Number sets/Overview/Section

Without further justification, we may say that mathematics deals among other things with numbers. We work with the following sets, we assume that the students know these.

${}\mathbb {N} =\{0,1,2,\ldots \}\,,$

the set of natural numbers (including ${}0$).

${}\mathbb {Z} =\{\ldots ,-2,-1,0,1,2,\ldots \}\,,$

the set of the integers,

${}\mathbb {Q} ={\left\{a/b\mid a\in \mathbb {Z} ,\,b\in \mathbb {Z} \setminus \{0\}\right\}}\,,$

the set of the rational numbers and the set of the real numbers ${}\mathbb {R}$.

These sets are endowed with their natural operations like addition and multiplication, we will recall its properties soon. We think of the real numbers as points on a line, on which all the described number sets lie. Also, one can think of ${}\mathbb {R}$ as the set of all sequences of digits in the decimal system (finitely many digits before the point, maybe infinitely many digits after the point). During the lecture we will encounter all the important properties of the real numbers, the so-called axioms of the real numbers, from which we can deduce all other properties in a logical way. Then we will be able to make our current viewpoint more precisely.