Lecture 1
Andrei Antonenko
January 30, 2003
1
Introduction
The first mathematical object which every person meets even in the childhood is a
number
.
So, the number is often considered as a main mathematical object, which is not really true
(though, it is not too far from reality).
In this lecture we’ll construct main groups (
sets
) of
numbers which we will use during the course, and figure out, what is the main algebraic object
of algebraic studies.
2
Different types of numbers
Numbers appeared during the time of ancient civilizations — but there numbers were simply
used for counting, so, only simplest numbers were used, like 1, 2, 3, 4, etc.
These numbers
are called
natural
. We can perform simple operations with these numbers — we can add them
together, and we can multiply them — we’ll still get natural number. But we can not subtract
them! How can we subtract, for example, 2 from 1? The answer will not be natural, so we
need to extend natural numbers by 0 and negative numbers, like 1, 2, 3, etc. So, we’ll get
integer
numbers. Now we can subtract, but there is another problem — we can not still divide
numbers! For example, we can not divide 1 by 2 — we will not get an integer number! So,
we have to introduce another class of numbers —
rational
numbers. Rational numbers are the
numbers which can be represented as
m
n
, where
m
is integer and
n
is natural, and we can always
find such numbers
m
and
n
that they do not have any common divisor greater than 1.
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 Spring '03
 Andant

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