Real Numbers

We review the numbers which will be used in this lecture. Natural numbers are the numbers used for counting such as $1,2,3,4,\ldots$ . Integers are the natural numbers with 0 and negative numbers such as $\ldots,-3,-2,-1,0,1,2,3,\ldots$ . Rational numbers are fractions of integers $\frac{m}{n}$ with non-zero denominator $n$

. Irrational number are numbers with non repeated infinite decimals such as $\sqrt{2}, \sqrt{3}, \pi$ . Real numbers are either rational numbers or irrational numbers. Furthermore, $i$

satisfying $i^2 = -1$

is called the imaginary unit.

Symbols ${\mathbb{N}}$ : Natural numbers
${\mathbb{Z}}$ : Integers
${\mathbb{Q}}$ : Rational numbers
${\mathbb{R}}$ : Real numbers
${\mathbb{C}}$ : Complex numbers

Complex numbers $a + bi$ with the imaginary unit $i$ and real numbers $a,b$ is called complex number.

The set of real numbers can be expressed either using the symbol ${\mathbb{R}}$ ${\mathbb{R}}$ or the interval $(-\infty,\infty)$ .

The real numbers can be thought of as points on a number line. In other words, every real number can be put into one-on-one correspondence with the point on the number line.

$\begin{displaymath}\begin{array}{lllllllll}\hline \hspace{1cm} & -3& -2 & -1 & 0 & 1 & 2 & 3 & \hspace{1cm}\\ \end{array}\end{displaymath}$

For $x \in {\mathbb{R}}$ , we define

$\displaystyle \vert x\vert = \left\{\begin{array}{ll} x & (x \geq 0)\\ -x & (x < 0) \end{array}\right.$

Absolute value $\vert x\vert = \sqrt{x^2}$

We call $\vert x\vert$ absolute value of $x$ . For example, $\vert 3\vert = 3, \vert-4\vert = 4$ . Now carefully look at $\sqrt{x^2}$ . For if $x \geq 0$ , then we have $\sqrt{x^2} = x$ . For if $x < 0$ , then we have $\sqrt{x^2} = -x$ . This means that $\sqrt{x^2}$ and $\vert x\vert$ give rise the same number. Thus, we can say $\sqrt{x^2} = \vert x\vert$ . For example, $\sqrt{3^2} = 3, \sqrt{(-4)^2} = -(-4) = 4$ . Element of the set A distinct object $a$ belongs to the set $A$ is called the element of the set $A$ and denoted by $a \in A$ .

**Figure 1:** Element
$\includegraphics[width=2cm]{SOFTFIG-1/calcfig1-1.eps}$

NOTE The set of natural numbers is ${\mathbb{N}}$ . Then we write $3 \in {\mathbb{N}}$ and say 3 is an element of ${\mathbb{N}}$ . 0 is not a natural number. Then we write $0 \not\in {\mathbb{N}}$ . Subset If all elements of $B$ are also elments of $A$ , then $B$ is subset of $A$ and denoted by $B \subset A$ .

**Figure 2:** Subset
$\includegraphics[width=2.5cm]{SOFTFIG-1/calcfig1-2.eps}$

NOTE For ${\mathbb{N}}, {\mathbb{Q}}, {\mathbb{R}}$ , we must have ${\mathbb{N}} \subset {\mathbb{Q}} \subset {\mathbb{R}}$ .

Union The Union of tow sets $A$ and $B$ is the collection of points which are in $A$ or in $B$ or in $A$ and $B$ , and denoted by $A \cup B$ .

NOTE The set $A \cup B$ consists of all elements of $A$ and elements of $B$ .

**Figure 3:** Union
$\includegraphics[width=3.0cm]{SOFTFIG-1/calcfig1-1-1.eps}$

Intersection The Intersection of two sets $A$ and $B$ is the collection of points which are in $A$ and $B$ and denoted by $A \cap B$ .

NOTE The set $A \cap B$ consists of elements which have both properties of $A$ and $B$ .

**Figure 4:** Intersection
$\includegraphics[width=3.0cm]{SOFTFIG-1/calcfig1-1-2.eps}$

Subsets of ${\mathbb{R}}$

$\begin{displaymath}\begin{array}{lll} & & \begin{array}{lllll} \hspace{1cm} &\h... ...cm} \bullet &\hspace{-0.3cm} \hspace{1cm}\end{array}\end{array}\end{displaymath}$