to each real number. Consider that we can assign the number 1 to 1.000... , 2 to 2.000 and so on for all whole numbers. Yet we still have numbers that have not been paired off with a whole number. This means that the real numbers are uncountably infinite.

The Irrational Numbers - An irrational number is any real number that goes on forever with no repeating pattern, that is any number that cannot be written as a fraction where the numerator and denominator are both whole numbers. If a number has a limited number of digits than it can be written as a fraction, i.e. 1.03 = 103 / 100.

A real number whose digits have a repeating pattern can also be expressed as a fraction. Consider that 1 / 9 = .111... , 2/9 = .222... , 9/9 = .999... = 1. Also 1/99 = .010101... and 1/999 = .001001001... So then we can write any repeating string of digits as such a fraction, and if we wish to create a real number that starts with a non-repeating string of digits than we can simply sum that fraction with the fraction of the repeating part of the number. Since the sum of any two fractions is also a fraction, we can conclude that any number that goes on forever with no pattern cannot be represented by a fraction.

We can represent any fraction m/n with two numbers m and n. We can count all possible combinations of m and n in this fashion:

m1, n1, m2, n2, m3, n3, m4, n4 ...

Therefore number of fractions is countably infinite. Since the rational numbers are countably infinite and the reals are uncountably infinite, the irrational numbers must be uncountably infinite.

The Transcendentals - A polynomial is any sum of expressions of the form ax^n where a is an integer and n is a positive integer. For example: 5x^2 + x + 3. The set containing all the possible roots of all possible polynomials contains all of the Algebraic Numbers. Any number which cannot be the root of a polynomial is a transcendental number. There is a proof that the transcendental numbers are uncountable, but I'd like to give it a go myself before I post it here.

What's so beautiful about this result is that it is very hard to prove that specific numbers are irrational or transcendental. In fact there are very few known transcendental numbers (pi is probably the most famous transcendental number), but any random number that we generate with an infinite number of digits is very likely to be transcendental.