 
Summary: Lecture 1.
Real numbers. Constants, variables, and
mathematical modeling.
In this lecture we briefly sketch the structure of Real numbers and recall properties of
addition and subtraction. Then we will discuss the notions of variables and constants and basic
ideas of mathematical modeling.
1.1. Real Numbers
We will try to sketch the structure of numbers by ranging them by the increase of com
plexity. Most people agree that the simplest are the counting numbers.
1, 2, 3, 4, 5
The next step is the whole numbers (a "zero" element was added)
0, 1, 2, 3, 4, 5
to make whole numbers even more useful the addition and multiplication were invented.
The addition and the multiplication were invented in such a way to satisfy commutative
properties
a + b = b + a
a · b = b · a,
and the associative properties
a + (b + c) = (a + b) + c
a · (b · c) = (a · b) · c,
