The set of transcendental real numbers

Real numbers are divided into rational and irrational numbers. Students learn about this division already in elementary school, but they become more familiar with it in secondary school. It is not so well known that real numbers are also divided into algebraic and transcendental numbers. Real numbers which are zeros of some polynomial with rational coefficients are called algebraic numbers. On the other hand, real numbers that are not zeros of any such polynomial are called transcendental numbers. This division of real numbers into algebraic and transcendental numbers represents the main topic of this diploma thesis. The set of algebraic real numbers and the set of transcendental real numbers are both infinite, but despite their infinity, we can still determine which is larger. The main goal of this diploma thesis is to present the proof that there are significantly more transcendental real numbers than algebraic real numbers, what may seem surprising, since we mostly operate with algebraic real numbers in elementary and secondary schools. Namely, almost the only transcendental real numbers that we learn about at school are the ratio between the circumference and the diameter of the circle, that is, the number π, and the basis of the natural logarithm, the number e.