What is Calculus?

 

While many people believe that calculus is supposed to be a hard math course, most don't have any idea of what it is about. The good news is that if you remember your algebra and are reasonably good at it then calculus is not nearly as difficult as its reputation supposes. This article attempts to explain just what calculus is about--where it came from and why it is important.

First, a little history leading up to the discovery of calculus, or its creation, depending on your philosophy.

The word "calculus" comes from "rock", and also means a stone formed in a body. People in ancient times did arithmetic with piles of stones, so a particular method of computation in mathematics came to be known as calculus.

Arithmetic and geometry are the two branches of mathematics originating in ancient times. Mathematicians attempted to do algebra in those days but lacked the language of algebra, namely the symbols we take for granted such as +, -, X, ÷ and =. Much of the world, including Europe, also lacked an efficient numbering system such as that developed in the Hindu and Arabic cultures. (Try long division, for example, using Roman numerals.) Algebra as a branch of mathematics can be said to date to around 825 A.D. when a Persian, al-Khwarizmi, wrote the earliest known algebra text. (The word "algebra" comes from a Persian word in the title, "al'jabr", which means "to restore". The English term for a systematic mathematical method, algorithm, was derived from al-Khwarizmi's name by way of a Latin translation.)

For over seven hundred years algebra and geometry coexisted but were not well linked. Geometry describes the physical nature of our world while algebra is a sophisticated tool for mathematical analysis. Due to the Greek influence on Persian (or Islamic) mathematics geometry was successfully used to verify some of their algebraic methods, but there was no known way to harness the analytical power of algebra to analyze geometry. In the late 1500's the French philosopher and mathematician, Rene Descartes, had a profound breakthrough when he realized he could describe position on a plane using a pair of numbers associated with a horizontal axis and a vertical axis. By describing, say, the horizontal measurement with x's and the vertical measurement with y's, Descartes was able to give geometric objects such as lines and circles representation as algebraic equations. This seminal construction of what we call graphs is, arguably, the cornerstone without which our modern technology would not be possible. Descartes thus united the analytical power of algebra with the descriptive power of geometry into a branch of mathematics he called analytic geometry. This term is sometimes seen in textbooks with titles such as "Calculus with Analytic Geometry."

Descartes, as philosopher, is also the author of the famous line, "Cogito, ergo sum," or, "I think, therefore I am." He was attempting to settle an argument about whether we exist independently of God's imagination.

The next major breakthrough in mathematics was the discovery (or creation) of calculus around the 1670's. Sir Isaac Newton of England, and a German, Gottfried Wilhelm Leibnitz, deserve equal credit for independently coming up with calculus. Each accused the other of plagiarism for the rest of their lives, but for what it's worth, the world largely adopted Leibnitz's calculus symbols. Calculus did allow Newton to establish physics principles which remained uncontested until the year 1900 and which in our ordinary scale world still suffice to explain physics to excellent accuracy.

Calculus was developed out of a need to understand continuously changing quantities. Newton, for example, was trying to understand the effect of gravity which causes falling objects to constantly accelerate. The speed of an object increases constantly every split second as it falls. How can one, for example, determine the speed of a falling object at a frozen instant in time, such as its speed when it strikes the ground? No mathematics prior to Newton and Leibnitz's time could answer such a question, which appeared to amount to the impossibility of dividing zero by zero. The solution to this type of issue came to be known as the derivative. Derivatives are slopes of particular lines called tangent lines, and the reader may recall that slope of a line is a concept from Descartes' graphing.

Differential calculus is one side of calculus, the part concerned with continuous change and its applications. By understanding derivatives the student has at his or her disposal a very powerful tool for understanding the behavior of mathematical functions. Importantly, this allows us to optimize functions, which means to find their maximum or minimum values, as well as to determine other valuable qualities describing functions. Real-world applications are endless, but some examples are maximizing profit, minimizing stress, maximizing efficiency, minimizing cost, finding the point of diminishing returns, and determining velocity and acceleration.

The other primary side of calculus is integral calculus. Integration is a process which, simplistically, resembles the reverse of differentiation. This amounts to efficiently adding infinitely many infinitely small numbers. This allows us, in theory, to find the area of any planar geometric shape, or the volume of any geometric solid. But the applications of integration, like differentiation, are also quite extensive.

Until the mid-1800's mathematicians were content to use calculus-style computations under the heuristic evidence that they seemed to work very well. This was a fragile house of cards increasingly based on the faith that what they saw would always work. Largely under the influence of Karl Friederich Gauss (1777 - 1855) the mathematical world gradually returned to the ancient Greek ideal of mathematical proof by logic found in their [Euclidean] geometry. Gauss' student, Bernhard Riemann (1826 - 1866), and some of his contemporaries established a rigorous logical foundation for calculus now known as real analysis. Their definitions and theorems greatly influenced the language and teaching of calculus today.

It was only through calculus and the rigorous treatment it received in the 19th century that mankind could really begin to grasp the difficult concepts of infinity and infinitesimal. Calculus also completes the link of algebra and geometry by providing powerful analytical tools that allow us to understand algebra functions through their related geometry.

We now realize that great thinkers in ancient times ran into calculus concepts. Archimedes used calculus thinking, for example, to establish the area of a circle and the volume of a sphere, borrowing his methods of exhaustion--essentially limits--from Eudoxus of Cnidus. Zeno of Elea proposed four famous paradoxes which caused Aristotle, centuries later, to grapple with calculus ideas in his failed attempt to resolve them.

Calculus, by tradition, is usually a one-year course (four quarters or three semesters). The first half is concerned with learning and applying the techniques of differentiation and integration. The second half is concerned with further applications, using both sides of calculus, to vectors, infinite sums, differential equations and a few other topics. The last term of calculus is sometimes known as multivariate calculus, which is an application of calculus to three or more dimensions.

Calculus provides the foundation to physics, engineering, and many higher math courses. It is also important to chemistry, astronomy, economics and statistics. Medical schools and pharmacy schools use it as a screening tool to weed out weaker aspirants under the assumption that people who are unwilling or unable to handle the rigors of calculus stand little chance of surviving the hard work of studying medicine or pharmacology.

There are three main facets to being a successful calculus student:

--You must be good at algebra skills. It is not enough to have passed algebra, you must also remember what you learned! If you have to relearn algebra while learning calculus then the burden can overwhelm.

--Memorization of computational patterns is not enough. Some people can get by in algebra by memorization without understanding. In calculus it is quite necessary to pay attention and learn the concepts in order to apply them. This is learning at a mature level.

--You must be dedicated to study. Don't skip any classes except for the most dire reasons. Take notes. Above all, practice lots of problems, without which those concepts will not be reinforced and learned.

Students who enjoy intellectual stimulation and the power of abstract thinking tend to enjoy the beauty of calculus the most, but there is much to appreciate for those who are looking for powerful tools which which to understand and create in the physical world.

Finally, a good reason to take calculus is that you will be more competitive and have more career opportunities. Many people avoid demanding challenges; those willing to face them head on tend to go much further in life.


To contact the author by e-mail click on this link: Jon Davidson