Geometry Versus Algebra – The Ultimate War

The Queen of Sciences once had a question posed to her: Which minister of hers was better? The Minister of lines, shapes, and solids, or the Minister of variables and equations? But who are the ministers here? None other than Geometry and Algebra respectively. This made all the citizens of the kingdom question which was better and the Ministers were asked to explain themselves, with their applications and unique elements. This debate would help the Queen in settling her doubt. While the field of Mathematics is vast and has a lot of applications, two of the most used and basic are Geometry and Algebra. Here we will see about Geometry Versus Algebra.

Geometry Versus Algebra - The Ultimate War

Geometry refers to the field of mathematics which deals with lines, shapes, points, and objects of various dimensions and is widely used ranging from basic mathematics to solving questions of the universe. Algebra, on the other hand, is a branch of mathematics that uses variables like letters and symbols in equations and expressions to arrive at the needed solution. Both have their pros and cons, which help users of mathematics in different ways. For instance, geometry uses less arithmetic and numerals, which can be easy for some, but also has more formulae to memorize than algebra. 

What is Geometry?

Geometry, as mentioned earlier, is a branch of mathematics that deals with spatial concepts like dimensions, angles, sizes, shapes, and positions of things. It is one of the oldest branches of mathematics and is derived from a Greek word, “geometry” which means ‘measurements of the universe’. Some of the popular formulae that one might know from geometry are Pythagoras’ Theorem, Euclid’s axioms, and areas of various shapes.

Although it was the Egyptians who initially developed formulae and rules in the field of geometry, the development is often credited to the Greeks as they perfected the formulae and had huge contributions to their names. The Father of Geometry, Euclid, himself was Greek. 

The applications of this field range from finding areas to being used in construction and other major fields. The development of pi and other significant symbols paved a way for geometry to attain more importance in the court of mathematics. 

Geometry is broadly divided into 2D and 3D geometry.

2D Geometry:

2D or 2 Dimensional geometry is a basic field that deals with points, lines and planes. This majorly has 2 axes: the x-axis and the y-axis, which are depicted on the Cartesian plane. This category of geometry is introduced at an early age to students as it is easy to understand and visualize. 

3D Geometry:

3D or 3 Dimensional geometry, as the name signifies, refers to the part of mathematics that deals with objects that have 3 dimensions, that is, length, breadth, and height. These objects cannot be depicted with the help of a plane and thus use the 3D Cartesian plane and have 3 axes: x, y, and z. 

Apart from this, with the onset of scientific development, it was found that even abstract objects could be depicted with the help of geometrical formulae and terms. 

Other Classifications of Geometry:

Euclidean Geometry: 

It is the study of geometrical shapes and objects, based on axioms and theorems laid down by Euclid in his book Elements. There are various postulates in Euclidean geometry and most of them are easily understood. One of them is the axiom which states that all right angles are equal. 

Analytic Geometry:

Analytic geometry is the branch of geometry that deals with the position of objects and is studied by using the coordinate system. It is also called coordinate geometry or cartesian geometry. The location of an object on a plane is represented as a combination of 2 numbers, which refer to the x and the y axes, in order. This was developed by French mathematician René Descartes.

Projective Geometry:

This deals with the properties of objects when they are projected onto another surface. The relationship between images, objects, and graphical notations is dealt with in this branch of geometry. Projective Geometry was developed by French mathematician Girard Desargues.

Differential Geometry:

This discipline of mathematics studies smooth manifolds or smooth surfaces like curves and surfaces. This was developed using differential calculus and hence the name. The German mathematician, Carl Friedrich Gauss, who developed differential calculus showed that the curvature of a cylinder can be shown as a plane by cutting the cylinder across the axis and flattening it. But the same cannot be said for a sphere since it cannot be flattened without distortion. 


Topology is the youngest and most sophisticated branch of geometry. It focuses on the properties of geometric objects that remain unchanged upon continuous deformation—shrinking, stretching, and folding, but not tearing. It was developed by Dutch mathematician L.E.J. Brouwer. 

What is Algebra?

Algebra is the branch of mathematics that represents problems and questions in the form of expressions with variables and numerals. This involves the usage of arithmetic operations like addition, and subtraction as well. Algebra is generally used to find one or more unknown values, which are represented by letters or symbols in expressions. 

2x-4=0 or 3Θ+57=0 are both algebraic equations. 

Algebra is also divided into various categories, based on the use:

Elementary Algebra:

The basic topics of algebra fall under elementary algebra and it is normally the first step in the exploration of the real number world. The concepts of variables, formulation of expressions and solving them, equalities and inequalities all fall under elementary algebra. 

Advanced Algebra:

This is the next level of difficulty and has a higher level of equations. 

Mastering this level of algebra will help in solving other algebra-related concepts like inequalities, and matrices among others. Trigonometry and graphical equations are also solved by concepts of advanced algebra. 

Abstract Algebra:

This division of algebra deals with the study of algebraic structures like rings and groups. Since this does not have the conventional properties of the operators, this is called abstract algebra. Some of the abstract algebraic concepts are sets, binary operations, identity elements, inverse elements, and associativity. Sets are a collection of objects that have a specific common property. Binary operations are the operatives of arithmetic being performed on two operands to give a single result. Identity elements are 0 and 1 for addition and multiplication operations respectively. An inverse number is a change in sign or the opposite. For a number P, the additive inverse is -P and the multiplicative inverse is P^-1. Associativity is the theorem that states that when integers are added, the grouping of the numbers does not affect the final result. 

Linear Algebra:

Linear algebra focuses on the representation of linear equations, linear mappings, and vectors. This branch of algebra applies to both pure and applied mathematics and is almost used in all areas of science. The concepts covered in this are linear equations, relations, matrices, and so on. 

Commutative Algebra:

The commutative rings and their ideals are studied under this discipline of algebra and include the rings of algebraic integers, polynomial rings, and others. This type of algebra plays a huge role in pure mathematics. 

The Final War

While geometry and algebra have their applications and pros and cons, most mathematics cannot be solved by isolating one among them. Both these disciplines not only deal with mathematical problems, but the skills learned in each of them would help in real-life implications as well. Both of them are correlated and the line of distinction becomes more blurred with the advancement of mathematics. Some algebraic equations are worked out with the help of geometrical planes and vice versa is true as well when some dimensions or other elements of geometry have to be found using algebraic equations. 

Considering the applications and the correlation of both her Ministers, the Queen of Sciences pronounced that neither Algebra nor Geometry wins the war. She also exclaimed that thanks to both of the Ministers, the world is a little less challenging and smoother to run. Apart from this, she also advised the Ministers and the citizens of her kingdom to not isolate each other and to work in peace. Through this article we have learned about Geometry Versus Algebra – The Ultimate War. 

  • What are the fields in which both algebra and geometry are used together?

Most of the advanced problems of mathematics involve both of these fields working hand in hand, and the most basic of them is the representation of algebraic equations on the 2D coordinate system as lines, curves, or shapes. 

  • Which is harder to learn, algebra or geometry?

It depends on each person. If one person has good spatial skills, geometry would be easier. Meanwhile, for someone with good arithmetic skills, algebra would be easier to learn and perfect. But both fields can be perfected with practice and diligence. 

  • What are the other branches of mathematics?

Other branches of mathematics include calculus, arithmetic, trigonometry, statistics, and analysis. Each of them plays a role in the sciences. Like algebra and geometry, they are correlated as well. 

Geometry Versus Algebra – The Ultimate War

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