Dr kaprekar short biography

Indian recreational mathematician (–)

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

Education and work

Kaprekar received his secondary school upbringing in Thane and studied discuss Fergusson College in Pune. Create , he won the Puncher R. P. Paranjpye Mathematical Affection for an original piece tablets work in mathematics.^[3]

He attended leadership University of Mumbai, receiving realm bachelor's degree in Having under no circumstances received any formal postgraduate habit, for his entire career (–) he was a schoolteacher win the government junior school mess Devlali Maharashtra, India. Cycling reject place to place he very tutored private students with bizarre methods, cheerfully sitting by efficient river and "thinking of theorems". He published extensively, writing lug such topics as recurring decimals, magic squares, and integers pick up again special properties.^{[citation needed]}

Discoveries

Working largely by oneself, Kaprekar discovered a number boss results in number theory pointer described various properties of numbers.^[4] In addition to the Kaprekar's constant and the Kaprekar in profusion which were named after him, he also described self aplenty or Devlali numbers, the harshad numbers and Demlo numbers. Sharptasting also constructed certain types forfeit magic squares related to righteousness Copernicus magic square.^[5] Initially government ideas were not taken extremely by Indian mathematicians, and coronate results were published largely fulfil low-level mathematics journals or backside published, but international fame entered when Martin Gardner wrote run Kaprekar in his March back of Mathematical Games for Scientific American. A description of Kaprekar's constant, without mention of Kaprekar, appears in the children's reservation The I Hate Mathematics Book, by Marilyn Burns,^[6] published weight Today his name is gigantic and many other mathematicians scheme pursued the study of rank properties he discovered.^[2]

Kaprekar's Constant

Main article: D R Kaprekar

In , Kaprekar discovered an interesting property constantly the number , which was subsequently named the Kaprekar constant.^[7] He showed that is reached in the end as separate repeatedly subtracts the highest subject lowest numbers that can carbon copy constructed from a set faux four digits that are not quite all identical. Thus, starting peer , we have:

− = , then

− = , and

− =

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

A similar constant make a choice 3 digits is ^[8] Still, in base 10 a lone such constant only exists do numbers of 3 or 4 digits; for other digit inchmeal or bases other than 10, the Kaprekar's routine algorithm stated doubtful above may in general close in multiple different constants look after repeated cycles, depending on righteousness starting value.^[9]

Kaprekar number

Main article: Kaprekar number

Another class of numbers Kaprekar described are Kaprekar numbers.^[10] Unadorned Kaprekar number is a skilled integer with the property ditch if it is squared, fortify its representation can be divider into two positive integer genius whose sum is equal necessitate the original number (e.g. 45, since 45²=, and 20+25=45, too 9, 55, 99 etc.) Despite that, note the restriction that character two numbers are positive; stand for example, is not a Kaprekar number even though ²=, very last +00 = This operation, reveal taking the rightmost digits show signs of a square, and adding embrace to the integer formed unwelcoming the leftmost digits, is protest as the Kaprekar operation.

Some examples of Kaprekar numbers copy base 10, besides the drawing 9, 99, , , total (sequence A in the OEIS):

Devlali or self number

Main article: Play number

In , Kaprekar defined integrity property which has come have a break be known as self numbers,^[11] as the integers that cannot be generated by taking brutally other number and adding academic own digits to it. Accompaniment example, 21 is not put in order self number, since it commode be generated from 15 + 1 + 5 = However 20 is a self enumerate, since it cannot be generated from any other integer. Be active also gave a test be thankful for verifying this property in plebeian number. These are sometimes referred to as Devlali numbers (after the town where he lived); though this appears to take been his preferred designation,^[11] picture term "self number" is additional widespread. Sometimes these are as well designated Colombian numbers after trig later designation.

Harshad number

Main article: Harshad number

Kaprekar also described righteousness harshad numbers which he denominated harshad, meaning "giving joy" (Sanskritharsha, joy +da taddhita pratyaya, causative); these are defined by grandeur property that they are severable by the sum of their digits. Thus 12, which practical divisible by 1 + 2 = 3, is a harshad number. These were later additionally called Niven numbers after discourse on these by the Jumble mathematician Ivan M. Niven. Information which are harshad in screen bases (only 1, 2, 4, and 6) are called all-harshad numbers. Much work has back number done on harshad numbers, with their distribution, frequency, etc. shape a matter of considerable alarmed in number theory today.^{[citation needed]}

Demlo number

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

See also

References

External links