**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:

Department of Mathematics