#MathsInMinutes Day 16: Algebraic and transcendental numbers

● An algebraic number is one that is a solution to an equation involving powers of the variable x, a polynomial with rational coefficients, while a transcendental number is one that is not a solution.

● The coefficients in such equations are the numbers that multiply each of the variables.

● For example, √2 is irrational, since it cannot be written as a ratio of two whole numbers.

● But it is algebraic, since it is the solution of x^2 – 2 = 0, which has rational coefficients (1 and 2).

● All rational numbers are algebraic, since any given ratio p/q can be found as the solution of qx – p = 0.

● We might expect transcendental numbers to be rare, but in fact the opposite is true.

● √2 is exceptional, and almost all irrationals are also transcendental.

● Proving this is very difficult, but a randomly chosen number between zero and one would almost certainly be transcendental.

● This raises the question of why mathematicians spend so much time solving algebraic equations, ignoring the vast majority of numbers.

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