Introduction to Algebra

Historical background of Algebra

Algebra is a branch of Mathematics that provides an easy solution to many complex mathematical problems especially when quantity is represented by a sign without any arithmetical value.

The word Algebra has been taken from the Arabic word Al-Jabar. A Muslim Persian mathematician Muhammad Ibn Musa Al-Khwarizmi wrote an Arabic book title Kitab al-muhtasar fi hisab al-gabr wa-l-muqabala in 820 AD in which he explained the techniques of solving complicated mathematical problems.

Afterward, this book was published in Europe with the title of “Algebra”, therefore, Al-Jabar turned into Algebra.

Real Number:

A real number is a value that can be represented along the number line as

Classification of Real Numbers:

Real Numbers are classified into the following,

Rational Numbers
- Fractions
- Integers
  - Whole Numbers
    - Natural Numbers
    - Zero
  - Negative Integers
Irrational Numbers

Now we discuss all of above types of real numbers one by one….

1 – Rational Numbers

Rational numbers can be written as a ratio of two integers in the form p/q, where p and q are both integers and q is not equal to zero. It is denoted by Q. few examples of rational numbers are,

$\displaystyle 12,\,\,\frac{3}{2},\,-5,\,\,2\frac{1}{4},\,\,0$

There are two kinds of decimal representation of rational numbers

Terminating decimal fractions – fixed number of digits in its decimal part. For example,

$\displaystyle \frac{3}{2}=1.5$

$\displaystyle \frac{6}{5}=1.2$

Non-terminating or recurring decimal fractions – no fixed number of digits in its decimal part or some digits are recurring again and again. For example,

$\displaystyle \frac{6}{7}=0.857142....$

$\displaystyle \frac{4}{9}=0.44444....$

1.1 – Fractions

A fraction can be written as,

$\displaystyle \frac{{Numerator}}{{Deno\min ator}}$

For examples,

$\displaystyle \frac{3}{2},\,\,\,2\frac{1}{4},\,\,\frac{{12}}{{25}}$

Types of fractions are;

Proper Fractions – In proper fractions, numerator is always less than denominator, e.g. $\displaystyle \frac{1}{4},\,\,\frac{{12}}{{25}}$
Improper Fractions – In improper fractions, numerator is always equal or greater than denominator, e.g. $\displaystyle \frac{5}{5},\,\,\frac{{21}}{4},\,\,\frac{{51}}{{25}}$
Mixed Fractions – In mixed fractions, there is a proper fraction and a whole number as well, e.g. $\displaystyle 2\frac{5}{{12}},\,\,6\frac{{21}}{{25}}$

1.2 – Integers

It can be a whole number or negative integer having no fractional value after decimal point. It is denoted by Z, i.e. $\displaystyle Z=\{........,-3,-2,-1,0,1,\,2,\,3,.........\}$

1.2.1 – Whole Numbers

Whole numbers are integers with no negative integer which start from zero to onward on a number line. Whole numbers are denoted by W. i.e.

$\displaystyle W=\{0,1,\,2,\,3,.........\}$

It is pertinent to mention here that, positive integers are just like whole numbers but these start from 1 to onward without 0. i.e.

$\displaystyle {{Z}^{+}}=\{1,\,2,\,3,4,5,.........\}$

1.2.2 – Natural Numbers

Natural numbers are all whole numbers start from 1 to onward i.e. 1, 2, 3, 4, 5,……………. Natural Numbers are denoted by N. i.e.

$\displaystyle N=\{1,\,2,\,3,4,5,.........\}$

Zero:

Zero is considered as a whole number but without having a positive or negative value.

Negative Integers:

Negative Integers start from minus one to onward, i.e.

$\displaystyle {{Z}^{-}}=\{..........,-5,-4,-3,-2,-1\}$

2 – Irrational Numbers:

Irrational numbers are real numbers that cannot be written as a ratio of two integers. It is denoted by Q’. Few examples of irrational numbers are,

$\displaystyle \,\pi \,,\,\sqrt{2},\,\sqrt{5}$

Mathematical Properties of Real Numbers

Mathematical properties of real numbers with respect to addition, subtraction, multiplication and division are:

Properties of Addition:

Closure Property:

If a, b are two real numbers and a + b = c, then c is also a whole number.

5 + 3 = 8. Where, 8 is also a whole number.

Commutative Property:

If a, b are two real numbers, then,

a + b = b + a.

For example,

a = 5, b = 3

⇒ 5 + 3 = 8 = 3 + 5

Associative Property:

Let, a, b, c are 3 whole numbers, then,

a + (b + c) = (a + b) + c = (a + c) + b.

For example,

a = 3, b = 5, c = 4

3 + (5 + 4) = (3 + 5) + 4 = (3 + 4) + 5 = 12

Additive Identity:

In additive identity, there exists a distinct real number i.e. 0, therefore,

a + 0 = a = 0 + a

For example,

3 + 0 = 3 = 0 + 3

Additive Inverse:

Additive inverse of a is denoted by – a, then

a + (–a) = 0 = (–a) + a

It means, additive inverse of 1 is – 1. For example,

5 + (–5) = 0 = (–5) + 5

Properties of Subtraction:

Closure Property:

If a, b are two real numbers and a – b = c, then c is not always a whole number. For example,

Let, a = 5 and b = 3, then

5 – 3 = 2 (a whole number) whereas,

3 – 5 = -2 which is not a whole number.

Commutative Property:

If a, b are two real numbers, then,

$\displaystyle a-b\ne b-a$

For example,

a = 5, b = 3

$\displaystyle 5-3=2\ne 3-5=-2$

Associative Property:

Let, a, b, c are 3 whole numbers, then,

$\displaystyle a-(b-c)\ne (a-b)-c$

For example,

a = 3, b = 5, c = 4

$\displaystyle 3-(5-4)=2\ne (3-5)-4=-6$

Properties of Multiplication:

Closure Property:

If a, b are two whole numbers and $\displaystyle a\times b=c$ , then c is also a whole number. For example,

Let, a = 5 and b = 3, then

$\displaystyle 5\times 3=15$ (a whole number) and

$\displaystyle 3\times 5=15$ , which is also a whole number.

Commutative Property:

If a, b are two real numbers, then,

$\displaystyle a\times b=b\times a$

For example,

a = 5, b = 3

$\displaystyle 5\times 3=15=3\times 5$

Associative Property:

Let, a, b, c are three whole numbers, then,

$\displaystyle a\times (b\times c)=(a\times b)\times c=(a\times c)\times b$

For example,

a = 3, b = 5, c = 4

$\displaystyle 3\times (5\times 4)=(3\times 5)\times 4=(3\times 4)\times 5=60$

Multiplicative property of zero:

Multiplicative property of zero is,

$\displaystyle a\times 0=0\times a$

For example,

$\displaystyle 3\times 0=0=0\times 3$

Multiplicative Identity:

Multiplicative identity is given by,

$\displaystyle a\times 1=a=1\times a$

For example,

$\displaystyle 5\times 1=5=1\times 5$

Properties of Division:

Closure Property:

This property tells that the result of division of two numbers is not always a whole number. For example,

$\displaystyle \frac{{25}}{5}=5$ (a whole number), whereas,

$\displaystyle \frac{{14}}{{28}}=\frac{1}{2}$

Commutative Property:

If a, b are two real numbers, then,

$\displaystyle a\div b\ne b\div a$

For example,

a = 20, b = 10

$\displaystyle \frac{{20}}{{10}}=2\ne \frac{{10}}{{20}}=\frac{1}{2}$

Associative Property:

Let, a, b, c are three whole numbers, then,

$\displaystyle a\div (b\div c)\ne (a\div b)\div c$

For example,

a = 3, b = 5, c = 4

$\displaystyle 3\div (5\div 4)=2.4\ne (3\div 5)\div 4=0.15$

Your might be interest : Types of Equations

Arithmetic Operations

Order of Operations:

Basic arithmetic operations are:

Addition
Subtraction
Multiplication
Division

It is pertinent to mention here that, if a single arithmetic expression contains more than one arithmetic operation then it is necessary to understand the order of arithmetic operations.

Usually, the following rules are applied to ensure the proper orders of operations.

Operations inside the Parenthesis should be performed first.
Operations of Exponents and roots should be performed earlier to Multiplication and Division.
Multiplication and Division are performed earlier to addition and subtraction.
Addition and Subtraction should be performed from left to right.

One way to remember these orders of arithmetic operations is to use the acronym formed by the red bold letters from the aforementioned rules.

PEMDAS (Parentheses, Exponents, Multiplication, Division, Addition, Subtraction).

Another way to remember these orders of arithmetic operations is to use this sentence.

“Please excuse my dear aunt Shahida”

Example 1:

2 + 3 + (5 + 2)

Rule No.1 (Operations inside the Parenthesis should be performed first) should be applied here, therefore,

2 + 3 + 7 = 12

Example 2:

2 × 3²

Rule No.2 (Operations of Exponents and roots should be performed earlier to Multiplication and Division) should be applied here, therefore,

2 x 9 = 18

Example 3:

2 x 3 + 2

Rule No.3 (Multiplication and Division are performed earlier to addition and subtraction) should be applied here, therefore,

6 + 2 = 8

Example 4:

2 + 3 – 2 + 5

Rule No.4 (Addition and Subtraction should be performed from left to right) should be applied here, therefore,

5 – 2 + 5

3 + 5 = 8

Example 5:

2 3 + (2 × 3²) – 3

Apply PEMDAS Rule here,

2 3 + (2 × 9) – 3

2 3 + 18 – 3

6 + 18 – 3

24 – 3 = 21

Positive and Negative Numbers:

As we have already read, the real number system contains the set of integers. Integer can be a whole number or negative integer having no fractional value after decimal point.

Here, it is necessary to understand the rules that handle the addition, subtraction, multiplication and division of positive and negative numbers. The following tables and examples facilitates you to understand these operations.

First Value	Operation	Second Value	Equals to	Final Value
+	×	+	=	+
+	×	–	=	–
–	×	–	=	+
–	×	+	=	–
+	÷	+	=	+
+	÷	–	=	–
–	÷	–	=	+
–	÷	+	=	–

Example 1:

2 3 = 6

Example 2:

2 (-3) = -6

Example 3:

-2 (-3) = 6

Example 4:

12 ÷ 3 = 4

Example 5:

12 ÷ (-3) = -4

First Value	Operation	Second Value (Smaller or Larger)	Equals to	Final Value
+	+	(Smaller)	=	+
+	+	– (Larger)	=	–
–	+	+ (Smaller)	=	–
–	+	+ (Larger)	=	+
–	–	– (Smaller)	=	–
–	–	– (Larger)	=	+

Example 1:

4 + (-2) = 4 – 2 = 2

Example 2:

2 + (-3) = 2 – 3 = -1

Example 3:

2 + (-5) = 2 – 5 = -3

Example 4:

12 + (-3) = 12 – 3 = 9

Example 5:

12 – (-3) = 12 + 3 = 15

Example 6:

2 – (-3) = 2 + 3 = 5