Math Musings

Enigma In 1918 , German engineer Arthur Scherbius applied for a patent for a machine that coded and decoded secret messages quickly and easily. Scherbius’s  Enigma  shifted letters similarly to a Caesar cipher, but Enigma constantly changed the order in which the letters were shifted. Such pattern changes mask the nature of the cipher by appearing random. Without knowledge of the algorithms generating the cipher patterns, code breaking methods require enormous amounts of computation. With around   possible configurations, the Enigma seemed a cryptographically safe bet. The Enigma worked quickly and accurately. While the Enigma had only modest commercial success, it became widely adopted by the German military. From the mid- until the end of World War II, Germany and its allies encrypted and decrypted important communications using various versions of the Enigma. Conquering Enigma During World War II, many of the world’s best problem solvers considered cracking Enigm

A prime number is a number that is divisible only by itself and 1. Some examples being  2, 3, 5, 7, 9, 11, 13.…   The great mathematician Euclid proved centuries ago that there are an infinite number of them continuing without pattern ad infinitum. However, their real importance is given by what is known as the Fundamental Theorem of Arithmetic. It states that every number can be uniquely expressed as a product of prime numbers. Let’s take the example of 364 = 2 × 2 × 7 × 13. This is the only way that 364 can be obtained by prime numbers! Therefore, they are the atoms or building blocks of numbers. But in 1859, the great German mathematician Bernhard Riemann hypothesized that the spacing of the primes logically follows from other numbers, now known as the “nontrivial zeros” of the Riemann zeta function. The Riemann zeta function takes inputs not only real numbers but also  complex numbers — meaning they have both “real” and “imaginary” components — and yields other numbers

Young Achiever: Gurugram teen wants to promote beauty of Math and its impact on everyday life Rohan Jha is the inaugural winner of Steven H Strogatz Prize for Math Communication conducted by National Museum of Mathematics in New York For Rohan Jha (16), the passion for  Math  was fuelled by a short video that he made at his architect mother’s request. It was the summer following his IXth grade at the Pathways School  Gurgaon , when Rohan made the short video on the Birch Swinnerton Dyer (BSD) Conjecture, a difficult millennium problem in Math, that changed his life’s course. His father Pankaj Jha, a startup owner in India who had worked at the Wall Street for over 20 years, sent the video to reputed mathematicians hoping one of them might respond with encouragement. Michael Harris, professor at Columbia university, wrote back saying that he would be delighted to have Rohan as a student when the time comes, while Benedict Gross, professor at Harvard University, sent him hi

The embarrassing thought about BSD conjecture is that it poses a question just beyond what we have learnt in high school, yet we are unable to prove it! It belongs to the area of mathematics known as number theory where we prefer to find solutions to equations in rational numbers (ratio of integers). In school, we learn to find solutions to equation in one variable by using the Rational Root theorem. We also learn how to solve equations of two variables in degree 1 . But what about equations in two variables of degree 2 ? Consider the equation x² +y² =1 This is known as the unit circle. The point (-1, 0) lies on the circle and consider the point P(x, y) Using all the rational slopes and finding the second point of intersection we are able to find all the rational points. However,   x² +y² =3 has no rational solutions! (For fun try proving that!) But what about cubic equations in two variables ? As it turns out we have n

The Millennium Prize Problems are seven problems in mathematics that were stated by the Clay Mathematics Institute in 2000. As of June 2020, six of the problems remain unsolved. A correct solution to any of the problems results in a US $1,000,000 prize sometimes called a Millennium Prize being awarded by the institute.  Solved Problem: -  Poincare conjecture. When the Clay institute first announced these seven problems, it was generally perceived by the mathematics community that these seven problems would remain unsolved for the foreseeable future. To their surprise, Grigori Perelman a recluse Russian mathematician solved the Poincare conjecture and in July 2010 he famously declined the 1 million dollar prize!   Furthermore, in August 2006, he declined the Fields medal which is considered even more prestigious than the Nobel Prize as it is given every four years to a mathematician under the age of 40. The Nobel Prize is given every year with no age limit. In a seri

Everybody will tell you that math is about numbers! Counting numbers 1, 2, 3 ………………….. Whole numbers 0, 1, 2, 3………………. Integers ……….. -3,-2,-1, 0, 1, 2, 3………………. Rational numbers 1/2, 1/3, -7/8 Real numbers …. √2 =  1.4142…… Complex numbers..... x + y √ -1 Where x and y are real numbers Each subsequent number system includes the previous number system. So far example integers are included in rational numbers and rational numbers are included in real numbers and so on. Each enlargement helps us to solve more difficult equations and something we could not solve earlier! x + 2 = 5 (needs negative numbers) 4x = 1 (needs rational numbers) x 4 = 7    (needs irrational numbers) x 4 = - 7  (needs complex numbers) With each of these enlargements we are able to solve harder problems but the math becomes more abstract and less intuitive For example, we understand when we have 5 apples but what does it mean to have   √-5    apples?