"Chapter 2 1/2": Custom Chapter Number

Question

I'm copying to LaTeX my notes on Maths subject. My professor is quite weird, and "invented" a Chapter 2 1/2 (two and a half) between chapter 2 and chapter 3. For example:

• Chapter 2 Vector spaces and linear equation systems
• Chapter 2 1/2 Orthogonalization and least squares
• Chapter 3 Diagonalization

I want to keep the chapter numbering, labeling it as 2 1/2 and, if possible, keeping sections like "2 1/2.1", "2 1/2.2", etc. I know it's weird and I've searched a lot. The only thing I've found is typing

\chapter*{Chapter 2 \nicefrac{1}{2} \vspace{10pt} Orthogonalization and least squares}

But that doesn't appear in ToC and doesn't keep section numbering. How can I solve this? Any ideas are apreciated.

PS: sorry for my terrible English, I'm Spanish!

EDIT: I've tried two different ways from Paulo Cereda's answer (Setting arbitrary chapter values), whose MWE are:

\documentclass[11pt]{book}
\usepackage[spanish]{babel}
\usepackage[utf8]{inputenc}
\usepackage{units}
\newcommand{\strangechapter}[1]{\renewcommand{\thechapter}{#1}\chapter}

\begin{document}
\tableofcontents
\chapter{Cálculo diferencial en R}
\chapter{Espacios vectoriales y sistemas de ecuaciones lineales}

\strangechapter{2 \nicefrac{1}{2}}{Ortogonalización y mínimos cuadrados}
\section{Producto escalar}
\setcounter{chapter}{2}
\renewcommand{\thechapter}{\arabic{chapter}}
\chapter{Matrices diagonalizables}

\end{document}

I've also used the same, but changing it a little bit:

\documentclass[11pt]{book}
\usepackage[spanish]{babel}
\usepackage[utf8]{inputenc}
\usepackage{units}
\newcommand{\strangechapter}[1]{\renewcommand{\thechapter}{#1}\chapter}

\begin{document}
\tableofcontents
\chapter{Cálculo diferencial en R}
\chapter{Espacios vectoriales y sistemas de ecuaciones lineales}
\clearpage% to flush out last chapter page with correct header and footer
\renewcommand{\thechapter}{2 \nicefrac{1}{2}}
\chapter{Ortogonalización y mínimos cuadrados}
\section{Producto escalar}

\setcounter{chapter}{2}
\renewcommand{\thechapter}{\arabic{chapter}}
\chapter{Matrices diagonalizables}
\end{document}

I've added \setcounter to keep the following chapters with correct numbering.

Both of them work great in the text, but the Table of Contents doesn't show well, as the '1/2' is printed over the chapter of section name. How can I solve this?