Are they irrational? Transcendental? | Epic Math Time

Описание: Showing that a number is transcendental can be difficult. While π and e have a deep connection involving exponentiation, other combinations of them, like π + e, are not as well understood.

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Definitions:

Algebraic number: A number that is a root of some polynomial with integer coefficients. Examples include 42 (a root of x - 42), √2 (a root of x² - 2), φ (The Golden Ratio, a root of x² - x - 1).

All rational numbers r = p/q are algebraic, because they are always the root of the equation qx - p. Irrational numbers can also be algebraic, such as the example of √2 above.

Transcendental number: A number that is not algebraic. Examples include π, e, cos(1), etc.

The usual transcendental numbers that one encounters cannot be expressed in terms of finitely many operations on integers. Practically speaking, this is why they often get their own symbol and name, like π, as any other way to express them (such as an infinite sum of rational numbers) is cumbersome.

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