Analysis I (part 1): Prelude, basic concepts, real numbers

Can you calculate the value of sin(π/4)? Does the identity ln(xy)=ln(x)+ln(y) sound familiar? In the prelude we recall and expand on a few such notions, in particular the trigonometric functions sin, cos and tan and their properties, the concept of reciprocal functions such as exp and ln, some calculation rules concerning powers, logarithms and roots, sets and functions. Next we discuss number systems: based on the "intuitive" notion of the natural integers N={0,1,2,3,...}, we rigorously define the rational numbers Q. We conclude our discussion of the rational numbers Q. In doing so, we note that some very simple equations, such as x^2=2, do not admit a solution in Q. This is one of the motivations for introducing a larger system of numbers: the real numbers R. We give the axiomatic definition of the real numbers and study their properties.