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01-29-03

# 01-29-03 - Lecture 1 Andrei Antonenko 1 Introduction The...

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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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