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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 mn , 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.
Example 2.1. 12 = 24 = 100200. Numbers 1 and 2 do not have any common divisors greater than 1, numbers 2 and 4 have common divisor which is equal to 2, and numbers 100 and 200 have common divisor which is equal to 100.
Now we have 4 elementary operations: “+”, “-”, “*”, “/”. Let’s define the operation of taking a power. By definition ab^ = a︸ · a · · · · ·︷︷ a︸ b
for any a and natural b. So, we can take a
second power of a number (square), but can we always find a number with the given square? For example, can we find a number x such that x · x = 2?
Theorem 2.2. There is no rational number such that its square equals to 2.
Proof. Let’s prove this by contradiction. Let α be such number that α^2 = 2, and m and n are minimal integer numbers such that α = mn (they are minimal, so they do not have any common divisor which is greater than 1). So,
α^2 = m
2 n^2 = 2.
So, m^2 = 2n^2. First of all, m should be even (Otherwise, if m is odd, then m^2 is odd, which is not true). So, m = 2k, and m^2 = 4k^2 , 4k^2 = 2n^2 , and 2k^2 = n^2. From this it follows that n is even too, thus it can be represented as n = 2l, and thus we see that α = mn = 22 kl = kl. So, we found another representation of α, and we have that k < m and l < n, which contradicts the initial assumption of the minimality of m and n. This contradiction proves that
2 can not be represented as a fraction, and thus is not a rational number.
So, we found a number which is not rational. Such numbers which can not be written as mn for integer m and natural n are called irrational. If we take these two groups of numbers — rational and irrational, and combine them to- gether, we will get the class of all numbers we can think of — class of real numbers. Actually, we can proceed with constructing different type of numbers, since, for example, we still can not find a number x, such that x^2 = −1. If we go by this way, we’ll come up with the notion of complex number. But we don’t need it right now.
3 Sets and operations
Bunches of similar objects are called sets. The exact mathematical definition of the set is very- very complicated, and is not really needed. But we will use the word “set” without this exact definition. Sets can contain objects of various types. We can introduce a set of all the students of Stony Brook University, or the set of all skyscrapers in the New York City. These sets are finite, since they contain only finite number of elements, but some sets are infinite, like a set of natural numbers. Different sets of numbers which we introduces above are denoted in the following way:
- N — set of natural numbers
- Z — set of integer numbers
