what is the last digit of pi

if pi is infinite, who makes each digit up

Guest Feb 15, 2017


Gee that's funny. It would have been even funnier if you had asked what is the last digit of π.

A number z is said to be algebraic if there is some polynomial p(x), with integer coefficients, for which p(z) = 0. So, for example, i is algebraic since i^2 + 1 = 0 (p(x) = x^2 + 1). All rational numbers a/b are algebraic since b(a/b) - a = 0 (p(x) = bx - a).

If a number is not algebraic, then it is said to be transcendental.

In 1882 Ferdinand von Lindemann and Karl Weierstrass proved that e^α is transcendental for every non-zero algebraic number α. So e is transcendental since e = e^1 and 1 is a non-zero algebraic number. Since e^(i π) = -1 and -1 is not transcendental, i π must therefore be transcendental. Since i is algebraic, it follows that π must be transcendental.

If π had a last digit, then it would be a rational number and, since rational numbers are algebraic, π cannot have a last digit.

Guest Feb 15, 2017

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