Assignment&Presentation:

Applied Mathematics

Prepared By:

Royal Benchers

Submitted To:

Sir Zeeshan Ghani Sahib

Royal Benchers

The

Group MembersZubair Mughal (Group Leader)

Nabeel AhmedWaqar AliAbdul QayyumKhadija MajeedZara MunirSoha Mazher

Algebra is the branch of mathematics that uses letters in place of some unknown numbers.

Basic AlgebraDefinition:

The part of mathematics in which letters and other general symbols are used to represent numbers and quantities in formulae and equations.

Basic Definition:

The word algebra originated from the title of the book ilm al-jabr

w'almuqabala, a book written during the ninth century the an Arabian

mathematician named al-Khworizimi. The original title was translated as

the science of restoration and reduction, basically meaning

transposing and combining similar terms of equations. Translated in

Latin toal-jabr (the union of broken parts) led to the term we now refer to as Algebra. Algebra was brought from

ancient.

Introduction To AlgebraHistory:

Fundamentals:Essentially, Algebra evolved from the

rules and operations of arithmetic which begins with the four

operations: addition, subtraction, multiplication and division of

numbers. Operations in algebra follow the same rules as those in arithmetic.

Algebra uses variables which is a symbol that represents a number and expressions which are mathematical statements that use numbers and or variables. The second example below

is an example of an expression. Algebra involves equations which are

statements that two numbers or expressions are equal.

5 + 5 = 2 x 55 + 5 = 10n + n = 2 x n5 + 5 = 2n (When 2n is written, it is understood that 2 is multiplied by n)

Example:

In Algebra, symbols or letters are introduced as a sort of shorthand which is used to abbreviate and

simplify long and often complicated statements.

Algebra is illustrated by many formulas used in science, computer

programs and the workplace industry.

- 2 = 4What is the missing number?

OK, the answer is 6, right? Because 6 − 2 = 4.Well, in Algebra we don't use blank boxes, we use a letter (usually an x or y, but any letter is fine). So we write x – 2 = 4It is really that simple. The letter (in this case an x) just means "we don't know this yet", and is often called the unknown or the variable.

X = 6

Puzzle:

Why Use a Letter?

It is easier to write "x" than drawing empty boxes (and easier to say "x" than "the empty box").

If there are several empty boxes (several "unknowns") we can use a different letter for each one.

How to Solve:

Algebra is just like a puzzle where we start with something like "x − 2 = 4" and we want to end up with something like "x = 6".But instead of saying "obviously x=6", use this neat step-by-step approach:

Work out what to remove get "x = ..."Remove it by doing the opposite (adding is the opposite of subtracting).

Do that to both sides.

Example:

We want to remove the "-2“

To remove it, do the opposite, in this case add 2:

Do it to both sides: Which is ...

Solved!

Why we add 2 to both sides?

To keep the balance,what we do to one side of the "=" 

we should also do to the other side!

In Balanc

eOut of Balanc

e

Balanc

e Again

X-2 = 4+2 4x-2

+2 +2x-2 4

Algebraic Expressions

Definition:

An algebraic expression is a mathematical phrase that can contain ordinary numbers, variables (like x or

y) and operators (like add,subtract,multiply, and divide).

Types of Algebraic Expressions

Monomial Polynomial Binomial Trinomial Multinomial

Monomial:

An algebraic expression which consists of two non-zero terms is called a binomial.

Examples of Monomial:

1) 10ab2 is a monomial in two variables a and b.

2) 5m2n is a monomial in two variables m and n.

3) -7pq is a monomial in two variables p and q.

Polynomial:

An algebraic expression which consists of one, two or more terms is called a polynomial.

Examples of polynomials:

1) 2a + 5b is a polynomial of two terms in two variables a and b.

2) 3xy + 5x + 1 is a polynomial of three terms in two variables x and y.

3) 3y4 + 2y3 + 7y2 - 9y + 3/5 is a polynomial of five terms in two variables x and y.

 Binomial:

An algebraic expression which consists of two non-zero terms is called a binomial.

Examples of binomials:

1) m + n is a binomial in two variables m and n.

2) a2 + 2b is a binomial in two variables a and b.

3) 5x3 – 9y2 is a binomial in two variables x and y.

Trinomial:

 An algebraic expression of three non- zero terms only is called a trinomial.

Examples of trinomial:1) x + y + z is a trinomial in three

variables x, y and z.2) 2a2 + 5a + 7 is a trinomial in one

variables a.3) xy + x + 2y2 is a trinomial in two

variables x and y.

Multinomial:

An algebraic expression of two terms or more than three terms is called a multinomial.

Examples of multinomial:1) p + q is a multinomial of two

terms in two variables p and q.2) a + b + c is a multinomial of three

terms in three variables a, b and c.3) a + b + c + d is a multinomial of

four terms in four variables a, b, c and d.

Important Terms

Integral:

If the algebraic expression doesn′t contain the division by variables (i.a. exponentiation with a fractional exponent), then it is called integral.

Fractional:

If algebraic expression consists of numbers and variables with the helping of operations of addition, subtraction, multiplication, exponentiation with the natural exponent and division, and besides division by expression with variables, then it is called fractional.

 Rational:

The integral and fractional expressions are called rational expressions.

Irrational:

If we use root extraction from variables (or raising to fractional power) in the algebraic expression, then such expression is called irrational.

Any Questions ?

Thank

You

The End