Why are transcendental functions called so?

[vktsn0303]

I have learned that they are called so because they cannot be expressed with the help of elemental methods of mathematics such as addition, subtraction, multiplication and division. But then isn't the whole of mathematics itself based on the elemental methods?

 

[pwsnafu]

Transcendental means in this context "to transcend algebra". At the time algebra meant studying polynomials. For example the fundamental theorem of algebra is a theorem about the roots of polynomials. Similarly an algebraic number is a number which is the root of a polynomial (usually with rational coefficients).
Transcendental functions are functions that cannot be expressed as polynomials, usually they are expressed as an infinite series. Even today, algebra is concerned with finite number of operations while analysis is concerned with infinite operations.

 

[vktsn0303]

Transcendental means in this context "to transcend algebra". At the time algebra meant studying polynomials. For example the fundamental theorem of algebra is a theorem about the roots of polynomials. Similarly an algebraic number is a number which is the root of a polynomial (usually with rational coefficients).
Transcendental functions are functions that cannot be expressed as polynomials, usually they are expressed as an infinite series. Even today, algebra is concerned with finite number of operations while analysis is concerned with infinite operations.

I see. Thank you pwsnafu.