Friday, March 02, 2012

The Cube Root: A New Short Cut?

In the first week February (2012), Nirbhay Singh Nahar, an Agra-based retired chemical engineer , who is also an amateur mathematician, claimed that he has figured out a formula to work out cube roots of any number using a simple calculator within a minute and a half by just addition and subtraction. Nahar has yet to disclose the algorithm that the scientific community needs to approve and incorporate. The development if proved is interesting, although not a path-breaking discovery.  However, for the student community, especially those planning to take competitive examinations it could prove to be a wonderful short-cut even using a calculator.
In a sense, Nahar has started to follow in the footsteps of the great Indian Mathematicians who made seminal discoveries that aided the development of architecture and engineering in a big way. Cubes have always fascinated mathematicians. Our beloved number theorist Ramanajuan, while admitted to hospital, on hearing that his mentor Hardy had arrived in a taxi with the number 1729, instantly replied that it was the smallest number that could be expressed as the sum of two distinct cubes (13+123=1729=93+103).
Deriving the cube root was an interesting cerebral challenge among all ancient mathematical traditions. Although the Chinese, Indian, Greek and Arabic mathematicians formulated several algorithms to find the cube root of any given number, an easy algorithm is still elusive. Chapters 35 and 36 of Vedic Mathematics (Motilal Banarsidas Publications) are dedicated only for methods to find cube roots. The Crest of the Peacock by George Gheverghese Joseph codifies all the ancient works towards this. Wikipedia provides a very simple method (refer article Cube Root) to compute cube roots using a non-scientific calculator, using only the multiplication and square root buttons.
Our world is a multi-dimensional one with most objects where, either animate or inanimate, possess a length, breadth and height/depth. Measurement each of these dimensions enables one to calculate the space or volume it occupies. What happens if the length, breadth and height/depth of some objects are similar? Then that object is called a cube. Historically, solids of similar kind are of great interest to man. The cube is also known as regular hexahedron, only because it has six equal faces. Considering ancient Greek philosopher Plato had extensively discussed cubes among others which are known as Platonic solids.
Therefore, depending on the total volume of the objects, architects and engineers seek to optimize the usage of space. Therefore, finding the cubic root proves relevant. Moreover with the availability of advanced computers it will not be a mind boggling exercise. But imagine a student getting a question in a test to solve the equation n3-k=0, where k is any given number, integer or non-integer. Owing to limited time in an examination environment, the best algorithm needs to be used to solve the problem.
Finding the cube root, without the help of a machine, of perfect cubes such as 8, 27, 64, etc itself is a complex problem in mathematics. Now, finding the cube roots of other integers which are not perfect cubes such as 2, 3, 4, 5, 6, etc becomes a harder problem. This is not the end. What about the non-integers? Suppose the volume of an object is something like 2.53, how does not figure out its length?
In day-to-day life one often gets these types of the numbers only. For instance, if a farmer wants to store his grains so that not a small space of his store house is to be wasted. Because a cube has the largest volume among all the cuboids, finding the cube root will solve the problem. In cities where space is limited and costly, cubical structures are the best to store things. Hopefully, Nirbhay Singh Nahar’s discovery should simplify life for those who choose to confront mathematics on a day-to-day basis, either as students tackling examinations or hi-tech engineers involved in design and development works.
(This was published as Equate and Apply in Deccan Herald on March 1, 2012)