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Number systems. Rational, irrational and real numbers. Open and closed intervals. Fields. Least upper bound. Greatest lower bound. Axioms, laws and properties.

Some numbers such as π, and e are not representable as the quotient of two integers. Thus they lie outside the system of rational numbers. These numbers are called irrational numbers. See the following:

The real number system. The real number system consists of the rational numbers and the irrational numbers. The set of all rational numbers plus the set of all irrational numbers gives the set of all real numbers. The real numbers correspond to points on a line called the real-number axis (see Fig. 1), with a one-to-one correspondence between the real numbers and points on the line.

Def. Rational number. A rational number is any number that can be expressed as the quotient of two integers. Thus the set of all rational numbers corresponds to the set of all quotients p/q where p and q are integers.

The following laws hold for rational numbers:

Irrational number. An irrational number is a number corresponding to any point on the real-number axis that is not representable by the quotient of two integers i.e. it is a number not expressible as the quotient of two integers. Examples: π, and e. The irrational numbers are in fact precisely those infinite decimals which are not repeating. There are two types of irrationals, algebraic and transcendental.

Def. Interval of real numbers. An interval of real numbers is the set containing all numbers between two specified numbers (the end points of the interval ) and one, both, or neither end point. Example: The set of all values of x such that 5 x < 8 is an interval denoted by [5, 8). An interval that contains both of its end points is called a closed interval and is denoted by [a, b], where a and b are the end points. One that contains neither of its end points is called an open interval and is denoted by (a, b).

We now introduce the concept of an abstract, axiomatically defined, mathematical system called a field. Both the rational numbers and the real numbers meet all the axiomatic requirements that a mathematical system must meet to qualify as a field.

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FIELDS

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Field. A set with two binary operations called addition and multiplication with the following abstract properties:

a,b in S implies a + b in S

2. Associative Law holds under addition.

a + (b + c) = (a + b) + c

3. Identity element called 0 exists under addition.

a + 0 = 0 + a = a for all a in S

4. Inverse exists for every element under addition.

a + (-a) = (-a ) + a = 0 for all a in S

5. Commutative Law holds under addition.

a + b = b + a

Multiplication.

6. Closed under multiplication.

a,b in S implies ab in S

7. Associative Law holds under multiplication.

a(bc) = (ab)c

8. Identity element called 1 (i.e. unit element) exists under multiplication.

1a = a1 = a for all a in S

9. Inverse exists for every nonzero element under multiplication.

aa-1 = a-1a = 1 for all a in S (except element 0 is excluded)

10. Commutative Law holds under multiplication.

ab = ba

11. Left Distributive Law holds -- multiplication over addition

a(b+c) = ab + ac

12. Right Distributive Law holds -- multiplication over addition

(a+b)c = ac + bc

13. No proper divisors of zero (i.e. there are no nonzero members a and b for

which ab = 0).

14. Cancellation Law under multiplication holds.

ax = bx implies a = b

Properties 1-11 are the defining properties for a field. Properties 12-14 are logical consequences of the first 11 properties. Thus any field has all of the above 14 properties.

Examples of fields: The set of all rational numbers, the set of all real numbers, and the set of all complex numbers.

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Both the rational and real numbers meet the following axioms of order:

Axioms of Order.

1. For any two numbers a and b one and only one of the following hold: a < b, a = b, a > b.

2. If a < b and b < c, then a < c

3. If a < b, then a + c < b + c

4. If a < b and c > 0, then ac < bc

As a consequence both the rational numbers and the real numbers are ordered fields.

Properties of sets of real numbers

The theory of functions of a real variable is concerned with a deep and careful investigation into functions, where and under what conditions they behave normally or abnormally, and of the sets of numbers on which they are defined.

Origin of idea of an open interval. Consider the function

What is the domain of definition of this function? We note that the denominator is zero at points x = 2 and x = 5. Thus the function is not defined at those two points (it becomes infinite at those two points). The domain of definition of the function is thus

- < x < 2, 2 < x < 5, 5 < x < .

In other words, the domain of definition consists of the open intervals

(- , 2), (2, 5), (5, ) .

Thus we see the source of the concept of an open interval and the reason for its importance. In investigating the behavior of functions and question of where they behave normally and where they behave abnormally we are led directly to the concept of an open interval.

Bounds. Let us now consider some examples of sets of real numbers.

1. The set

E1 = { [2, 5] , 9, 11 }

2. The set

E2 = { (3, 6) }

3. The set

Let us ask some simple questions about these sets that ought to be easy to answer. What is the smallest number in set E1? The answer: 2. What is the smallest number in set E3? Reflection will show that that question is not so easy to answer. It is not 0 because 0 is not in the set. How about set E2? What is the smallest number in E2? It is not 3 because 3 is not included in the set. What is the largest number in E2? It is not 6. The number 6 is not in E2. We can see that infinite sequences of numbers and the concept of open intervals present complications for us. One would think that if we are talking about sets of numbers we could answer simple questions like what is the smallest and largest number in the set. In response to this particular problem it has been necessary to introduce the somewhat awkward concepts of the least upper bound and greatest lower bound of a set of numbers. Before doing so we must first define the concepts of upper bound and lower bound.

Def. Upper bound of a set of numbers. A number (any number) that is greater than or equal to every number in the set. In other words, a number u is an upper bound of a set S of numbers if for all x ε S, x u.

Example. An upper bound for the set (3, 6) could be any number that is greater than 6, or the number 6 itself. Therefore the numbers 7, 10 , 20, etc. all qualify as upper bounds of the set (3, 6).

Def. Lower bound of a set of numbers. A number (any number) that is less than or equal to every number in the set. In other words, a number u is a lower bound of a set S of numbers if for all x ε S, x u.

Example. A lower bound for the set (3, 6) could be any number that is less than 3, or the number 3 itself. Therefore the numbers 1, 2, -7, etc. all qualify as lower bounds of the set (3, 6).

Def. Least upper bound or supremum (abbreviated l.u.b. or sup) of a set of numbers. The smallest of its upper bounds. If Q is the set of all upper bounds of set S, the least upper bound of S is the smallest number in Q. This is either the largest number in S or the smallest number that is greater than all the numbers in S.

Example. The least upper bound of the set (3, 6) is 6.

Def. Greatest lower bound or infimum (abbreviated g.l.b. or inf) of a set of numbers. The largest of its lower bounds. If Q is the set of all lower bounds of set S, the greatest lower bound of S is the largest number in Q. This is either the smallest number in S or the largest number that is less than all the numbers in S.

Example. The greatest lower bound of the set (3, 6) is 3.

Least upper bound axiom. A set of numbers that has an upper bound has a least upper bound.

Laws and properties. If x1, x2, ... ,xn are real numbers then the following laws hold:

1]        | x1x2 | = |x1||x2| or |x1x2 ... xn | = |x1||x2| ..... |xn|

References

James and James. Mathematics Dictionary

Spiegel. Real Variables (Schaum)

Mathematics, Its Content, Methods and Meaning. Vol. III