The decimalsystem is a positionsystem (based on powers of ten) which means that the value of a number is determined by its position in the number.
e.g. 333 = 3• 100 + 3• 10 + 3•1.
Thanks to this ingenous system it is possible to express all rational numbers with just 10 digits (indo-arabian) 0 – 9. Equalling the number of fingers.
The natural numbers i.e. the positive integers are infinitely many.
This statement an be proved by adding 1 to any given candidate to being the biggest integer.
A peculiar fact is that the number of real-numbers (rational numbers + irrationalnumbers i.e. all the numbers between the real numbers) is higher than the number of integers. But how can anyyhing be bigger than infinity?
Georg Cantor solved this problem by dividing infinity into different categories (Cardinalities): the natural numbers are countably infinite ℵ(Cardinal number)=0 wheras the real numbers are uncountably infinite ℵ=1.
The transcendental numbers however such as π and e cannot be be explicitly written with these integers and therefore one must use special signs for them.
A mathematical function is a rule that tells you how to caculate a value from a given variable. This value must be unambiguous: only one function value must correspond to a given variable value.
A continuous function is a function that can be drawn in a coordinate system without lifting the pencil from the paper.