Much Less Than in LaTeX (\ll)

Where it is used

Much-less-than relation, saying the left quantity is smaller than the right by orders of magnitude.

Physics and engineering use \ll for approximation regimes: v \ll c for non-relativistic motion, or x \ll 1 to justify \sin x \approx x. Analytic number theory uses Vinogradov notation f \ll g, meaning f = O(g), and measure theory writes \mu \ll \nu for absolute continuity. It is a single glyph, so do not type << or < <, which give two separated relation symbols with wrong spacing. The triple version \lll exists for even stronger inequality.

Also listed under: Inequalities & Order

Minimal compilable example

Paste this into an empty .tex file and compile it with pdfLaTeX, XeLaTeX or LuaLaTeX.

\documentclass{article}
\usepackage{amsmath}
\begin{document}
Much Less Than: $\ll$

\[ v \ll c \implies \gamma = \frac{1}{\sqrt{1 - v^2/c^2}} \approx 1 \]
\end{document}

Related symbols

• Much Greater Than\gg
• Not Parallel\nparallel
• Precedes\prec
• Parallel\parallel
• Succeeds\succ
• Perpendicular\perp

Frequently asked questions

How do I write Much Less Than in LaTeX?

Type \ll in math mode, for example $v \ll c \implies \gamma = \frac{1}{\sqrt{1 - v^2/c^2}} \approx 1$. It works in plain LaTeX without loading any package.

Which package do I need for \ll?

None. \ll is part of standard LaTeX, so it compiles with nothing in the preamble but the document class.

Can I use \ll in text mode, or only in math mode?

Only in math mode. Outside $...$ LaTeX stops with "Missing $ inserted", so write $\ll$ in running text. If you only need the glyph in text, paste the Unicode character ≪ instead (with XeLaTeX or LuaLaTeX and a font that has it).