Fundamentals of Algebra - 01

Real Number System

Introduction to Sets

A set is a well-defined collection of objects. The objects in the collection are called the elements or members of the set.

In set notation, $a\in A$ means a is an element of the set A (the symbol $\in$ means 'belongs to').

Likewise, $b\notin A$ means b is not an element of the set A.

A set A is a subset of a set B if every element of A is also an element of B, and this relationship is represented as $A\subseteq B$. The symbol $\subseteq$ means 'is a subset of' or 'is contained in'.

Set Representation

The Number Sets

Counting numbers are called Natural Numbers ($\mathbb{N}$ ), represented in set notation as:

$$\mathbb{N}=\{1,2,3,\dots \}$$

Adding the number 0 to the set of natural numbers gives the set of Whole Numbers ($𝕎$ ):

$$𝕎=\{0,1,2,3,\cdots \}$$

Adding all the negatives of natural numbers to the set of whole numbers gives the set of Integers ($\mathbb{Z}$ ):

$$\mathbb{Z}=\{\cdots ,-2,-1,0,1,2,\cdots \}$$

Rational numbers ($\mathbb{Q}$ ) are those numbers which can be expressed as a fraction of integers, provided the denominator is not zero. In decimal representation, rational numbers are either terminating or repeating decimals.

$$\mathbb{Q}=\left\{\frac{p}{q}:p\in \mathbb{Z},q\in \mathbb{Z},q\ne 0\right\}$$

Irrational numbers ($𝕋$ ) are those which cannot be expressed in fractional form like rational numbers. They are represented as non-terminating and non-repeating decimal numbers. e.g. $\pi \text{,}\sqrt{2}$. In fact, if n is a positive integer, then $\sqrt{n}$ is irrational unless n belongs to the sequence of perfect squares.

$$𝕋=\{x:x\in ℝ,x\notin \mathbb{Q}\}$$

Real numbers ($ℝ$ ) have decimal representations, and comprise the combined set of the rational and the irrational numbers.

The relationships among the number sets are as follows: $$\mathbb{N}\subseteq 𝕎\subseteq \mathbb{Z}\subseteq \mathbb{Q},\mathbb{Q}\subseteq ℝ,𝕋\subseteq ℝ$$

A set of numbers is said to be closed under an operation if performing the operation on numbers of the set always produces another number in the same set. For example, the set of odd integers is closed under multiplication, but is not closed under addition.

The Real Number Line

A one-to-one correspondence exists between the set of real numbers and the set of points on a line. A line with a real number associated with each point, and vice versa, as shown in Fig. , is called a real number line, or simply a real line. Each number associated with a point is called the coordinate of the point.

Intervals and Inequalities

A subset of the real line is called an interval. To express intervals, we write an inequality in the form $a\le x\le b$.

For example, the set of all real numbers x such that $x>4$ is an interval, as is the set of all x such that $-3\le x\le 7$.

The set of all nonzero real numbers is not an interval; since 0 is absent from the set, the set fails to contain every real number between −1 and 1 (for example).

Geometrically, intervals correspond to rays and line segments on the real line. Intervals of numbers corresponding to line segments are finite intervals; intervals corresponding to rays and the real line are infinite intervals.

A finite interval is said to be closed if it contains both of its endpoints, half-open if it contains one endpoint but not the other, and open if it contains neither endpoint. The endpoints are also called boundary points; they make up the interval's boundary.

Interval notations are used to define intervals. Specifically (with reference to Fig. ),

• The finite interval $[a,b]$ represents a closed interval.
• The finite interval $(a,b)$ represents a open interval.
• The finite intervals $[a,b)$ and $(a,b]$ represent half-open intervals.
• Infinite intervals are closed if they contain a finite endpoint, and open otherwise.
• The entire real line is an infinite interval that is both open and closed.
• The interval notation for a single point a would be simply [a].

Properties of Real Numbers

Commutative Properties

• a + b = b + a
• ab = ba

Associative Properties

• a + (b + c) = (a + b) + c
• a (bc) = (ab) c

Distributive Properties

• a (b + c) = ab + ac

Identities

• For any real number a there exists a unique real number 0 such that a + 0 = 0 + a. The number 0 is called the additive identity.
• For any real number a there exists a unique real number 1 such that a ⋅ 1 = 1 ⋅ a. The number 1 is called the multiplicative identity.

Inverses

• The additive inverse of a real number a is −a, since a + (−a) = 0.
• The multiplicative inverse of a real number a, $a\ne 0$, is ${\displaystyle \frac{1}{a}}$ , since $a\cdot {\displaystyle \frac{1}{a}}=1$.

Absolute Value

The absolute value of a number a, denoted by $\left|a\right|$, is defined by the formula

$$\left|a\right|=\{\begin{array}{c}a\text{,}a\ge 0\\ -a\text{,}a<0\end{array}$$

The absolute value of a number is also known as its magnitude, and is never negative.

Properties of absolute values

• $$\left|a\right|\ge 0$$

  .......(7)

• $$\left|-a\right|=\left|a\right|$$

  .......(8)

• $$\left|ab\right|=\left|a\right|\left|b\right|$$

  .......(9)

• $$\left|{\displaystyle \frac{a}{b}}\right|={\displaystyle \frac{\left|a\right|}{\left|b\right|}}, b\ne 0$$

  .......(10)

Absolute Value as distance from origin

Geometrically, the absolute value of a real number a is the distance from the origin to the real number a on the number line. If a and b are two points on the real number line, then the distance between them is given by $\left|b-a\right|$ or $\left|a-b\right|$.