Numbers and symbols of inequality etc.

The conceptualization of Numbers and Number Line is important in order to understand the concepts of Limits, Functions and Calculus. Those who want to develop a clear concept of Calculus must assimilate the basics of Numbers and Number line etc. discussed in the previous two post along with this one.

Understanding Equality and Inequalities:

Symbols:

=      >      <      ≤     ≥

In the previous post, the basics of Numbers and Number Line were discussed, continuing from where we left of, let us bring back the diagram, Fig. 9.01, of the Number Line and try to understand a few more points. 

Real Number Line Fig 9.01

Talking about real numbers, we say the real numbers are 'ordered'. It means they follow certain principles, in the sense that, when we say a is less than b, which is generally written, using symbols, as a < b ,then by subtracting a from b  i.e. b - a we would get a positive number.
The above statement can also be also expressed as:  b is greater than a , written as b > a

Now let's understand how and where can we use the symbols   ≤      ≥  

The symbol  a ≤  b  or ( b ≥  a) means that either a < b or a = b , which is read as " a is less than or equal to b"

Some of the example of inequalities are as follows:

7 < 7.4 < 7.5     - π <  π  < 3    √2 < 2     2 ≤  2

If we show these numbers on these numbers on the number line they will look like this, refer to Fig. 9.02:  

The positioning of various numbers according to the numerical values Fig. 9.02

Conventionally the  set of real numbers is denoted by ℝ .

Also conventionally, by using the word number ,we mean the real number. The various types of numbers, discussed this post, and the previous two posts can be represented in the from the Diagram, 
Fig. 9.03 posted here under:

Diagram showing various Numbers Fig 9.03

Some of the properties of the real numbers are;

Each real number has a decimal representation. In case the number is rational it has repeating decimals, for example:

1/4  = 0.2500..... = 0.250 (zero keeps repeating)

 9/7 = 1,285714285714..... where .258714 ( repeats itself)

The irrational numbers such as  √2  or π  have decimal representation which is non-repeating, shown below:

√2 = 1.414213562373095........... 

The numbers have other properties which they exhibit during multiplication, division etc.