# Meaning of $\geqslant$, $\leqslant$, $\eqslantgtr$, $\eqslantless$

What do slanted inequality signs mean? Specifically, these are $\geqslant$, $\leqslant$; and the variation: $\eqslantgtr$, $\eqslantless$.

Is there any place I can look this up? I've searched Wikipedia and the web and can't find anything about them. The last two were found when looking up the first two.

$\geqslant$ is an alternative to $\geq$ and means the same: $a \geqslant b$ = "$a$ is greater than or equal to $b$".

Likewise, $\leqslant$ is an alternative notation for $\leq$, with $a \leqslant b$ = "$a$ is less than or equal to $b$".

I haven't encountered $\eqslantgtr$ or $\eqslantless$, but given that the former is formatted eqslantgtr and the latter eqslantless, I would venture to guess that they likewise denote "greater than or equal to" and "less than or equal to", respectively. Perhaps with these symbols, where the emphasis appears to be on the "equals" component, they are read as "equal to or greater than" and "equal to or less than", respectively.

• Thanks. The context in which I saw $\geqslant$ and $\leqslant$ did make it look like they meant what their unslanted counterparts do, but I couldn't tell for sure.
– anon
Nov 30 '12 at 1:36

In Russia $\geqslant$ and $\leqslant$ are used instead of $\geq$ and $\leq$.
We don't use $\geq$ and $\leq$ at all.
But I've never seen $\eqslantgtr$ and $\eqslantless$ in Russian math texts.

P.S.
1. There are more interesting distinctions between Russian and English math notations.
For example, we occasionally using $\ n\vdots d\$ instead of $\ d|n$.
2. There are a lot of strange math symbols exists: see here and here.

• in Vietnam we use ⩾ and ⋮ too, so it confuses me occasionally when seeing d | n. I think it's due to the influence of France. ≥ is only seen when typing Jul 30 '18 at 2:50

I haven't seen $\eqslantless$ or $\eqslantgtr$ used. A Google search for "eqslantless" turned up http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=tmf&paperid=3775&option_lang=eng but the actual paper (both the Russian original and the English translation) used $\leqslant$.

• I have: here. In particular the definitions of minimizing and maximizing orders. Of course I was forced by sheer shortage of still available symbols... ;-)
– WimC
Aug 28 '14 at 17:19

In France too, we use $\geqslant$ and $\leqslant$, at least in high school teaching.

The $\geq$ and $\leq$ signs are understandable though, and used by pocket calculators.

• In Vietnam ⩾ and ⩽ are also used. In fact most of the Vietnamese mathematics and physics are heavily influenced by France (like U for voltage, tg for tangent, sin for waveforms) and Russia (as old Communist allies). In the first half of the 20th century, most students have to learn French, and in the next quarter century they have to learn Russian. However recently the new textbooks changed to the more US-centric style (tg → tan, sin → cos for wave function) Jul 30 '18 at 3:01

Since people agreed with $\leq$ vs $\leqslant$ is a style issue. I checked a list of books from various disciplines,

the "$\leq$" group:

R. Shankar, Principle of Quantum Mechanics.

O.C. Zienkiewicz, et al. The Finite Element Method.

J. Munkres, Topology.

S. Boyd, et al. Convex Optimization.

S.M. Carroll, Spacetime and Geometry

R.W. Hamming, Numerical Methods for Scientists and Engineers

T. Sauer, Numerical Analysis.

R.A. Horn, et al. Matrix Analysis

G.H. Golub, et al. Matrix Computations

W. Fulton, et al. Representation Theory

T.W. Hungerford, Algebra

N. Jacobson, Basic Algebra

I. Goodfellow, Deep Learning

the "$\leqslant$" group:

D.V. Hutton, Fundamentals of Finite Element Analysis.

S.J. Farlow, Partial Differential Equations for Scientistes and Engineers

R.L. Graham, Concrete Mathematics

C.M. Bishop, Pattern Recognition and Machine Learning

It seems that most gurus preferred "$\leq$".

• it depends on the gurus' language. It seems the slanted version is preferred in European countries and their colonies Jul 30 '18 at 2:53
• I encountered the slanted version in The C Programming Language second edition by B. Kernighan and D. Ritchie, which led me to research and end up here. Aug 10 '18 at 20:46

As other answers have remarked, $$\eqslantless$$ and $$\eqslantgtr$$ are rarely seen, and the difference between $$\leqslant$$ and $$\leq$$, and similarly between $$\geqslant$$ and $$\geq$$, is just a matter of style. I would like to add that $$\leqslant$$ and $$\geqslant$$ were the preferred standard of many publishers in the UK and the US until recent decades when the direct setting of authors' LaTeX files took over in books and journals. In LaTeX, the default characters are $$\leq$$ and $$\geq$$, the slanted versions requiring an extra five keystrokes each for the author to type.

Personally, I prefer the aesthetics of the slanted version, to the extent that I am willing to type the extra letters.