01-29-03 - Lecture 1 Andrei Antonenko January 30, 2003 1...

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Unformatted text preview: 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 well 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 well 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, well 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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This note was uploaded on 05/28/2011 for the course AMS 2010 taught by Professor Andant during the Spring '03 term at SUNY Stony Brook.

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01-29-03 - Lecture 1 Andrei Antonenko January 30, 2003 1...

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