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Transcendental numbers

Transcendental Numbers

Transcendental number is possibly a complex number, which is not an Algebraic number. It means not a root of polynomial equation with Rational coefficients. The note worthy example of Transcendental numbers are Π and e. The name Transcendental was first given by Leibniz in 1682 while he proved sin x is not an Algebraic function of x. Euler was the first person to define Transcendental number in modern sense.

Its difficult to show whether the given number is transcendental number. Because they are mostly real and complex numbers which is uncountable,while the Algebraic numbers are countable. These numbers are also irrational as rational numbers are Algebraic.

Transcendental number cannot be constructed in the finite number of steps from the elementary functions and their inverses. The first 'non-constructed' number was proved transcendental after 1873, when Charles Hermite proved e as transcendental. In 1884, Π was proved transcendental by Ferdinand von lindemann.

In 1884, Liouville found a theorem called Liouville's approximation theorem. It's used to determine whether any special numbers are transcendental. He specifically proved that any number with rapidly converging sequence of rational approximations should be transcendental.

Gelfond's theorem gives the general rule to determine whether the given special numbers of the form αβ are transcendental. In transcendental theory, Baker gives the lower bound for linear combination of logarithms of algebraic numbers.

Transcendental numbers plays significant role in the history, because it was the first proof to solve the ancient puzzle, called as 'Squaring the circle'. These numbers are used to formulate the close approximations to Squaring the Circle.

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Daily Maths Topic Today - transcendental numbers.


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