Equate and apply

Understanding Math

The cube and its algorithms have fascinated mathematicians for centuries. Joseph Varghese attempts to explain the relevance of the cube root in creating practical solutions for everyday problems

In the first week of February, Nirbhay Singh Nahar, an Agra-based retired chemical engineer and amateur mathematician, announced that he had 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 path-breaking. 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 some sense, Nahar is following 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 Ramanujam, while admitted in hospital, came upon a theory that claimed that the number 1729 was the smallest number that could be expressed as the sum of two distinct cubes (13+123=1729=93+103).

Deriving the cube root has been an interesting cerebral challenge among all ancient mathematical traditions. Although the Chinese, Indian, Greek and Arabic mathematicians have 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 theories created towards this. Wikipedia provides a very simple method (refer to 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, either animate or inanimate, possessing a length, breadth and height/depth. Measurement of each of these dimensions help 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 have been of great interest to man. The cube is also known as a regular hexahedron, only because it has six equal faces. The ancient Greek philosopher, Plato had extensively discussed cubes among others which are known as Platonic solids.

Depending on the total volume of the objects, architects and engineers seek to optimise 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. But this is not the end. What about non-integers? Suppose the volume of an object is 2.53, how do we figure out its length?

In day-to-day life one often comes across these types of numbers. For instance, if a farmer wants to store his grains so that not even a small space of his store house is wasted, it can be solved by finding the cube root.

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.

(The writer is Assistant Professor at the Department of Mathematics, Christ University, Bangalore)