D. R. Kaprekar
Encyclopedia
Dattaraya Ramchandra Kaprekar (1905 – 1986) was an India
n mathematician
who discovered several results in number theory
, including a class of numbers
and a constant named after him. Despite having no formal postgraduate training and working as a schoolteacher, he published extensively and became well-known in recreational mathematics
circles.
and studied at Fergusson College
in Pune
. In 1927 he won the Wrangler R. P. Paranjpe Mathematical Prize for an original piece of work in mathematics.
He attended the University of Mumbai
, receiving his bachelor's degree in 1929. Having never received any formal postgraduate training, for his entire career (1930–1962) he was a schoolteacher at Nashik in Maharashtra
, India. He published extensively, writing about such topics as recurring decimals, magic squares, and integers with special properties.
s which were named after him, he also described self number
s or Devlali numbers, the Harshad number
s and Demlo numbers. He also constructed certain types of magic squares related to the Copernicus magic square. Initially his ideas were not taken seriously by Indian mathematicians, and his results were published largely in low-level mathematics journals or privately published, but international fame arrived when Martin Gardner
wrote about Kaprekar in his March 1975 column of Mathematical Games for Scientific American
. Today his name is well-known and many other mathematicians have pursued the study of the properties he discovered.
Repeating from this point onward leaves the same number (7641 − 1467 = 6174). In general, when the operation converges it does so in at most seven iterations.
A similar constant for 3 digits is 495. However, in base 10 a single such constant only exists for numbers of 3 or 4 digits; for more digits (or 2), the numbers enter into one of several cycles.
Some examples of Kaprekar numbers in base 10, besides the numbers 9, 99, 999, …, are :
s which he named harshad, meaning "giving joy" (Sanskrit
harsha, joy +da taddhita pratyaya, causative
); these are defined by the property that they are divisible by the sum of their digits. Thus 12, which is divisible by 1 + 2 = 3, is a Harshad number. These were later also called Niven numbers after a 1997 lecture on these by the Canadian mathematician Ivan M. Niven
. Numbers which are Harshad in all bases (only 1, 2, 4, and 6) are called all-Harshad numbers. Much work has been done on Harshad numbers, and their distribution, frequency, etc. are a matter of considerable interest in number theory today.
s 1, 11, 111, ….
India
India , officially the Republic of India , is a country in South Asia. It is the seventh-largest country by geographical area, the second-most populous country with over 1.2 billion people, and the most populous democracy in the world...
n mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
who discovered several results in number theory
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
, including a class of numbers
Kaprekar number
In mathematics, a Kaprekar number for a given base is a non-negative integer, the representation of whose square in that base can be split into two parts that add up to the original number again. For instance, 45 is a Kaprekar number, because 45² = 2025 and 20+25 = 45. The Kaprekar numbers are...
and a constant named after him. Despite having no formal postgraduate training and working as a schoolteacher, he published extensively and became well-known in recreational mathematics
Recreational mathematics
Recreational mathematics is an umbrella term, referring to mathematical puzzles and mathematical games.Not all problems in this field require a knowledge of advanced mathematics, and thus, recreational mathematics often attracts the curiosity of non-mathematicians, and inspires their further study...
circles.
Biography
Kaprekar received his secondary school education in ThaneThane
Thane , is a city in Maharashtra, India, part of the Mumbai Metropolitan Region, northeastern suburb of Mumbai at the head of the Thane Creek. It is the administrative headquarters of Thane district. On 16 April 1853, G.I.P...
and studied at Fergusson College
Fergusson College
Fergusson College is a degree college in western India, situated in the city of Pune. It was founded in 1885 by the Deccan Education Society and at that time was the first privately governed college in India. It is named after Sir James Fergusson, the Governor of Bombay, who donated a then...
in Pune
Pune
Pune , is the eighth largest metropolis in India, the second largest in the state of Maharashtra after Mumbai, and the largest city in the Western Ghats. Once the centre of power of the Maratha Empire, it is situated 560 metres above sea level on the Deccan plateau at the confluence of the Mula ...
. In 1927 he won the Wrangler R. P. Paranjpe Mathematical Prize for an original piece of work in mathematics.
He attended the University of Mumbai
University of Mumbai
The University of Mumbai , is a state university located in Mumbai, Maharashtra, India. It was known as the University of Bombay until 1996 when the city of Bombay was renamed as Mumbai. The affiliated colleges of the university are spread throughout the city of Mumbai and four coastal districts in...
, receiving his bachelor's degree in 1929. Having never received any formal postgraduate training, for his entire career (1930–1962) he was a schoolteacher at Nashik in Maharashtra
Maharashtra
Maharashtra is a state located in India. It is the second most populous after Uttar Pradesh and third largest state by area in India...
, India. He published extensively, writing about such topics as recurring decimals, magic squares, and integers with special properties.
Discoveries
Working largely alone, Kaprekar discovered a number of results in number theory and described various properties of numbers. In addition to the Kaprekar constant and the Kaprekar numberKaprekar number
In mathematics, a Kaprekar number for a given base is a non-negative integer, the representation of whose square in that base can be split into two parts that add up to the original number again. For instance, 45 is a Kaprekar number, because 45² = 2025 and 20+25 = 45. The Kaprekar numbers are...
s which were named after him, he also described self number
Self number
A self number, Colombian number or Devlali number is an integer which, in a given base, cannot be generated by any other integer added to the sum of that other integer's digits. For example, 21 is not a self number, since it can be generated by the sum of 15 and the digits comprising 15, that is,...
s or Devlali numbers, the Harshad number
Harshad number
A Harshad number, or Niven number in a given number base, is an integer that is divisible by the sum of its digits when written in that base. Harshad numbers were defined by D. R. Kaprekar, a mathematician from India. The word "Harshad" comes from the Sanskrit + , meaning joy-giver. The Niven...
s and Demlo numbers. He also constructed certain types of magic squares related to the Copernicus magic square. Initially his ideas were not taken seriously by Indian mathematicians, and his results were published largely in low-level mathematics journals or privately published, but international fame arrived when Martin Gardner
Martin Gardner
Martin Gardner was an American mathematics and science writer specializing in recreational mathematics, but with interests encompassing micromagic, stage magic, literature , philosophy, scientific skepticism, and religion...
wrote about Kaprekar in his March 1975 column of Mathematical Games for Scientific American
Scientific American
Scientific American is a popular science magazine. It is notable for its long history of presenting science monthly to an educated but not necessarily scientific public, through its careful attention to the clarity of its text as well as the quality of its specially commissioned color graphics...
. Today his name is well-known and many other mathematicians have pursued the study of the properties he discovered.
Kaprekar constant
Kaprekar discovered the Kaprekar constant or 6174 in 1949. He showed that 6174 is reached in the limit as one repeatedly subtracts the highest and lowest numbers that can be constructed from a set of four digits that are not all identical. Thus, starting with 1234, we have- 4321 − 1234 = 3087, then
- 8730 − 0378 = 8352, and
- 8532 − 2358 = 6174.
Repeating from this point onward leaves the same number (7641 − 1467 = 6174). In general, when the operation converges it does so in at most seven iterations.
A similar constant for 3 digits is 495. However, in base 10 a single such constant only exists for numbers of 3 or 4 digits; for more digits (or 2), the numbers enter into one of several cycles.
Kaprekar number
Another class of numbers Kaprekar described are the Kaprekar numbers. A Kaprekar number is a positive integer with the property that if it is squared, then its representation can be partitioned into two positive integer parts whose sum is equal to the original number (e.g. 45, since 452=2025, and 20+25=45, also 9, 55, 99 etc.) However, note the restriction that the two numbers are positive; for example, 100 is not a Kaprekar number even though 1002=10000, and 100+00 = 100. This operation, of taking the rightmost digits of a square, and adding it to the integer formed by the leftmost digits, is known as the Kaprekar operation.Some examples of Kaprekar numbers in base 10, besides the numbers 9, 99, 999, …, are :
| Number | Square | Decomposition |
|---|---|---|
| 703 | 703² = 494209 | 494+209 = 703 |
| 2728 | 2728² = 7441984 | 744+1984 = 2728 |
| 5292 | 5292² = 28005264 | 28+005264 = 5292 |
| 857143 | 857143² = 734694122449 | 734694+122449 = 857143 |
Devlali or Self number
In 1963, Kaprekar defined the property which has come to be known as self numbers, which are integers that cannot be generated by taking some other number and adding its own digits to it. For example, 21 is not a self number, since it can be generated from 15: 15 + 1 + 5 = 21. But 20 is a self number, since it cannot be generated from any other integer. He also gave a test for verifying this property in any number. These are sometimes referred to as Devlali numbers (after the town where he lived); though this appears to have been his preferred designation, the term self number is more widespread. Sometimes these are also designated Colombian numbers after a later designation.Harshad number
Kaprekar also described the Harshad numberHarshad number
A Harshad number, or Niven number in a given number base, is an integer that is divisible by the sum of its digits when written in that base. Harshad numbers were defined by D. R. Kaprekar, a mathematician from India. The word "Harshad" comes from the Sanskrit + , meaning joy-giver. The Niven...
s which he named harshad, meaning "giving joy" (Sanskrit
Sanskrit
Sanskrit , is a historical Indo-Aryan language and the primary liturgical language of Hinduism, Jainism and Buddhism.Buddhism: besides Pali, see Buddhist Hybrid Sanskrit Today, it is listed as one of the 22 scheduled languages of India and is an official language of the state of Uttarakhand...
harsha, joy +da taddhita pratyaya, causative
Causative
In linguistics, a causative is a form that indicates that a subject causes someone or something else to do or be something, or causes a change in state of a non-volitional event....
); these are defined by the property that they are divisible by the sum of their digits. Thus 12, which is divisible by 1 + 2 = 3, is a Harshad number. These were later also called Niven numbers after a 1997 lecture on these by the Canadian mathematician Ivan M. Niven
Ivan M. Niven
Ivan Morton Niven was a Canadian-American mathematician, specializing in number theory. He was born in Vancouver. He did his undergraduate studies at the University of British Columbia and was awarded his doctorate in 1938 from the University of Chicago. He was a member of the University of Oregon...
. Numbers which are Harshad in all bases (only 1, 2, 4, and 6) are called all-Harshad numbers. Much work has been done on Harshad numbers, and their distribution, frequency, etc. are a matter of considerable interest in number theory today.
Demlo number
Kaprekar also studied the Demlo numbers, named after a train station where he had the idea of studying them. These are the numbers 1, 121, 12321, …, which are the squares of the repunitRepunit
In recreational mathematics, a repunit is a number like 11, 111, or 1111 that contains only the digit 1. The term stands for repeated unit and was coined in 1966 by Albert H. Beiler...
s 1, 11, 111, ….