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# pi q and a

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what is the last digit of pi

if pi is infinite, who makes each digit up

Guest Feb 15, 2017
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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