what are transcendental numbers?

Answer 1

a transcendental number is a (possibly complex) number that is not the root of any integer polynomial, meaning that it is not an algebraic number of any degree. every real transcendental number must also be irrational, since a rational number is, by definition, an algebraic number of degree one. the most well known example of a transcendental number, of course, being pi.
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Answer 2

In mathematics, a transcendental number is any complex number that is not algebraic, that is, not the solution of a non-zero polynomial equation with integer (or, equivalently, rational) coefficients. The most prominent examples of transcendental numbers are π and e.

Although transcendental numbers are never rational, some irrational numbers are not transcendental: the square root of 2 is irrational, but it is a solution of the polynomial x2 − 2 = 0, so it is algebraic.

The transcendental numbers are uncountable. The proof is simple: Since the polynomials with integer coefficients are countable, and since each such polynomial has a finite number of zeroes, the algebraic numbers must also be countable. But Cantor's diagonal argument proves that the reals (and therefore also the complex numbers) are uncountable; so the set of all transcendental numbers must also be uncountable. In a very real sense, then, there are many more transcendental numbers than algebraic ones. However, only a few classes of transcendental numbers are known and proving that a given number is transcendental can be extremely difficult.

Answer 3

Numbers that cannot be obtained by a finite or infinite sequence of elementary operations. Examples include pi and e. Some
irrational numbers can be generated by a sequence of elementary operations, (add, mult, sub, div).

All transcendental numbers are irrational but not the converse.