## Real Number System

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# 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, $a ∈ A$ means a is an element of the set A (the symbol $∈$ means 'belongs to').

Likewise, $b ∉ A$ 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 $A ⊆ B$. The symbol $⊆$ means 'is a subset of' or 'is contained in'.

#### Set Representation

Two methods of representing a set are:
• Roster form: The elements of the set are listed out, e.g.
$A = { a , e , i , o , u }$; $B = { 1 , 3 , 5 , 7 , 9 , 11 , … }$
The ellipses '$…$ ' stand for continuation of the pattern of numbers shown earlier.
• Set-builder notation: The elements of the set are defined in terms of their common property, e.g.
In the above example the braces stand for 'the set of all' and the colon stands for 'such that'. The above is read as – "C is the set of all x such that x is an even integer and x is greater than 0". The pipe symbol '|' can also be used in place of the colon ':'

#### The Number Sets

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

$ℕ = { 1 , 2 , 3 , … }$

.......(1)

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

$𝕎 = { 0 , 1 , 2 , 3 , ⋯ }$

.......(2)

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

$ℤ = { ⋯ , − 2 , − 1 , 0 , 1 , 2 , ⋯ }$

.......(3)

Rational numbers ($ℚ$ ) 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.

$ℚ = { p q : p ∈ ℤ , q ∈ ℤ , q ≠ 0 }$

.......(4)

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. . In fact, if n is a positive integer, then $n$ is irrational unless n belongs to the sequence of perfect squares.

$𝕋 = { x : x ∈ ℝ , x ∉ ℚ }$

.......(5)

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: $ℕ ⊆ 𝕎 ⊆ ℤ ⊆ ℚ , ℚ ⊆ ℝ , 𝕋 ⊆ ℝ$

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 $a ≤ x ≤ b$.

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 $− 3 ≤ x ≤ 7$.

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, $a ≠ 0$, is $1 a$ , since $a ⋅ 1 a = 1$.

### Absolute Value

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

.......(6)

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

##### Properties of absolute values

• $| a | ≥ 0$

.......(7)

• $| − a | = | a |$

.......(8)

• $| a b | = | a | | b |$

.......(9)

• .......(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 $| b − a |$ or $| a − b |$.

## Solved Examples

List the subsets of $S = { − 3 , − 2 3 , 0 , 1 , 3 , 9 5 , 144 }$ consisting of (A) natural numbers, (B) integers, (C) rational numbers, and (D) irrational numbers. (Adapted from: Barnett et al. 2011, Algebra with Trigonometry, p. 10 #35)
$A = { 1 , 144 }$
$B = { − 3 , 0 , 1 , 144 }$
$C = { − 3 , − 2 3 , 0 , 1 , 9 5 , 144 }$
$D = { 3 }$
▼(show solution)
Convert the repeating decimal $0.090909 …$ into a fraction. (Adapted from: Barnett et al. 2011, Algebra with Trigonometry, p. 10 #49)
▼(show solution)
Describe the following graphs using interval notation:
(a)
(b)
(c)
(d)
(Adapted from: Narasimhan 2009, Algebra & Trigonometry, p. 10 #25,27,29,31)
(a) $[ − 3 , 4 ]$   (b) $[ 0 , ∞ )$   (c) $( 2 , 4 )$   (d) $( 1 , ∞ )$
▼(show solution)
Find the distance between the numbers on the real number line for the following:
(a) 12, 7.5  (b) 4.3, 7.9  (c) $− 4 5$ , $1 3$
(Adapted from: Narasimhan 2009, Algebra & Trigonometry, p. 10 #59,61,65)
(a) With a = 12 and b = 7.5, distance is given by $| a − b | = | − 12 − ( − 7.5 ) | = | − 4.5 | = 4.5$
(b) distance $= | − 4.3 − 7.9 | = | − 12.2 | = 12.2$
(c) distance $= | − 4 5 − 1 3 | = | − 17 15 | = 17 15$
▼(show solution)
Use absolute value notation to describe the following situations: (a) The distance between x and 5 is no more than 3; (b) The distance between x and 10 is at least 6. (Adapted from: Larson 2014, Algebra, p. 12 #51,52)
(a) $| x − 5 | ≤ 3$
(b) $| x − ( − 10 ) | ≥ 6 ⇒ | x + 10 | ≥ 6$
▼(show solution)

# List of References

Barnett AB, Ziegler MR, Byleen KE & Sobecki D, College Algebra with Trigonometry, 9th edn, NY, USA: McGraw-Hill, 2011.
Beecher, JA, Penna, JA & Bittinger, ML, Algebra and Trigonometry, 4th edn, NY, USA: Addison Wesley, 2012.
Coburn, JW, College Algebra, NY, USA: McGraw-Hill, 2006.
Larson R, College Algebra, 9th edn, MA, USA: Brooks/Cole, 2014.
Narasimhan R, College Algebra and Trigonometry: Building Concepts and Connections, Boston, USA: Houghton Mifflin Harcourt Publishing, 2009.

# Bibliography

Barnett AB, Ziegler MR, Byleen KE & Sobecki D, College Algebra with Trigonometry, 9th edn, NY, USA: McGraw-Hill, 2011.
Narasimhan R, College Algebra and Trigonometry: Building Concepts and Connections, Boston, USA: Houghton Mifflin Harcourt Publishing, 2009.
Thomas, GB, Weir, MD, Hass, J & Giordano, FR, Thomas' Calculus – including second-order differential equations, 11th edn, USA: Pearson Addison-Wesley, 2004.