Motion of a moving particle is considered to be one of the most instructive and useful physical systems one can study. In a real world case, such systems may exhibit immense complexity and intractability. But if we are lucky, we may be able to isolate the moving particle from unwanted effects coupled to the system, and may consider the system of a moving particle, a simple problem in physics. The standard approach to completely solve the problem of a moving particle has been known for a very long time - the methods given to us by Isaac Newton. In modern literature, these methods employ simple ..read more

From elementary mathematics, we are familiar with the four basic mathematical operations associated with numbers - addition, subtraction, multiplication and division. There are many kinds of numbers in mathematics with their own set of unique operations. Here we are concerned about the most commonly used numbers that we deal with everyday, which are technically called "real" numbers - like $43, 14/223, 7.381, 0.001$ and so on. The use of these four operations have been taught to us at a very young age, which makes the use of such operations a trivial task for most of us. It's possible that a ..read more

Leonhard Euler (15 April 1707 – 18 September 1783) was a Swiss mathematician, who is considered to be one the greatest mathematicians of all time. He is also considered to be the most prolific mathematician ever, who wrote more than 500 books and papers during his lifetime, filling more than 90 volumes, comprising of over 25000 pages! It would be impossible for us to pick out his most important work, but one of his most celebrated results is - $$e^{i\theta} = \cos{\theta} + i \sin{\theta},$$ where $i = \sqrt{-1}$. This formula is known as the Euler's Formula. In his famous Feynman Lectures on ..read more

Albert Einstein has been one of the most celebrated scientists of all time. It is funny, that his ideas are also, among the most difficult to understand, and many people do not quite know what he did, and how he did all those things. It is the mystery of not knowing how the mind of a scientist works, that creates an aura of greatness. It takes years of study, everlasting passion and a great deal of luck, to come up with something worth celebrating. One thing that is common to all great scientific breakthroughs is that - most new ideas are almost always driven by old ideas that have been worki ..read more

One of the classic applications of equations of motion while studying Newtonian mechanics is to find the time required by a person running after an accelerating bus to catch. The algebra involved in this problem is not too hard and there is quite instructive. Such problems contain various physical parameters like distance, speed, acceleration and time. The parameters which are physically measurable are usually denoted by real numbers. It is believed that complex numbers cannot directly be associated with quantities that can be measured in the real world. If there is ever a scenario where thes ..read more

Introduction The idea of an angle is one of the most fundamental in geometry. It comes naturally while studying properties of triangles where we establish the connection between its sides and angles. These connections are systematically constructed in the subject called Trigonometry. What is an angle and how do we measure it? Although the idea of an angle is visually quite intuitive, its mathematical definition is not so straightforward. Let us try to develop a visual method to understand how we define an angle. Consider the following diagram The "spread" between the lines $OA$ and $OB$ is w ..read more

In the previous part, we obtained a mathematical constant $\mathcal{C}$ for an equilateral triangle, and its value was approximately $5.196$. In the second part, we will continue the calculations in the same manner for a 4-sided polygon, which is a square. The Sqaure A square is a closed geometrical shape having four sides having equal lengths, making an angle of $90^{\circ}$ with adjacent sides. Let us write down some basic facts that we obtained by careful observation All the angles of a square have the same value, which is $90^{\circ}$. The geometrical center of a square is at the same di ..read more

One of the best mathematical tools ever developed is the logarithm of a number. It has been used extensively for the simplification of lengthy arithmetic calculations. The standard way of using the technique is via tables of common logarithms. The method of using these tables is well known and has been in use for decades. Today we will talk about a lesser known method of finding log of a number without using the tables. This method has proven to be very useful when a log-table or an electronic calculator is not around. So, here it is... Things to memorize We choose to work with base 10 \(\lo ..read more

Linear algebraic equations are one of the simplest equations that we can solve. If there is only one variable, then the solutions are trivial to obtain, while for a system of linear equations, there are many ways to find unique solutions. In this article, we are interested in a special linear equation, with many variables. It is well known that such an equation may have an infinite number of solutions. So, we are going to put certain restrictions, and bring down the number of solutions to a great extent. The general form of the equation, that we are interested in, is given by $$x_1 + x_2 + x_ ..read more

The classic 1864 science fiction novel by Jules Verne, titled "Journey to the Center of the Earth" has mesmerized many generations. The desire to discover the secrets inside the earth is perhaps as old as the desire to discover the secrets of the stars. In reality, the mysteries of the earth are not as fantastic as envisioned by Jules Verne in his novel, and in this article we are going to take a very simple mathematical journey inside the earth, following a set of basic rules. The Fantasy Ride Our plan is to discuss a rather nice (and old) result from Newton's theory of Gravitation, using a ..read more