- #1
murshid_islam
- 457
- 19
can anyone tell me why the transcendental numbers are uncountable?
HallsofIvy said:Since a polynomial of degree n has exactly n coefficients, there are a countable number of such polynomials.
benorin said:You can prove A is countably infinite by:first defining the height of a polynomial as the sum of the absolute values of its coefficients and its degree, e.g., |P(z)|=|a0|+|a1|+...+|an|+n for the above polynomial.
Then prove that there are finitely many polynomials of a given height, and that each such polynomial has finitely many roots (use the fundamental theorem of algebra for the second part).
murshid_islam said:i didn't understand this. could you elaborate a little?
Transcendental numbers are real numbers that are not algebraic, meaning they cannot be expressed as the root of a polynomial equation with rational coefficients.
Transcendental numbers are different from algebraic numbers because they cannot be expressed as a finite combination of rational numbers using the basic arithmetic operations (addition, subtraction, multiplication, division) and taking roots. Algebraic numbers, on the other hand, can be expressed in this way.
The concept of transcendental numbers was first introduced by the mathematician Joseph Liouville in 1844. However, the first explicit example of a transcendental number was given by Johann Heinrich Lambert in 1761.
The most famous transcendental number is pi (π), which is the ratio of a circle's circumference to its diameter. It is approximately 3.14159265358979323846 and has been studied and approximated by mathematicians for centuries. Another well-known transcendental number is e, the base of the natural logarithm.
Transcendental numbers have a significant impact on mathematics and science as they provide a way to describe and understand the physical world with greater precision. They are used in various mathematical and scientific fields, such as physics, engineering, and cryptography. The discovery of transcendental numbers also led to the development of new mathematical concepts and techniques.