Category Archives: mathematics

Calculus – A Game Changer

In modern day mathematics, calculus is probably one of the most frequently used tools and it’s development came after centuries of rigorous studies. During the ancient period any mathematical concept was termed to be calculus or in simpler words can be said that calculus was assumed to be a synonym of mathematics. However different ideas put forth by different scientists from different regions gave birth to the evolution of modern day calculus. Calculus was used by ancient people as a tool to calculate areas and volumes. For instance, early Egyptians used formulas which resemble integral calculus to calculate areas of solid shapes, Archimedes used the method of exhaustion to compute the area inside a circle and the Indians related calculus with trigonometry to give certain differentiation methods as early as the 8th century BC.

The 17th century saw the dawn of modern calculus with various mathematicians such as Blaise Pascal, Pierre De Fermat and Rene Descartes putting forth the concept of derivative. Certain concepts related to differentiation such as maxima, minima and tangential equations were created around this time especially by Fermat. Cavalieri’s theorems and methodology provided a more holistic approach to integral calculus and a refined version of the method of exhaustion. Cavalieri’s principle gave other mathematicians a foundation to work with especially after the computation of area under the x^n curve or a curve with higher degree. Also around this time Fermat created the base for integral calculus by giving the two fundamental theorems of calculus.
Majority of the modern day calculus was created by two men: Newton and Leibniz, who independently developed its foundations. Although they both were instrumental in its creation, they thought of the fundamental concepts in very different ways. While Newton considered variables changing with time, Leibniz thought of the variables x and y as ranging over sequences of infinitely close values. He introduced dx and dy as differences between successive values of these sequences. Leibniz knew that dy/dx gives the tangent but he did not use it as a defining property. On the other hand, Newton used quantities x’ and y’, which were finite velocities, to compute the tangent. Of course neither Leibniz nor Newton thought in terms of functions, but both always thought in terms of graphs. For Newton the calculus was geometrical while Leibniz took it towards analysis however the controversy surrounding them was both of them religiously preached infinitesimal calculus.
Although one could not argue with the success of calculus, this concept of infinitesimals bothered mathematicians. Lord Bishop Berkeley made serious criticisms of the calculus referring to infinitesimals as “the ghosts of departed quantities”. Ultimately, Riemann reformulated Calculus in terms of limits rather than infinitesimals. Thus the need for these infinitely small (and nonexistent) quantities was removed, and replaced by a notion of quantities being “close” to others. The derivative and the integral were both reformulated in terms of limits and definite integrals.
So when do you use calculus in the real world? In fact, you can use calculus in a lot of ways and applications. Among the disciplines that utilize calculus include physics, engineering, economics, statistics, and medicine. It is used to create mathematical models in order to arrive into an optimal solution. For example, in physics, calculus is used in a lot of its concepts. Among the physical concepts that use concepts of calculus include motion, electricity, heat, light, harmonics, acoustics, astronomy, and dynamics. In fact, even advanced physics concepts including electromagnetism and Einstein’s theory of relativity use calculus. In the field of chemistry, calculus can be used to predict functions such as reaction rates and radioactive decay. Meanwhile, in biology, it is utilized to formulate rates such as birth and death rates. In economics, calculus is used to compute marginal cost and marginal revenue, enabling economists to predict maximum profit in a specific setting. In addition, it is used to check answers for different mathematical disciplines such as statistics, analytical geometry, and algebra.
signing off…….Akarsh b Vasisht

Geometry Today

hello guys it’s great to be back. Today I am going to talk about one of the most hated topic especially but school students but probably one of the most interesting topics when we dig deep into the subject.

Math and many of its aspects are a major part of everyday life. We spend the majority of our school years studying and learning the concepts of it. Many times, the question of ‘why do we need to know these things?’ has been asked. So, in this article I am going to emphasise on the development of one subconcept of mathematics which is called “geometry” and it’s ever growing usage.

‘Geometry’ technically means ‘measure of the earth’. Geometry is the mathematics of the properties, measurement, and relationship of the points, lines, angles, surfaces, and solids.    Pythagoras emphasised the study of musical harmony and geometry. His theorem was that the square of the length of the hypotenuse is equal to the sum of the other two sides. Johannes Kepler, formulator of the laws of planetary motion is quoted as saying, ‘Geometry has two great treasures: one of them is the theorem of Pythagoras, the other the division of a line into mean and extreme ratios, that is the Golden Mean.’ As time moved on new concepts came out from various scientists.

An ancient Greek mathematician, named Euclidean, was the founder of the study of geometry. Euclid’s Elements is the basis for modern school textbooks in geometry. He created various axioms and postulates which then went on to shape modern geometry. His axioms and postulates were less of mathematics and more of mere common sense and simple logic. One such postulate was the parallel postulate: “That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.” People like Euclid and Pythagoras stated geometric concepts way before modern geometry had evolved and their theories formed a basis for modern geometry to develop. However, once various scientists came and went fresh ideas and approaches were needed to keep up with the quickly evolving science.

Some notable contributors to geometry were Rene Descartes, Isaac Newton, Leonhard Euler, and Albert Einstein.

Rene Descartes invented the methodology of analytic geometry, also called Cartesian geometry after him, which comprises of coordinate geometry as it’s called today. Blaise Pascal worked on projective geometry which is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary geometry, projective geometry has a different setting, projective space, and a selective set of basic geometric concepts. Leonhard Euler who is considered to be the greatest mathematicians ever was credited for successfully relating algebra to geometry using the graph theory and number theory. Newton and Einstein contributed to develop non-Euclidean geometry and embedded calculus into geometry to find areas and volumes of irregular geometrical shapes.

The reason geometry is such an important part of mathematics is because mathematics is all about visualisation and without a visual idea it would have been nearly impossible to create mathematics as it stands today. To put this into perspective, if a person wants to measure the length between the foot of a ladder lying on a wall and the base of a wall, he would first imagine the scenario and visualise what this would look like which then makes it easier to calculate using the Pythagoras theorem. So, to answer the question asked at the start of this essay ‘why do we need to know these things?’    it is because although geometry may not be directly applied in real life it forms the basis for other mathematical operations.

 

signing off………..Akarsh