# Real Numbers and Their Properties

Real Numbers and the Number Line

A description of a number line: A number line is a straight line. After fixing any point on the line as the origin which corresponds to the number 0, for any real number there is a point on the line corresponding to the number and for each point on the line there is a real number corresponding to the point. In another word, there is a one-to-one correspondence between the points on the number line and the set of real numbers. A real number is the displacement from the origin to the point on the number line that corresponds to the number and it is called the coordinate of the point.

Exs. From the above figure, the coordinate of the point is the number 0, the coordinate of the point is .

Commonly used sets of numbers:

Natural Numbers
Whole Numbers
Integers
Even Numbers
Odd Numbers
Rational numbers     and     are integers and
Real Numbers     corresponds to a point on a number line
Irrational numbers     is real but not rational

Operations of Real Numbers

 Operation Inverse Operation Addition: Subtraction: and Multiplication: Division: , if and , if Power: Root: if is odd or if is even and

Order of Operations When evaluating an expression which contains more than one operations, the order of operations is as follows:

1: Simplify all operations inside parentheses, brackets.
2: Simplify all exponents, roots working from left to right.
3: Perform all multiplications and divisions, working from left to right.
4: Perform all additions and subtractions, working from left to right.

Ex.

Properties of Operations Let , and be real numbers. Then the following properties hold:

1. Closure: and are real numbers.
2. Commutative: and
3. Associative: and
4. Identity: and
5. Inverse: and
6. Distributive:

Subsections

Department of Mathematics