What is the Pi value and how is it derived?

This is an important result since it answers the ancient question of squaring the circle.  That question asks: given a circle, can you construct a square with the same area as that circle using the Euclidean tools of straightedge and compass

Answer by David Joyce:

You ask if π can be expressed simply as a rational number or in surd form.

By surd form, you mean an expression in terms of roots along with the arithmetic operations of addition, subtraction, multiplication, and division, starting with whole numbers.  An example of a surd form is

                                           [math]\displaystyle \frac{10-\sqrt3}{\sqrt[3]7+\sqrt{5+\sqrt {11}}}.[/math] 

Numbers that can be so expressed are all algebraic numbers, that is, roots of polynomial equations with integer coefficients. Numbers that are not algebraic are called transcendental numbers.  Rational numbers are included among the algebraic numbers.

Lindemann (1852–1939) proved in 1882 that π is a transcendental number.  Therefore, π cannot be expressed as a surd.

This is an important result since it answers the ancient question of squaring the circle.  That question asks: given a circle, can you construct a square with the same area as that circle using the Euclidean tools of straightedge and compass.  The ancient Greek geometers conjectured it couldn't be done, but they had no proof.  Any construction would show that π could be expressed in surd form.  Lindemann's 1882 theorem, therefore, finally proved this ancient conjecture that a circle can't be squared using Euclidean tools.

             Ferdinand von Lindemann

What is the Pi value and how is it derived?

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