# Difference Between Algebra and Calculus

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## Introduction

As a student, some subjects are made compulsory for education. Science, History, Language and Mathematics. These are compulsory because they are considered vital for survival in our day-to-day life. Take science, for example – everything you need from the moment you wake up needs science – toothpaste on your toothbrush, your sleepy staring reflection in the mirror, the mechanism of the water tap – all require science. Similarly, we require language for simple communication, reading and understanding. We need History for knowledge of the past so that we don’t make the same mistakes in the future. Mathematics is for basic calculations. It is also a significant part of science.

How many times have you openly cursed it? Maybe not everyone, but surely there are quite a few people in this world that have declared open repulsion towards the subject. It starts easy with simple addition and subtraction but over time gets so complicated that it is hard to make sense of the start and end of the problems. For example – A hen lays ‘x’ number of eggs and a crow flicks off ‘y’ number of eggs every month. A snake then steals ‘z’ number of eggs from the crow. Find out how many eggs the snake eventually steals off the hen every year. An easy problem, yes. But why is it necessary to know what the birds and the snake are up to? Its nature. Okay, from a scientific point of view it might seem important to keep tabs on the habitat but not to a 6th grader. Endless hours have been sacrificed by students to get a grip on such problems and their significance has deemed the problems preposterous, to say the least. Why are we made to endure this torment?

## Algebra vs. Calculus

Mathematics does contribute rather considerably to our daily life and while simple math itself is hard for some, it is further divided into branches that have specific functions. That may seem more agonizing but broken to its elements, it does have its benefits. Two such convoluted branches of math are algebra and calculus. The basic difference between the two is that algebra is more about operations and calculus is more about the rate of change. They are both essential and essentially different.

## Differences Between Algebra and Calculus in a Tabular Form

 PARAMETER ALGEBRA CALCULUS Definition Algebra is a branch of mathematics that deals with finding out the unknown variables using operations. Calculus is a branch of mathematics that deals with the rate of change. Origin of word The word ‘Algebra’ is derived from the Arabic word ‘al-jabr’, which translates to ‘reunion of broken parts’. The word ‘Calculus’ is derived from the Latin word, ‘calculus’, which means ‘pebble’. It depicts the pebbles used for calculation using an abacus. Origin of branch The earliest known equations in algebra were found in ancient Egypt in 1650 BC. Calculus originated in the 17th century. Domain Algebra operates within a known domain and yields results belonging to the same domain. Calculus does not have a specific domain. One can start in one domain and the results might lead to new discoveries that are not in the same domain. Level of complexity In comparison to Calculus, it is not as complex. Calculus is very complex. Main operations Algebra uses basic arithmetic operations like additions and subtraction to find the unknown variable. Calculus performs the functions of differentiation and integration, which are rather more complex. Branches There are two main branches of algebra. They are – elementary algebra and advanced algebra. There are two branches of calculus. They are – differential calculus and integral calculus. Uses Algebra is employed in everyday problems like finding the distance, weight or slope of the line. Calculus is employed for higher functions in advanced sciences.

## What is Algebra?

Algebra is a branch of mathematics that in its purest forms helps solve daily problems that involve a quest for the unknown. These unknown factors are called variables. Algebra employs arithmetic operations like addition, subtraction, multiplication and division to find these unknown variables.

Proof of algebra dates back to the medieval era. The earliest text was found in ancient Egypt – the Rhind Papyrus in 1650 BC. It gave evidence that Egyptians used algebraic linear equations to find the unknown variable. Evidence from 300 BC showed that Egyptians also employed two equations for two unknown quantities along with quadratic equations in their problems. Along with Egypt, algebra was found even in the historic accounts from Babylon, Greece, India and China.

The term ‘algebra’ itself is derived from the Arabic ‘al-jabr’, which translates to ‘reunion of broken parts’. A Persian mathematician Muhammed ibn Musa al-Khwarizmi, also known as the ‘Father of Algebra’, wrote a book titled – ‘Kitab al Muhtasar fi Hisab Al Gabr Wa I Muqabla’, which translates to ‘The Compendious Book on Calculation by Completion and Balancing’. He also described algebra as the ‘reduction’ and ‘balancing’ of subtracted terms.

There are two main branches of algebra – Elementary algebra and Advanced algebra. But over time, there were more additions or rather, divisions made of the branches.

• Elementary algebra:

It is the most standard branch of algebra and uses basic arithmetic operations and symbols like +, -, x and. The variables to denote the unknown quantities are mostly – a, b, x, y etc. some of the concepts covered in elementary algebra are variables, evaluation of equations, positive rational roots, exponents etc.

It is also known as the intermediate level of algebra. The concepts covered here are matrices, solving systems of linear equations, polynomial equations, trigonometry etc.

• Abstract algebra:

This branch is advanced and explores algebraic systems that are independent of the specific nature of basic operations. For example – vectors, groups, modules, fields, lattices etc. The concepts studied here are sets, binary operations, associativity etc. Such topics help in understanding ideas like the motion of an object through space or even understanding software codes in engineering.

• Linear algebra:

Linear algebra studies both pure mathematics and applied mathematics. It is used to study planes, lines and vector spaces. Linear equations are employed for linear functions of vector spaces and matrices. Linear algebra finds wide applications in the field of gaming, optimization of computer programming etc.

• Commutative algebra:

Commutative algebra is for commutative rings and their ideals. It is essentially part of algebraic geometry to study ring structures like polynomial rings, rings of algebraic integers etc.

Algebra is very useful in our daily life. We often employ it without even realizing it. Additionally, it has a wider scope than calculus.

## What is Calculus?

Calculus is a branch of mathematics that studies the rates of change. Before calculus, one could only calculate the value of the variable and not the rate at which it changed. With the invention of calculus, this was possible.

In the 17th century, Isaac Newton of England and Gottfried Wilhelm Leibniz of Germany independently developed calculus. It was considered an important invention in modern mathematics. Today, calculus is used in many advanced courses and forms the basis for advanced studies in the fields of physics, chemistry, technology, biology, medicine etc., where optimum solutions are a requirement.

It is used in mathematical problems that cannot be solved by simple algebra. Since it helps determine the rate of change it also finds applications in problems of supply and demand, cost production, temperature, pressure etc., where the math needs to be further analyzed.

### There are branches of calculus:

• Differential calculus:

In essence, the rules of calculus are to find the formula for a slope of a tangent to a curve at any point whilst having only the formula for the curve. The rate of change of function is denoted by ‘f’ and is called the derivative. Finding the formula of the derivative function is known as differentiation. The rules involved in this complex procedure form the basis of differential calculus.

• Integral calculus:

It is the branch of calculus that calculates integrals, which are infinitely small parts. Integral calculus is essentially the summation of these integrals and determining the whole. It has applications in the work systems, for example – the calculation of the pressure behind a dam at a certain depth.

Calculus is a complex branch of mathematics and is harder to understand which makes it less popular than algebra. But it is nevertheless, important. Complex fields like engineering, statistics, computer sciences etc. apply calculus in a way that makes our lives easier.

## Differences Between Algebra and Calculus in Points

Following are the main differences between Algebra and Calculus:

1. Algebra is a branch of mathematics that deals with solving equations to find out the unknown variable while calculus is a branch of mathematics that deals with finding out the change in the rates.
2. Algebra was invented long before calculus. The earliest known records of algebra date back to 1650 BC. Calculus was invented in the 17th century by Isaac Newton and Gottfried Wilhelm Leibniz.
3. The word ‘algebra’ is derived from the Arabic ‘al-jabr’, which translates to ‘reunion of broken parts.’ ‘Calculus’, on the other hand, is derived from the Latin word ‘calculus’, meaning ‘pebble’ for the pebble used for calculations with an abacus.
4. Algebra uses simple arithmetic mathematics to find out the unknown variables in the equation. Calculus uses differentiation and integration to find out the change in rates.
5. Algebra mainly operates in a single domain and the results obtained also belong to the same domain whereas, with calculus, the problem can lead to discoveries in other domains as well.
6. Algebra can be further divided into elementary algebra and advanced algebra. Calculus can be further divided into differential calculus and integral calculus.
7. Algebra is used in day-to-day life for finding distance, weight, circumference etc. Calculus is specifically used for complicated and advanced sciences.
8. In comparison to calculus, algebra is fairly easy to understand and use while calculus is rather difficult to comprehend.

## Conclusion

Algebra and calculus are branches of mathematics that find applications in our lives. Algebra is an old branch of pure mathematics that originated in the medieval era. Evidence of algebraic formulations was found in the ancient accounts of Egyptians, Greek, Babylonians, Indians and the Chinese. Since then, algebra has evolved and been modified. It is used to find the unknown variable, mostly denoted by a letter like a, b, x or y. Introduced in elementary school, it uses simple mathematical operations like addition, subtraction, multiplication and division. There are two broad categories of algebra – elementary algebra, which uses basic operations for unknown quantities and advanced algebra, which involves more complex functions and variables. Algebra finds applications in all our daily pursuits and has a broader scope than calculus.

Calculus was developed independently by Isaac Newton and Gottfried Wilhelm Leibniz in the 17th century and is considered a historic milestone of modern mathematics. It is the branch of mathematics that finds out the rates of change of variables concerning other variables. Since it goes the further step of determining the rate of variables, it is more complex than algebra. Yet, its applications are not limited. It is used in the advanced sciences – physics, chemistry, medicine, demographics and statistics. There are two basic types of calculus – differential calculus, which determines the rate of change and integral calculus, which calculates the sum of integrals.

While both these branches of mathematics are maddeningly obtuse at times, their use is what makes current lives a cakewalk. These branches of mathematics are often integrated with other branches of mathematics like geometry or trigonometry to solve intricate mathematical problems. So, instead of cursing the snake and the birds for their interest in eggs, we should simply solve for the unknown and call it a day because algebra is not going anywhere. We need algebra and calculus to increase our knowledge to help create epic inventions, generate newer ideas, figure out more about the unknowns and master the tough art of mathematics. It is hard, yes, but the solutions are also just as rewarding. Therefore, the goal is to never give up.

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