Algebraic expressions that we see and encounter today

is way different than it was centuries ago. Until 19^{th} century, they

consisted of theories. The father of algebra Muhammad ibn Musa al-Khwarizmi

described algebra to be reduction and balancing of terms that is a

transposition to other sides of the equation. Algebra went through different

stages to development along the centuries. Along which the three stages that

gives us the history of symbolic algebra are to be discussed in this article.

### 1. RHEOTORICAL STAGE

In this stage, the algebraic expressions were written in

sentences. For example, an equation as: x+ 24= 36 would be described as

‘something added to 24 gives us 36’. It was developed by Babylonians back in

the 16^{th} century.

### 2. SYNCOPATED ALGEBRA

In this stage, the symbolism was used but it didn’t

contain all of the characteristics of symbolic algebra, there were some

restrictions involved. Syncopated algebra made its first appearance in the 3^{rd}

century AD through Diophantus Arithmetic, followed by Brahmgupta’s Brahma

Sphuta Siddhanta back in the 7^{th} century.

### 3. SYMBOLIC ALGEBRA

Symbolic algebra made its appearance through Islamic

mathematicians such as Ibn al-Banna in 13-14^{th} centuries and

al-Qalasadi in 15^{th} century in their works. And it was fully

developed by Francois Viete in the 10^{th} century. Later, Rene

Descartes by 17^{th} century, introduced the use of ‘x’ in an equation.

He also showed that problem in geometry can be expressed and solved in terms of

algebra. Quadratic equations played an important role in early algebra and it

is to be noted that earlier only three types of quadratic equations existed:

Since math was used in commerce there were no use of

negative terms. ‘p’ and ‘q’ was positive in each of the above cases and so were

the roots.

On to most interesting part where mathematical

expressions were solved with geometric practices. Between the rheotorical and

syncopated stage of symbolic algebra, the geometric constructive algebra’ was

developed by classical Greek and Vedic Indian Mathematicians in which equations

were solved through geometry.

Now let’s look at this equation:

Since we already know how to find roots of quadratic

equation, i.e. by using,

As they didn’t consider the negative part, let us

exclude that and put the respective values, we will get

And now let’s solve it with the help of figures. Taking

the x^2 as a square of side x and thus the term gives its area.

And the 26x can be denoted as rectangle with sides 26 and x and the term being

the area.

Now, if we divide the rectangle into two equal parts

which will give us two rectangle of side x and 13.

By placing the rectangles on two adjacent sides of the

square, we will see a L shaped figure which can we converted into a square if

we added a square of side 13 in the space.

And since we added a square, we now have to add 169, which

is the area of the additional square, on each sides of the equation. It gives

us

The left-hand

side of the equation represents the sum of areas of the 2 squares and 2 rectangle

and the right-hand side gives the total area. Since all the shapes collectively

makes a square, the number 246 denotes the area of the biggest square. And hence

14 will be the length of the sides. And now we know that the value of x must be

1.

While this method will only work on positive

entities and will only give the positive part of the solution, we can now see the quadratic equations from a whole new perspective.

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