I am trying to compile a list of mind-boggling/curious historical facts in mathematics that will inspire and attract young people (9–11 years old) to the discipline of Mathematics. Do you have one fact to share? Multidisciplinary historical facts are welcome!
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asked Jul 8 at 18:18
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$\begingroup$ Sokrates explaining to a serf how to double a square, The greeks finding that the diagonal of a sqaure can not divided in small parts of the side . $\endgroup$– trulaJul 13 at 16:50
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$\begingroup$ The entire life of Evariste Galois. That and the Italians that discovered the cubic and quartic formula in the 1500s. $\endgroup$– ruferdJul 14 at 19:07
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$\begingroup$ Perhaps, also you can include the fact that the Four Color Theorem was the first proof aided by computers. This might get them interested in how computers can help us prove things and also get them to question how we trust the computer proof (if the humans that wrote it were flawed then wouldn't the proof be flawed too? So how do we trust the results of technology?) $\endgroup$– ruferdJul 14 at 19:10
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3 Answers
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I recall being inspired, at about the age of 12, by the widely-known anecdote of how the mathematical talent of the famous mathematician Carl Friedrich Gauss was first recognized at the age of nine. His teacher put his students to work on summing the first one hundred integers, and young Gauss came up with the answer almost immediately: 5050.
Our math teacher provided some historical context: Gauss had been born into a working-class family in Brunswick in Northern Germany in 1777. He was the only child of his father's second marriage. His father worked as a bricklayer and gardener, and while he could read, write, and perform simple arithmetic, Gauss's mother was functionally illiterate. From the age of seven Gauss attended a public school: A one-room school house with a schoolmaster in charge of one hundred students. Corporal punishment was common. Students used slate tablets as the use of paper was too expensive (the use of slate tablets remained common in German elementary schools until after World War II).
Our teacher did not tell us right away how Gauss had figured out the result almost instantaneously; instead he challenged us to ponder how he might have done it. I spent an entire afternoon trying to figure it out, and I seem to recall I worked out a formula based on averaging. The math teacher revealed Gauss's (supposed) technique the next day: One can form pairs $100+1$, $99+2$, etc and in this way generate fifty pairs each summing to $101$, thus the final sum is $50 \times 101 = 5050$.
It should be noted that the specifics of this computation seem to be a bit of a modern invention. The earliest biography of Gauss does include this anecdote and claims Gauss himself as the source, but it just mentions that Gauss's teacher tasked his students with summing an unspecified arithmetic series during instruction in arithmetic. It further states that it was the custom at that school that when finished with a task students would drop off their slate tablet at the teacher's desk, one on top of the other. Gauss turned in his slate almost immediately with the words "There it lies" and then sat quietly until the end of class, when the teacher flipped the stack of slate tables around, inspected the top-most slate and to his astonishment found that Gauss had written down the correct answer and nothing else:
Wolfgang Sartorius von Waltershausen, Gauss zum Gedächtniss. Leipzig: S. Hirzel 1856 (online)
A recent publication has tried to trace how details were added to this story over time as it was retold numerous times:
Brian Hayes, "Gauss's Day of Reckoning." American Scientist, Vol. 94, No. 3, May 2006, pp. 200-205 (online)
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1$\begingroup$ Hayes also has a list of "Versions of the Gauss Schoolroom Anecdote" here: bit-player.org/wp-content/extras/gaussfiles/gauss-snippets.html (last updated 2018-06-25 with 145 items) $\endgroup$– user18187Aug 15 at 2:01
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$\begingroup$ (According to Hayes, the earliest version of the story with the arithmetic series being 1-to-100 was by Mathé in 1906.) $\endgroup$– user18187Aug 15 at 2:34
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The mathematician Hardy told the following story when he went to visit the mathematician Ramanujan: "I remember once going to see him when he was ill at Putney. I had ridden in taxi cab number 1729 and remarked that the number seemed to me rather a dull one, and that I hoped it was not an unfavourable omen. "No," he replied, "it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways." (Indeed: $1729 = 10^3+9^3 = 12^3+1^3$)
It's amazing that Ramanujan was able to come up with this instantly.
I am adding a second story. In 1939, George Bernard Dantzig arrived late for a graduate-level statistics class. There were two problems written on the board. Because he was late he didn't know they were examples of "unsolved" statistics problems. Instead he took them for part of a homework assignment, copied them down, and solved them. He used these solutions as his PhD thesis. I
It's said that if he had known they were unsolvable he wouldn't have tried to solve them.
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5$\begingroup$ I saw this variant on the (now hidden) MO joke thread; "I went to visit him while he was lying ill at the hospital. I had come in taxi cab number 14 and remarked that it was a rather dull number. 'No' he replied, 'it is a very interesting number. It's the smallest number expressible as the product of 7 and 2 in two different ways.' $\endgroup$ Jul 10 at 12:54
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Factoring numbers is usually taught at around that age, and students are taught about prime and composite numbers and how to find all the divisors of a number. It's not too much additional work to discuss perfect numbers and amicable numbers and to demonstrate why 220 and 284 are an amicable pair.
The curious historical fact is that after centuries of brilliant mathematicians exploring number theory, it wasn't until 1867 that a sixteen year old boy in Italy, Nicolò I. Paganini, discovered the second smallest pair, that 1184 and 1210 are amicable.
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3$\begingroup$ I have doubts that this is a good topic to "inspire and attract young people [...] to the discipline of Mathematics". It might (or might not) be a nice story for those who are already attracted to mathematics - for most others it will probably reinforce the notion that maths were a somewhat esoteric and pointless endeavour where people discuss strange questions about numbers. $\endgroup$ Jul 11 at 11:14
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2$\begingroup$ A relative gave me a copy of a Martin Gardner book a long time ago when I was in fourth grade. That's when I learned about Nicolò I. Paganini in Gardner's chapter on "Perfect, Amicable, and Sociable." I thought aliquot sequences were beautiful back then and still think so today. I think it's sad that many who are in mathematics as a career in the 21st century seem to have lost the sense of simple beauty that led to the ancients labeling some whole numbers as being perfect even though I recognize that the topic has no use in applied science and seems esoteric at the moment. $\endgroup$ Jul 11 at 21:09
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1$\begingroup$ Thanks for your reply! I'm a bit under the impression that you are reading a few things into my comment that I actually have not said. Please note I did not draw any connections between (non-)usability in applied sciences and being esoteric (in fact much of my own research in math is very abstract). I did not say either that I find perfect and amicable numbers non-beautiful or esoteric or pointless. Instead, I said that people who are not already attracted to mathematics are likely to find them esoteric and pointless. My concern is not that those people will consider things without [...] $\endgroup$ Jul 13 at 14:52
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1$\begingroup$ applications as necessarily esoteric. Rather, my concern is that they will find perfect and amicable numbers esoteric because (a) to most people those concepts will not appear particularly natural to come up with at first glance (e.g., why do we consider the sum of all divsisors? Why not, say, the sum of all prime divisors only?) and (b) when brought up anyway, those concepts do not provide particularly clear or far reaching insight (e.g., we don't even know today if infinitely many perfect numbers exist). Compare this, for instance, to the sum of the first n positive [...] $\endgroup$ Jul 13 at 14:57
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1$\begingroup$ integers (mentioned in another answer). To me this is a wonderful example of, as you call it, "simple beauty": it is (i) a very natural question to ask; it has an answer that is (ii) non-obvious but (iii) intriguingly clear once you see it; and (iv) it can teach a lot about how "finding the right structure" can help to simplify a seemingly tedious task. I think when we try to "inspire and attract young people" (in particular those who are not intrisically attracted), (i)-(iv) are extremely valuable. On the other hand, I don't see any of these four points when discussing, say, perfect numbers. $\endgroup$ Jul 13 at 14:59