What is the difference between Algebra and Calculus?

Algebra and calculus are two fairly distinct branches of mathematics; to say that they are not related is probably not strictly right in that most branches of mathematics are somehow inter-related – but this does not mean the two are very closely associated.

ALGEBRA:

Students are introduced to algebra at quite an early stage – usually in junior high schools. In the initial stages of algebra studies, the idea of a symbol is introduced. For instance, algebraic symbols are used to represent certain quantities which may or may not vary. Say a student is required to cut out a rectangle from manila cardboard, given that the length of the rectangle must be 3 times its breadth. In such a situation, algebra teaches us that we should let x be the breadth of the rectangle, in which case the length of the rectangle would be 3 x, and therefore the area of the rectangle shall be (length) times (breadth) which translates to (3x )(x) or 3×2 , read as ‘three x squared’. And if, in addition, it is a requirement that the rectangle in question must have an area equal to a certain constant number, then, unless the number is specified, algebra says that we could write the area of the rectangle as 3 x2 = a, where a is the symbol representing the constant. Of course, if we are given that the area of the rectangle has the value 50 cm2 then our formula for the area becomes 3×2 = 50.

Now, the mathematical statement 3×2 = 50 is what is known as an algebraic equation and more specifically it is a quadratic equation in that one could solve this equation to find the values of the unknown x, in this illustration representing the breadth of the rectangle in cm.

Generally speaking, algebra is the branch of mathematics studying the various algebraic expressions and functions, classifying and manipulating them and discovering their attributes. In fact, the term ‘function’ is itself an important math concept, which we shall not delve into deeply except to say that it is a way of showing the relationship between variables, quantities which change in values and which are usually represented by the symbols x, y, z, etc.

Algebra can be studied from a very elementary level up to advanced, undergraduate and even graduate levels. If you have done some algebra in junior to senior high school, you might have been quite familiar with terms like ‘linear function’ or ‘linear expression’, ‘quadratic/cubic expressions’, ‘quadratic equations’ etc and the various methods of factorizing the said ‘quadratic/cubic expressions’. You could have also studied the graphing of certain expressions/functions, given their algebraic equations. And in advanced courses at college and graduate levels, one could study in depth algebraic concepts like linear systems – systems of linear equations involving many unknowns and one would study the various techniques of ‘manipulating’ these systems. Of course, this is but one of the myriad topics in algebra. There are many other branches of algebra e.g. Boolean algebra, Algebraic structures, fields, and rings, to mention a few.

CALCULUS:

It is safe to say that not all high school students who take up math courses have studied calculus, a branch of mathematics which is both intellectually stimulating and conceptually challenging. Because of its perceived difficulty and complexity, not all high school students opt to study this branch of mathematics.

The fundamental concept in calculus is the idea of a ‘rate of change’, a quantity which measures how two or more variables change in relation to the others. In mathematical parlance, to compute the rate of change of say, two variables x, y (represented by two algebraic symbols, x and y ) a new math concept is introduced, denoted by dy/dx , known as the derivative of y with respect to x. And the process of finding this derivative is one of the most important objectives in the study of calculus.

Of course, it is imperative to be able to find the derivatives of mathematical functions such as y = x3 – x2; from the techniques of differentiation ( the techniques of finding the derivatives ) one obtains the expression dy/dx = 3×2 – 2x. No doubt these are algebraic expressions and in order to derive general rules of differentiation, the use of general algebraic expressions is unavoidable. This is where algebra meets calculus but the techniques of calculus are quite different from that of algebra. But calculus is much more than just finding the derivatives of algebraic functions – other functions like trigonometric functions also come into the picture.

The application of calculus is found in many fields: engineering, economics and other social sciences, the physical and biological sciences where a question with given parameters ( e.g temperature, pressure, humidity; population, costs of production, supply and demand etc ) is analysed and the parameters represented by algebraic symbols and the question becomes one of mathematics, specifically calculus. Depending on the question at hand, the engineer, economist or scientist will have to be able to form some mathematical equations ( for instance, differential equations ) and, given certain initial conditions about the values of the parameters, the question can be solved by solving the differential equation using the technique of finding the ‘anti-derivatives’ or ‘integration’.

CONCLUSION : DIFFERENCE BETWEEN ALGEBRA AND CALCULUS

I have allocated more space discussing algebra than calculus because most readers will be more familiar with ( elementary ) algebra rather than calculus. Most students would have some rudimentary knowledge of algebra – algebraic fractions; multiplication and division of algebraic expressions; factorization etc. Some simple questions of everyday occurrence require simple knowledge of how to apply algebra to these questions – the aforementioned question of the rectangle is one illustration. You should be able to find many relatively simple, everyday situations in which to usefully apply the knowledge acquired in your high school algebra course.

Most people would agree that unlike algebra, calculus is perhaps a more ‘advanced’ branch of mathematics in the sense that you don’t really need to know calculus to be able to solve simple, everyday questions that may arise. It would not be inaccurate to state that only in more complex scenarios occurring in the fields of engineering, the sciences that the techniques of calculus – differentiating various functions and integrating them – come into play.

To put in another way, algebra, at least at an elementary level, is the study of symbolic representation of variable and invariable quantities and the various operations involving them. Calculus, on the other hand, has no ‘elementary’ stage; even at the beginning level, one has to grasp the relatively more difficult concept of rates of change of variables. Both algebra and calculus can be studied up to highly advanced levels but perhaps calculus is slightly more challenging to grasp, even at the introductory level.