A Man of Numbers

Biography

Ravi Vishwakarma

This is a biographical article on **Dattatreya Ramachandra Kaprekar** and his works. Although not widely known, the kind of work he did was not only significant but also fascinating and elementary to understand.

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Mathematics, as a subject, can mean trouble for many while, for others, it remains an absolute favourite. Irrespective of their opinions on maths though, most people are aware of the legacy Ramanujan left behind, in this world of numbers.

Let us begin with an anecdotal conversation between Ramanujan and another mathematical genius, G. H. Hardy. When G.H. Hardy was visiting Srinivasa Ramanujan at the hospital, on his way he came across 1729 on a taxi number plate. He described 1729 as a ‘dull’ number, to which Ramanujan replied that it, in fact, is very interesting, as it is the smallest number that can be expressed as the sum of two cubes in two different ways.

1729 = 1^{3} + 12^{3} and 1729 = 10^{3} + 9^{3}

This might be one of the more familiar stories; but this article, however, is focussed on another man who played with numbers. Our number of interest will be calculated by you; so be ready for some simple high school arithmetic. Select any four-digit number (I choose, say 1674). Don't take a number where all 4 digits are the same (like 1111, etc.). Once you have selected a four-digit number, arrange the numbers to find the largest number (7641 in my case) and the smallest number (1467 in my case) from these four digits. Subtract these two digits from each other, and you again obtain a number (6174). Again find the largest and smallest number and continue the same procedure. It should not take more than seven steps before you get surprised. You will end up with the same number, which is 6174. This number is also called **Kaprekar constant**.

Any four-digit number, when put through the same arithmetic treatments will beget 6174. If you try the same with a three-digit number, similarly, what you will get is the number 495. You might be intrigued to know that this interesting discovery was made by an Indian mathematician Dattatreya Ramachandra Kaprekar.

Kaprekar was born on 17 January 1905 in Dahanu, a small town near Mumbai, Maharashtra. He lost his mother early and was brought up and nurtured by his father, a government clerk, and an astrologer. Thus young Kaprekar grew up with astrology and mathematics. His fascination with numbers grew with age, and he spent most of his time with numbers, which even prompted his classmates to ridicule him often. After completing his bachelor's in 1929, he joined as a school teacher in Devlali, a town close to Nashik. His obsession with numbers persisted throughout his life. “A drunkard needs to go on drinking wine to live in that pleasurable state. The same is true of me so far as numbers are concerned.” he would say about himself. He would often lecture and motivate students about his unique methods at local colleges where he was invited for talks Most of Kaprekar's work went unrecognized during his initial living years. He spent most of his life teaching in Devlali, until he retired at the age of 58 in 1962. Soon afterwards he lost his wife and his life took a grim turn with financial instability raising its head. Thus, he spent the latter part of his life providing private tuition to local children.

Kaprekar is best known for his discovery of the Kaprekar Constant (6174) and a class of numbers that are now known as **Kaprekar numbers**. A number is a Kaprekar number if it is positive and, when squared and split into two parts, the sum of these two parts gives back the original number. An example will make it easier to understand. For example, 55 is a Kaprekar number as 55^{2} = 3025 and 30 + 25 = 55. Some Kaprekar numbers are 1, 9, 45, 55, 99, 297, etc.

Another class of numbers described by Kaprekar are called the **Harshad numbers**. A number is a Harshad number if it is divisible by the sum of their digits. For example, 12 is divisible by 1 + 2 = 3. It is noticed that 80 and 81 are a pair of consecutive Harshad numbers, whereas 110, 111, 112 are three consecutive Harshad numbers. It was further proved that a sequence of twenty-one consecutive numbers that are all Harshad numbers cannot exist. Another interesting property is that 2!, 3!, 4!, 5!, 6! ,etc. are all Harshad numbers, where ‘!’ is the symbol for factorial function. Factorial denotes nothing but the product of all the whole numbers from our chosen number after which the ‘!’ symbol is written up till one. For example, 3! would indicate the product of 3, 2 and 1. Now before we are tempted to conjecture that n! is a Harshad number, for every value of n, we need to remember that the first or the smallest factorial which is not a Harshad number is 432!. Harshad numbers were later also called *Niven numbers* named after Canadian mathematician Ivan M. Niven. The first few Harshad numbers are 10, 12, 18, 20, 21, 24, 27 and so on…

Kaprekar has multiple sets of unique numbers to his credit - Demlo numbers, self numbers, junction numbers, are a few. He has contributed to many published books on mathematics in addition to articles in journals. He came into the limelight when Martin Gardner wrote a column about Kaprekar in Mathematical Games for the Scientific American in 1975. Today, Kaprekar and his works are well known, and many mathematicians are fascinated by his ideas about numbers. His name is now written in a Swedish publication - *The World Directory of Mathematicians*, as an eminent mathematician.

References

- https://mathshistory.st-andrews.ac.uk/Biographies/Kaprekar/
- https://www.academia.edu/16470864/Unsung_Hero_of_Mathematics_D_R_Kaprekar

S Ravi Vishwakarma, an alumnus of the Department of Physical Sciences, IISER Kolkata, loves to try his hands at different stuff. He tends to be captivated by the science hidden in history and tries to protect them from getting lost in time. Now with this first article of his, he plans to keep writing!

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