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Newsletter July-August-2019 Volume 7, Issue 1 : Patron : Prin Sir P. T. Sarvajanik College of Science, Surat Department of Mathematics Mthematics Newsletter July-August-2019 Volume 7, Issue 1 : Patron : Prin. Dr. Pruthul R. Desai : Editorial Board : Dr. K. J. Chauhan, Dr. J. M. Desai, Ms. K. J. Contractor, Ms. P. D. Dave This newsletter is meant for private circulation only D. R. Kaprekar Early Life Kaprekar received his secondary Born: 17 January 1905 in Dahanu, school education in Thane and Maharashtra, India studied at Fergusson College in Died: 1986 (aged 81) in Devlali, Pune. In 1927 he won the Wrangler Maharashtra, India 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 Mthematics Newsletter Page 1 special properties. He is also known Kaprekar constant as "Ganitanand" (गणितानंद) In 1949, Kaprekar discovered an Mathematical Work interesting property of the number 6174, which was subsequently Working largely alone, Kaprekar named the Kaprekar constant. He discovered a number of results in showed that 6174 is reached in the number theory and described limit as one repeatedly subtracts the various properties of numbers. In highest and lowest numbers that addition to the Kaprekar constant can be constructed from a set of four and the Kaprekar numbers which digits that are not all identical. were named after him, he also described self- numbers or Devlali Thus, starting with 1234, we have numbers, the Harshad numbers and 4321 − 1234 = 3087, then Demlo numbers. He also constructed certain types of magic 8730 − 0378 = 8352, and squares related to the Copernicus 8532 − 2358 = 6174. magic square. Initially his ideas were not taken seriously by Indian Repeating from this point onward mathematicians, and his results leaves the same number (7641 − were published largely in low-level 1467 = 6174). mathematics journals or privately In general, when the operation published, but international fame converges it does so in at most seven arrived when Martin Gardner wrote iterations. about Kaprekar in his March 1975 column of Mathematical Games for A similar constant for 3 digits is 495. Scientific American. Today his name However, in base 10 a single such is well-known and many other constant only exists for numbers of mathematicians have pursued the 3 or 4 digits; for more digits (or 2), study of the properties he the numbers enter into one of discovered. several cycles. Mthematics Newsletter Page 2 Kaprekar number 27282728² = 7441984744 + 1984 = 2728 52925292² = 2800526428 + 005264 = 5292 Another class of numbers Kaprekar described are the Kaprekar 857143857143² = 734694122449734694 numbers. A Kaprekar number is a + 122449 = 857143 positive integer with the property Devlali or Self number that if it is squared, then its representation can be partitioned In 1963, Kaprekar defined the into two positive integer parts property which has come to be known as self- numbers, which are whose sum is equal to the original integers that cannot be generated by number. taking some other number and (e.g. 45, since 452 = 2025, and 20 + adding its own digits to it. For 25 = 45 , also 9, 55, 99 etc.) example, 21 is not a self- number, However, note the restriction that since it can be generated from the two numbers are positive; for 15: 15 + 1 + 5 = 21. But 20 is a self- number, since it cannot be example, 100 is not a Kaprekar generated from any other integer. number even though 1002 = He also gave a test for verifying this 10000, and 100 + 00 = 100. This property in any number. These are operation, of taking the rightmost sometimes referred to as Devlali digits of a square, and adding it to numbers (after the town where he the integer formed by the leftmost lived); though this appears to have digits, is known as the Kaprekar been his preferred designation, the operation. term self-number is more widespread. Sometimes these are Some examples of Kaprekar also designated Colombian numbers numbers in base 10, besides the after a later designation. numbers 9, 99, 999,…, are (sequence A006886 in OEIS): Harshad number Kaprekar also described the Number Square Decomposition Harshad numbers which he named 703703² = 494209494 + 209 = 703 harshad, meaning "giving joy" Mthematics Newsletter Page 3 (Sanskrit harsha, joy +da taddhita light, nor did Einstein need Statistics pratyaya, causative); these are for the theory defined by the property that they are divisible by the sum of their of relativity. Thermodynamics and digits. Thus 12, which is divisible by quantum mechanics are 1 + 2 = 3, is a Harshad number. fundamentally statistical, but lots of These were later also called Niven progress could have been made in numbers after a 1977 lecture on these areas without Statistics. The these by the Canadian second law of thermodynamics is an mathematician Ivan M. Niven. observable fact; ditto the two-slit Numbers which are Harshad in all experiment and various bases (only 1, 2, 4, and 6) are called experimental results revealing the all-Harshad numbers. Much work nature of the atom. has been done on Harshad numbers, What about the A-bomb and, almost and their distribution, frequency, certainly, the H-bomb? Maybe these etc. are a matter of considerable would never have been invented interest in number theory today.∎ without Statistics – but, on balance, A world without Statistics I think most people would feel that the world would be a better place -Andrew Gelman without these particular developments. Not so long ago, I was asked by a reporter for a quote regarding the At a more applied level, Statistics importance of Statistics. This got me helped to win the Second World War thinking: Is Statistics really that – most notably in cracking the important? Would a world without Enigma code, but also in various Statistics be much different from the operational research efforts. And it one we have now? is my impression that “our” Statistics were better than “their” What would be missing? Science Statistics. So that is something. would be pretty much OK. Newton did not need Statistics for his But where would civilian theories of gravity, motion, and technology be without Statistics? I Mthematics Newsletter Page 4 am not sure. Without Statistics we But, on balance, I doubt these would would not have modern quality be huge mistakes, and the big ones control, so maybe we would still be would eventually get caught with driving around in AMC Gremlins and careful record-keeping – even the like. That is a scary thought, but without statistical inference and not a huge deal. adjustments. In a world without Statistics, would Of course, without Statistics, the study of quantum physics have biologists would not be able to progressed far enough so that sequence the gene, and I assume transistors were invented? That they would be much slower at would make a difference in my life. developing tools such as tests that No transistors mean no blogging – allow you to check for chromosomal and I guess we could forget about abnormalities in amnio. other unequivocally beneficial technological innovations such as I doubt all these things add up to modern pacemakers, hearing aids, much yet, but I guess there is cochlear implants, and Microsoft’s promise for the future. Clippy. Statistics is also necessary for a lot Modern biomedicine uses lots of of drug development – right now my Statistics, but would medicine be so colleagues and I are working on a much worse without it? I do not pharmacodynamics model of dosing think so – at least not yet. You do not – but, again, without any of this, it is need Statistics to see that penicillin not clear the world would be so works, nor to see that mosquitoes much different. transmit disease and that nets keep Take another one: polling. You mosquitoes out. cannot do that well without Statistics. But would a world Without Statistics, I assume that without polling be so horrible? I various mistakes would get into the think polling is generally a good system – ineffective treatments that thing – I agree with George Gallup people think are effective. that measurement of public opinion Mthematics Newsletter Page 5 is an important part of the modern experimentation and inference to democratic process – but I would get p-values for tabloid-bait not want to hang too much of the scientific papers; its use by Google benefits of Statistics on this one use. and Amazon to perfect their techniques for squeezing money out A deeper understanding of their customers; or, at best, to test Perhaps the most important a medical treatment that increases contribution of Statistics comes not survival rate for some rare disease from the direct use of statistical by two percentage points. methods in science and technology, But Statistics is central to how we but rather in helping us learn about think about the world. I still think the world. Statisticians from Francis that it is much Bayesian inference, Galton and Ronald Fisher onwards and the rest: here I am simply have used Statistics to give us a talking about the nature of much deeper understanding of correlation and variation. human and biological variation. I For a humbler example, consider the cannot see how any non-statistical, baseball historian and statistician mechanistic model of the world Bill James. Baseball is a silly could reproduce that level of example, to be sure, but the point is understanding. Forget about p- values, So here is one strong benefit to see how much understanding has to the formal study of Statistics: been gained in this area through without Statistics, there would still Statistical measurement and be numbers, along with people comparison.
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