Another Derivation of Euler’s Fabulous Formula

Leonhard Euler (15 April 1707–18 September 1783) was a Swiss mathematician, who is considered to be one of 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:

Let’s Design A Simple Algorithm

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 particular calculation involving these operations might be time consuming, but the rules are simple to follow. In particular, we find it quite easy to perform the operations of addition and subtraction even on big numbers. Unfortunately, we cannot say that for multiplication and division operations — they are notoriously hard to understand and execute even for relatively small numbers. …

A Practical Guide To Making A Scientific Breakthrough

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 working well for a long time, and somehow begin to fail under special conditions. A scientific progress is always made by small incremental changes. The only exception is — when we make an accidental scientific discovery, which usually opens up a Pandora’s box, often leading to significant advancements in a very short amount of time. …

A Method of Counting The Number of Solutions

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…

A 42 Minute Fantasy Ride

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 hypothetical situation. …

Another Proof Of Zero Equals One

The first experience with calculus is usually quite memorable for most of us. For some it introduces a tool of immense beauty and seemingly infinite potential to solve problems — like a super power. For others, it seems like a lifetime of suffering and madness. The reason is simple — there just too many traps and pitfalls in calculus to worry about, and even seasoned mathematicians are sometimes found scratching their heads.

In this article, we are going to see a demonstration of how things can go horribly wrong if we are not careful, while dealing with even the simplest of integrals. …

Ptolemy’s Incredible Theorem

Ptolemy was an ancient astronomer, geographer, and mathematician who lived from (c. AD 100 — c. 170). He is most famous for proposing the model of the “Ptolemaic system”, where the Earth was considered the center of the universe, and the stars revolve around it. This was the generally accepted view for many centuries, until Copernicus (19 February 1473–24 May 1543) proposed the theory where the sun was the center with earth an other planets revolving around it.

Although, Ptolemy’s most famous idea turned out to be wrong, he did a lot of really useful work in geometry and we are going to discuss one of those very nice ideas. …

A Formula For Fibonacci Sequence

Fibonacci numbers are one of the most captivating things in mathematics. They hold a special place in almost every mathematician’s heart. Throughout history, people have done a lot of research around these numbers, and as a result, quite a lot of interesting facts have been discovered.

Let us see how they look like -

The methods of finding roots of a quadratic equations are quite easy and are very well understood. There are all kinds of approaches available, like algebraic, geometrical, graphical etc. The most common approach is the algebraic method, where we use the well known quadratic formula, given by

The “Unreal” Math Of Chasing A Bus

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 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 these parameters somehow assume complex values, we would have to rethink about it. …