Why is the famous number "e" considered transcendental instead of just irrational?

Question

What makes something transcendental? Why is e transcendental? Just wondering

Answer 1

A transcendental number is any real or complex number that is not an algebraic number, which is defined to be any root of any algebraic polynomial equation with rational real or complex coefficients. Roots of such equations are commonly irrational, but certain numbers such as e or π cannot ever be the root of any such equation.

However, that does not mean that e and π cannot be expressed together in a simple equation, as with the famous Euler equation:

e^(iπ) = -1

This is not a counterxample, because the power is a transcendental complex number, namely iπ.

Answer 2

A transcendental number is a number that is not the root of any polynomial with integer coefficients. (A number is a "root" of a polynomial if, when we plug that number in for x in the polynomial, we get zero.)

To understand what a transcendental number is, it is probably easier to first understand what a transcendental number isn't.

Any rational number is not transcendental, because it's easy to find a polynomial which has it as a zero. For example, 2/3 is the root of 3x - 2; 7/12 is the root of 12x - 7; and, in general, a rational number p/q is the root of qx - p.

The number √2 is not transcendental, because it is the root of x^2 - 1.

The number √2 + √3 is not transcendental, because it is the root of x^4 - 10x^2 + 1.

The number e, on the other hand, is NEVER the root of ANY polynomial with integer coefficients. (This is very hard to prove, but it has been proven.)

This is what it means to say e is transcendental.

The number π is also transcendental.

Very few other numbers have been shown to be transcendental.

You can read more about transcendental numbers on Wikipedia at http://en.wikipedia.org/wiki/Transcendental_number . Be aware that math articles on Wikipedia are usually very technical. (Most of them are written for mathematicians by mathematicians, with little to no effort to make things understandable for an average reader.)

(N.B. others have said that a number is transcendental if it is not the root of any polynomial with *rational* coefficients, whereas I said *integer* coefficients. These two definitions are equivalent. It turns out that if something is the root of a polynomial with rational coefficients, then it is also the root of a polynomial with integer coefficients, and vice versa.)

Answer 3

A number is transcendental if it is not the root of any polynomial equation with rational coeffiecients. In plain English, that means it will never arise as the solution of any ordinary algebraic equation. You will have noticed, on the other hand, that irrational numbers such as the square root of 2 often crop up in solving quadratic equations.

Answer 4

π is another one. It is transcendental because it is one of the irrationals whose pattern of numerals never repeats. These numbers have been taken out to over a million decimal places, and no pattern has been detected. 1/3 is irrational, 0.333333..., but of course the pattern repeats. Euler named it.

Answer 5

Because it has been proved ! (The proof uses maths way beyond high-school and many university maths courses !)

There are many proofs available on the web: e.g.

http://www.math.sc.edu/~filaseta/gradcourses/Math785/Math785Notes6.pdf