What is Algebra? || Why do we study Algebra? || History of Algebra || Applications of Algebra || (Constant, variable, Coefficient, Algebraic Expression/Expression, Term)

Difference between Arithmetic and Algebra

Did you understand the difference between Arithmetic and Algebra?

What is Algebra?

• The term algebra is an extension of arithmetic in which letters, symbols, or variables replace numbers or quantities.

• Such a problem in which at least one number is unknown is called Algebra.

E.g. $x+5$,   $8y+4$,   $△+3$,   $◻+3=3$

Why do we study Algebra?

• Algebra provides methods to transform and solve difficult and complex mathematical problems in an easy way.

History of Algebra

• Algebra is the invention of the Muslims.
• In $820$ $AD$, a Muslim mathematician and astronomer Muhammad ibn Musa Al-Khwarizmi $(780-850)$ wrote a book in Arabic named (Al-Kitab al-mukhtasar fi hisab al-jabr wal-muqabala)
• In English book will be "The compendious book on calculation by completion and balancing".

Al-jabr $=$ Restoration/Completion

$x+2=8$

$x=8-2$

$x=6$

Muqabala $=$ Balancing

$x+y$ $=y+3$

$x$ $=y-y+3$

$x$ $=3$

• Algebra is taken from the Arabic language Al-jabr which means "bringing together broken parts".
• The first translation of this book named after the title of this book Al-jabr is Algebra which means "Restoration" and was published in Europe in the Latin language which is the base of the subject "Algebra".
• The letter "$x$" is used in almost every equation to find the unknowns.
• When Al-Khwarizmi wrote a book in Arabic, He used some letters as in Arabic letter "ش" and in English "sh".
• In Arabic, the word "الشیء" means "a thing" which we don't know the value or "unknown".
• This book was translated into many languages such as Latin, Greek, Spanish, etc.
• When this book was translated from Arabic into Spanish language then there was a difficulty in the pronunciation of "sh" ("ش") because ("ش") "sh" does not exist in the Spanish language.
• This difficulty was solved by the Greeks were from taken the word "ck" and using the symbol "$ϗ$" which is the Greek letter "Kai".
• After some time when the books were translated into the Latin language from the Spanish language, they used (ax) $x$, instead of $ϗ$ (kai) because this is similar to $x$, which is used for unknown, or variable that is the reason and up to now we use the letter $x$ (ax) in every expression and equation.

Applications of Algebra

Muslims first time used algebra in inheritance

let an old man has $24,000$ and he has two sons and two daughters he has to divide among them.

In Islam, every daughter gets half of the son

so,

If the sharing of each son is $2x$ and the daughter's sharing is $x$

Two sons get $=2x+2x$ $=4x$

Two daughters get $=x+x$ $=2x$

As

$2x+2x+x+x$ $=24,000$

$6x$ $=24,000$

$x$ $=\frac{24,000}{6}$

$x$ $=4,000$

Each son gets $=2x$ $=2(4,000)$ $=8,000$

Each daughter gets $=x$ $=4,000$  Ans.

$1)$ Variable

• A variable is a symbol used to denote an uncertain value.
• an amount or symbol whose value may fluctuate.
• In equations, formulae, and expressions, variables represent letters.

E.g. your class, your age, your pocket money,

In algebra, variables are used to identify the unknowns in an equation.

Most of used variables are

$x$, $y$, $z$,….$a$,  $b$,  $c$.  etc.

Variable $=$ Vary $+$ able

    $2.$ Constant

• A symbol having a fixed numerical value.
• any quantity or symbol that is not vary called a constant.

e.g.

$0$,  $1$,  $2$,  $3$,  $4$,…;  $\sqrt{2}$,  $-1$,  $\frac{1}{2}$,  $-4\sqrt{3}$

Real-life examples of Constant

Father, Mother, Gender, Colors, Constant Names, every day of the week, etc.,

 

    $3.$ Coefficient

• A coefficient is a constant number that is used to multiply a variable.

Example:

$5x^2$,  $5$ is the coefficient of $x^2$

$3x^2$ $-4x$, $3$ is the coefficient of $x^2$ and $-4$ is the coefficient of $x$.

 

    $4.$ Algebraic Expression/Expression

• Any collection of constants and variables joined by fundamental operations $(+, -, ×, ÷)$, roots, and powers is called an algebraic expression.

Thus,

$3a+5b$,  $4x^2$ $-\frac {2}{x}$ $+xy-2$,  $\frac {1}{\sqrt {x}}$ $+\frac {2xy}{z}$ $-x^2$ $+2$,  $4x$ $+\sqrt[3]{2xy}$ $-\left( x \right)^{3/2}$ $-\frac {1}{x}$ all are algebraic expressions.

 

    $5.$ Term

• The terms of the expression are the components of an algebraic expression joined by "$+$" or "$-$" signs.
• A term is a part of an algebraic expression that is separated by the sign “$+$” or “$-$” is called a term.

e.g.

$4x^2$ $+\frac {2\sqrt{2x}}{y}$ $-2xyz$ $+\left( xy \right)^2$ $+5$ has five terms

$4x^2$ $+2x$ $-2$ has three terms

$2x^2y^2$ $\sqrt {xyz}$ $+2xy$ $-2y$ $+8$ has four terms

 

$i)$    Like Terms

• Like terms are those terms with the same variable and the same exponent.

$x^2$,  $-2x^2$,  $9x^2$, $\sqrt {2}x^2$ are like terms

$xy^2z^2$,  $-3xy^2z^2$,  $4\sqrt {2}xy^2z^2$ are like terms

$x^{3/2}y^{2/7}z^{3}$,  $4x^{3/2}y^{2/7}z^{3}$ are like terms

 

$ii)$   Unlike Terms

Those terms having different variables or the same variables with different exponents are called, unlike terms.

e.g.

$2x$,  $2y$,  $4z$,  $2y^2$,  $-3x^2$,  $xy^2z$,  $-xyz^2$,  $2ab$,  $2bc$,  $7ac$,  $3xy$ are all unlike terms.