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.

Figure: A number line
\includegraphics[width=4in,height=0.5in]{line.eps}

Exs. From the above figure, the coordinate of the point $ O$ is the number 0, the coordinate of the point $ A$ is $ -2$.

Commonly used sets of numbers:

$ \diamond$ Natural Numbers $ \{1, 2, 3, 4,...\}$
$ \diamond$ Whole Numbers $ \{0, 1, 2, 3, 4,...\}$
$ \diamond$ Integers $ \{...,-3, -2, -1, 0, 1, 2, 3,...\}$
$ \diamond$ Even Numbers $ \{...,-4, -2, 0, 2, 4,...\}$
$ \diamond$ Odd Numbers $ \{...,-5, -3, -1, 1, 3, 5,...\}$
$ \diamond$ Rational numbers $ \Big{\{}\frac{p}{q}\Big\vert p$    and $ q$    are integers and $ q\ne 0\Big{\}}$
$ \diamond$ Real Numbers $ \{x\vert x$    corresponds to a point on a number line$ \}$
$ \diamond$ Irrational numbers $ \{x\vert x$    is real but not rational$ \}$

Operations of Real Numbers

Operation Inverse Operation
Addition: $ a+b=c$ Subtraction: $ c-a=b$ and $ c-b=a$
Multiplication: $ ab=c$ Division: $ c\div b =a$, if $ b \ne 0$ and $ c\div a=b$, if $ a \ne 0$
Power: $ a^n (=a\cdot a \cdot a \cdot\cdot\cdot a) =c$ Root: $ a=c^\frac{1}{n}$ if $ n$ is odd or if $ n$ is even and $ a\geq 0$



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. $ (2+5)\cdot 3-4\cdot6+1=7\cdot 3-24+1=21-4\cdot6+1=21-24+1=-2$

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

1. Closure: $ a+b$ and $ ab$ are real numbers.
2. Commutative: $ a+b=b+a$ and $ ab=ba$
3. Associative: $ (a+b)+c=a+(b+c)$ and $ (ab)c=a(bc)$
4. Identity: $ a+0=a$ and $ a \cdot 1=a$
5. Inverse: $ a+(-a)=0$ and $ a \cdot \frac{1}{a}=1 $
6. Distributive: $ a(b+c)=ab+ac$



Subsections


Department of Mathematics
Last modified: 2005-08-30