Vinogradov introduced the notation $$f(x) \ll g(x)$$ to denote that for some $C>0$, we have $|f(x)|\leqslant C\,g(x)$ for all $x$ under consideration; usually for all $x$ larger than a fixed constant (equivalently $f(x) = O(g(x))$ is used).

I've also been made aware that this is sometimes used in less formal settings to mean $A$ is "much less than" $B$ ($A\ll B$). My question is, where did this notation come from, and what was it used for first? Did Vinogradov invent it, or was it used in this informal sense prior to that?


2 Answers 2


Vinogradov likely adapted $\ll$ from Poincaré and Borel, who used it for asymptotic series in 1890s (Cajori cites Borel, *Leçons sur les séries 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 Poincaré's symbol or simply adapted what was typographically available is unclear. Here is the entry on the subject from Cajori's History of Mathematical Notations, v.II, §486:

"The sign $\ll$ was introduced by H. Poincaré and by E. Borel in comparing series like: $$u_0+u_1z+u_2z^2+\dots\ll M\left[1+\frac{z}{R^1}+\left(\frac{z}{R^1}\right)^2+\dots\right],$$ where the second series has positive coefficients; the modulus of each coefficient of the first series is less than the corresponding coefficient of the second series. The signs $\ll$ and $\gg$ are also used for "much less than" and "much greater than". Zeitschrift für Physik, Vol. XIII (1923), p. 366 (the sign was used by H. A. Kramers and W. Pauli); Torsten Heurlinger, Untersuchungen über die Struktur der Bandenspectra (Lund, 1918), p. 39".

The Kramers-Pauli paper is Zur Theorie der Bandenspektren, uses it as a matter of course without comment or explanation. Heurlinger is not cited, but, given the shared topic, they might have inherited the symbol through intermediaries.

  • $\begingroup$ So it seems that $\ll$ was used initially in the asymptotic sense $\endgroup$ Feb 25, 2020 at 15:47
  • $\begingroup$ @LukeCollins It was. One would like to speculate that it was transferred by analogizing the indeterminate to base 10, and powers to orders of magnitude (as happened in other contexts), but I am not aware of any evidence to support that here. $\endgroup$
    – Conifold
    Feb 25, 2020 at 23:39
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    $\begingroup$ The cited Borel link introduces the sign on p.142. A footnote says "Le signe $\ll$, introduit par M. Poincaré...Voir Poincaré, Les méthodes nouvelles de la Mécanique Céleste, t. I." $\endgroup$ Feb 28, 2020 at 23:35
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    $\begingroup$ See Poincaré, Les méthodes nouvelles de la Mécanique Céleste, vol 1, p. 48. $\endgroup$ Feb 28, 2020 at 23:42

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 $$\lim\frac{f(x)}{\phi(x)}=\infty, \qquad\lim\frac{f(x)}{\phi(x)}\text{ is finite }, \qquad\lim\frac{f(x)}{\phi(x)}=0.$$

A brief look at some 1880 papers by Poincare, Borel, and Stieltjes (found mentioned in Edelyi's little Asymptotic Expansions) did not show any version of the $\ll$ sign.


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