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12

On page 10 of that book the author wrote The most important example of a ground field is the field of common rational numbers for which I use the freely invented symbol 9... where he uses the symbol in question instead of the 9 that I used above.

10

Vinogradov likely adapted $\ll$ from Poincare and Borel, who used it for asymptotic series in 1890s (Cajori cites Borel, *Lecons sur les series divergentes", 1901). Physicists used it for vague "much less than" as early as 1918 (Heurlinger's doctoral dissertation Untersuchungen über die Struktur der Bandenspektra). Whether they reinterpreted Poincare's ...

9

When introducing the older terminology in the previous sentence, Peano describes it thus: ... signifie "il y a des a", "les a existent"... It seems likely this is the source of the inverted "E".

7

According to Cajori's History of Mathematical Notations, v.2, §502 (1929) the overbar, overarrow and boldface notations are older, which may be one reason why they are more common. Shaw's 1912 survey of vector notations also mentions boldface, italics, Gothics, overbar, etc., but not underbar. Judging by How to represent a tensor/matrix/vector/array in ...

6

North Korea has its own academy of science in which there is a mathematics institute. Its mathematicians are trained (mostly) in Russia (formal Soviet Union) and other Eastern European countries like Hungary, East Germany and Poland. North Korea mathematics is far worse than South Koreans in term of research, but is OK in term of application of mathematics ...

6

One would think that Russian usage stems from Kolmogorov's seminal works on probability. However, in Über die Summen durch den Zufall bestimmter unabhängiger Größen (1928) he uses $\mathfrak{M}$ to denote probability (presumably from messen, measure), not mathematical expectation. In the famous Grundbegriffe der Wahrscheinlichkeitsrechnung (1933), which gave ...

5

In 1795 lower case abbreviations were proposed for the prefixes myria, kilo, hecto, deca, deci, centi, milli: m, k, h, d, d, c, m. They were rarely used until after 1840, when the temporary mesures usuelles were replaced by the original unit names of the metric system. By then capitals were often, but not always, used for the multiples (myria, kilo, hecto, ...

5

This is not so straightforward, see Peirce, Frege, the Logic of Relations, and Church's Theorem by Dipert for a sketch of history. The notation Russell used was not created by Peano, and certainly not by Frege. Nobody used Frege's convoluted notation except his own Begriffsschrift (1879), not even Frege himself afterwards. Peano did contribute, but the basis ...

4

Lucky you! (or me :-) ). This question was answered a while back on Math.SE I found his original thoughts in the translated version of "Institutiones calculi differentialis cum eius usu in analysi finitorum ac doctrina serierum, volume 1", chapter 7. The translation is called "Foundations of Differential Calculus" and a link is found here https://...

4

Wikipedia: Languages using duodecimal number systems are uncommon. Languages in the Nigerian Middle Belt such as Janji, Gbiri-Niragu (Gure-Kahugu), Piti, and the Nimbia dialect of Gwandara and the Chepang language of Nepal are known to use duodecimal numerals. Also: It is thought that Nimbia, which is isolated from the rest of Gwandara, acquired its ...

3

Likely relevant resources are Konrad Zuse, "Der Computer ― Mein Lebenswerk" (Springer, Berlin 1984) and Raúl Rojas (ed.), "Die Rechenmachinen Konrad Zuses" (Springer, Berlin 1998). F.L. Bauer, H. Wössner, The "Plankalkül" of Konrad Zuse: A Forerunner of Today’s Programming Languages. Communications of the ACM, Vol. 15, No. 7, July 1972, pp. 678-685 ...

3

It comes from Bernstein's Habilitation dissertation Untersuchungen aus der Mengenlehre (1901, published 1905), where he also introduced the now common symbolism for cardinal arithmetic. The exponential notation is introduced in §2 as follows (my translation): "If $M$ and $N$ are two sets, we call that set which - in the sense of a known expression - ...

2

Jeff Miller's very valuable collection of the origins of mathematical expressions has the entrie "Integration around a closed path": Dan Ruttle, a reader of this page, has found a use of the integral symbol with a circle in the middle by Arnold Sommerfeld (1868-1951) in 1917 in Annalen der Physik, "Die Drudesche Dispersionstheorie vom ...

2

It is possible that $U$ stands for "utility" (in mechanics), and it is due to Hamilton and not to Rankine. Possibly, Rankine wished to be consistent with Hamilton's notation, but, in any case, I suspect hat the modern use in mechanics has more to do with following Hamilton than with following Rankine, whose musings are very abstract and ...

2

A Bourbaki "Note Historique" led me to this: Du Bois-Reymond, P. Sur la grandeur relative des infinis des fonctions. Annali di Matematica 4, 338–353 (1870), available behind a paywall, uses the notations $$f(x)\succ\phi(x),\qquad f(x)\sim\phi(x), \qquad f(x)\prec\phi(x)$$ to mean \lim\frac{f(x)}{\phi(x)}=\infty, \qquad\lim\frac{f(x)}{\phi(x)}\...

1

The earliest reference I can find to "banana brackets" is in: G. Malcolm. Data structures and program transformation. Science of Computer Programming, 14(2-3):255-280, October 1990. Where they are clearly crescent-moon/banana shaped symbols: ⦅...⦆. The later style using $($ and $|$ seems to be a typographic practicality, and is used by Malcolm ...

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