The Set of Real Numbers - The Real Number System
    Sets of Numbers:
      Natural numbers
      Whole numbers
      Rational numbers
      Irrational numbers
      Real numbers
    The real number line and relations
      One-dimensional coordinate system
      Unit interval
      Relations, less than and greater than
Sets of Numbers:
The set of natural numbers 
         N = {1, 2, 3,  . . . , n, n + 1,  . . . }, the positive integers used for counting.
The set of whole numbers      
         N0 = { 0, 1, 2, 3,  . . . }, is just like the set of natural numbers except that it also includes zero.
The set of integers 
         Z = { . . . , 3, 2, 1, 0, 1, 2, 3,  . . . }, consists of all natural numbers, negative whole numbers and
zero. This means that the set of natural numbers is a subset of integers, i.e., N is a subset  Z.
The set of rational numbers
         Q = { a/b | a, b Z, b is not 0 }, is the set of all proper and improper fractions. That is, a ratio or
quotient of two integers a and b, where b is not zero.
All integers are in this set since every integer a can be expressed as the fraction a/1 = a. Thus, the set of all natural numbers N is proper subset of integers Z and the set of integers is proper subset of the set of rational numbers, N is a subset  Z is a subset  Q. Rational numbers can be represented as integers, fractions, terminating decimals and recurring or repeating decimals.
The set of irrational numbers, denoted I, is the set of numbers that cannot be written as ratio of two integers. 
An irrational number expressed as a decimal never repeat or terminate. The irrational numbers are precisely those numbers whose decimal expansion never ends and never enters a periodic pattern, such as 0.1020030004..., p, 2, 3, or any root of any natural number that is not a perfect root is an irrational number.
The set of real numbers, denoted R, R = Q U I
is the set of all rational and irrational numbers,  R = Q U I.

The real numbers or the reals are either rational or irrational and are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line, the number line.

The Real Number Line, and Relations


The real number line is an infinite line on which points are taken to represent the real numbers by their distance from a fixed point labeled O and called the origin
We use the variable x to denote a one-dimensional coordinate system, in this case the number line is called the x-axis.
The line segment  OE  denotes unit length, that is, | OE | = 1. The absolute value (or modulus) of a real number x, denoted | x | is its numerical value without regard to its sign.
For example, | + 5 | = 5 and | 5 | = 5, and 0 is the only absolute value of 0. The absolute value of a real number a is its distance from the origin.

The unit interval is the interval [ 0, 1 ] that is the set of all real numbers x such that  0 x 1, we say x is greater than or equal to zero and x is less than or equal to one, meaning x is between 0 and 1 including the endpoints.

A rational number - a/b, in the above picture, corresponds to the point  A' which is symmetrical regarding the origin to the point A, which denotes the rational number a/b.
Therefore, | OA | = | OA' | = a/b | OE | = a/b.  
For each real number x, there is a unique real number, denoted x, such that x + ( x ) = 0. In other words, by adding a number to its negative or opposite, the result is 0.
Every point of the number line corresponds to one real number. 

Relations, less than and greater than
Let a and b are distinct real numbers. We say that a is less than b if a b is a negative number, and write  a < b, i.e.,  a < b means a b is negative.
On the x-axis, a < b is represented as the number a lies to the left of b.
We say that a is greater than b if a b is a positive number, and write a > b, i.e.,  a > b means is positive. On the x-axis, a > b is represented as the number a lies to the right of b.
Similarly, a b denotes that a is less than or equal to b, and a > b, a is greater than or equal to b.
Beginning Algebra Contents A
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