Irrational numbers
         Rationalizing a denominator
    The real number line, and relations
         Inequalities of real numbers
         Relations, less than and greater than
      Interval definition and notation
         Closed and open intervals (unbounded intervals)
      Distance and absolute value
         Properties of absolute value
Irrational numbers
Irrational numbers are numbers that can be written as decimals but not as fractions. Irrational numbers have decimal expansion that neither terminate nor become periodic.
Any number on a number line that isn't a rational number is irrational. For example, 2, 3, and 5 are irrational numbers because they can't be written as a ratio of two integers. Only the square roots of square numbers are rational.
For any integers m and n nm is irrational unless m is the nth power of an integer. Or, any root that is not a perfect root is an irrational number.
Rationalizing a denominator
Rationalizing a denominator is a method for changing an irrational denominator into a rational one.
Examples:  
   
   
To rationalize a denominator or numerator of the form multiply both numerator 
and denominator by a conjugate, where are conjugates of each other.
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 which is symmetrical regarding the origin to the point A, which corresponds to 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.
Inequalities of real numbers
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 a b 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.

Interval definition and notation

An interval is the set containing all real numbers (or points) between two given real numbers, a and b, where
a < b.
A closed interval [a, b] includes the endpoints, a and b, and it corresponds to a set notation
{x | a < x  < b}, while an open interval (a, b) does not include the endpoints.
On the real line, the half-closed (or half-opened) interval from a to b is written [a, b) or (a, b], where square brackets indicate inclusion of the endpoint, while round parentheses denote its exclusion.
Thus,          [a, b] = {x | a < x  < b}  - a closed interval
                  [a, b) = {x | a < x  < b}  - an interval closed on left, open on right 
                  (a, b] = {x | a < x  < b}  - an interval open on left closed on right
                  (a, b) = {x | a < x  < b}  - an open interval
Unbounded intervals or intervals of infinite length are also written in this notation, thus [a, oo ) is unbounded interval x > a, which is regarded as closed, while (a, oo ) is the open interval x > a.
Hence,        ( oo , a] = {x | x < aand   [a, oo ) = {x | x > a},
                  ( oo , a) = {x | x < aand   (a, oo ) = {x | x > a}.
The real line R = ( oo , oo ).
Distance and absolute value
The distance between a number x and 0 equals x if x > 0, and equals  x if x < 0, therefore the absolute value of a number x, denoted | x |, is x if x > 0, and x if x < 0.
From the definition of the absolute value it follows that the distance between two numbers a and b,
  d(a, b) = | a b |.  
     Properties of absolute value             Examples   
 1  1
2 2
3 3
4 4
5 5
6 6
7 7
8 8
Functions contents A
Copyright 2004 - 2020, Nabla Ltd.  All rights reserved.