D. r. kaprekar birth and death

Indian recreational mathematician (–)

Dattatreya Ramchandra Kaprekar (Marathi: दत्तात्रेय रामचंद्र कापरेकर; 17 January &#;&#; ) was an Indian recreational mathematician who described several classes of natural numbers including the Kaprekar, harshad and self numbers and discovered the Kaprekar's constant, named after him.^[1] Despite having no formal postgraduate training and working as a schoolteacher, he published extensively and became well known in recreational mathematics circles.^[2]

Education and work

Kaprekar received his secondary school education in Thane and studied at Fergusson College in Pune.

In , he won the Wrangler R. P. Paranjpye Mathematical Prize for an original piece of work in mathematics.^[3]

He attended the University of Mumbai, receiving his bachelor's degree in Having never received any formal postgraduate training, for his entire career (–) he was a schoolteacher at the government junior school in Devlali Maharashtra, India.

Cycling from place to place he also tutored private students with unconventional methods, cheerfully sitting by a river and "thinking of theorems". He published extensively, writing about such topics as recurring decimals, magic squares, and integers with special properties.^{[citation needed]}

Discoveries

Working largely alone, Kaprekar discovered a number of results in number theory and described various properties of numbers.^[4] In addition to the Kaprekar's constant and the Kaprekar numbers which were named after him, he also described self numbers or Devlali numbers, the harshad numbers and Demlo numbers.

He also constructed certain types of magic squares related to the Copernicus magic square.^[5] 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 column of Mathematical Games for Scientific American.

A description of Kaprekar's constant, without mention of Kaprekar, appears in the children's book The I Hate Mathematics Book, by Marilyn Burns,^[6] published in Today his name is well-known and many other mathematicians have pursued the study of the properties he discovered.^[2]

Kaprekar's Constant

Main article: D R Kaprekar

In , Kaprekar discovered an interesting property of the number , which was subsequently named the Kaprekar constant.^[7] He showed that is reached in the end 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 , we have:

− = , then

− = , and

− =

Repeating from this point onward leaves the same number ( − = ). In general, when the operation converges it does so in at most seven iterations.

Dattatreya ramachandra kaprekar biography books That is. If I take the number in the first step, then rearranging in descending order gives back Reversing them gives me the number If by remainder, you mean , I get I rearrange and reverse to get back , and the new remainder is , a dead-end.

A similar constant for 3 digits is ^[8] However, in base 10 a single such constant only exists for numbers of 3 or 4 digits; for other digit lengths or bases other than 10, the Kaprekar's routine algorithm described above may in general terminate in multiple different constants or repeated cycles, depending on the starting value.^[9]

Kaprekar number

Main article: Kaprekar number

Another class of numbers Kaprekar described are Kaprekar numbers.^[10] 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=, and 20+25=45, also 9, 55, 99 etc.) However, note the restriction that the two numbers are positive; for example, is not a Kaprekar number even though ²=, and +00 = 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, , , are (sequence A in the OEIS):

Number	Square	Decomposition
	² =	+ =
	² =	+ =

Devlali or self number

Main article: Self number

In , Kaprekar defined the property which has come to be known as self numbers,^[11] as the 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 + 1 + 5 = 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,^[11] the term "self number" is more widespread.

Sometimes these are also designated Colombian numbers after a later designation.

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D.r. kaprekar contribution

Harshad number

Main article: Harshad number

Kaprekar also described the harshad numbers which he named harshad, meaning "giving joy" (Sanskritharsha, 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 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.

Dattatreya ramachandra kaprekar biography books youtube View one larger picture. He was brought up by his father after his mother died when he was eight years old. His father was a clerk who was fascinated by astrology. Although astrology requires no deep mathematics, it does require a considerable ability to calculate with numbers, and Kaprekar's father certainly gave his son a love of calculating. Kaprekar attended secondary school in Thane sometime written Thana , which is northeast of Mumbai but so close that it is essentially a suburb.

are a matter of considerable interest in number theory today.^{[citation needed]}

Demlo number

Kaprekar also studied the Demlo numbers,^[12] name of which was derived from the name of a train station Demlo (now called Dombivili) 30 miles from Bombay on the then G. I. P. Railway where he had the idea of studying them.^[2] The best known of these are the Wonderful Demlo numbers 1, , , , , which are the squares of the repunits 1, 11, ,, ^[13]

See also

References

External links