How to switch easily between normal font and sans-serif font for maths?

The sansmath package provides a command (\sansmath) and environment (sansmath) for exactly this purpose:

\documentclass{article}
\usepackage[T1]{fontenc}
\usepackage{sansmath}

\def\mySfFamily{\fontfamily{cmbr}\selectfont\sffamily\sansmath}
\def\N{\mathbf{N}}
\def\R{\mathbf{R}}
\def\Q{\mathbf{Q}}
\def\Z{\mathbf{Z}}
\def\C{\mathbf{C}}


\begin{document}

\begingroup
\mySfFamily
This should be sans-serif : $\R, \Q, \N, \Z, \C$ are sets.
\endgroup

\bigskip

This should be serif : $\R, \Q, \N, \Z, \C$ are sets.

\end{document}

To apply this "automagically", I would hack it into the \sffamily and \rmmfamily commands. (I prefer \appto from the etoolbox package for this, but one could as well use the usual \let\OLDmacro\macro and \renewcommand\macro{\OLDmacro <extension>} idiom.):

\documentclass{article}
\usepackage[T1]{fontenc}
\usepackage{sansmath,etoolbox}

\renewcommand{\sfdefault}{cmbr}

\appto{\sffamily}{\sansmath}
\appto{\rmfamily}{\unsansmath}


\def\N{\mathbf{N}}
\def\R{\mathbf{R}}
\def\Q{\mathbf{Q}}
\def\Z{\mathbf{Z}}
\def\C{\mathbf{C}}


\begin{document}


\sffamily

This should be sans-serif : $\R, \Q, \N, \Z, \C$ are sets.

\rmfamily

This should be serif : $\R, \Q, \N, \Z, \C$ are sets.


\end{document}

Some time ago I asked nearly the same question, the answer was to define a new math alphabet and hack it into the definition of \sffamily. So I’d suggest to close this question as duplicate but it can’ be done until the bounty is closed.

---

This is the code a build upon the answer of egreg.

\documentclass[parskip=half]{scrartcl}

\usepackage[T1]{fontenc}
\usepackage{lmodern}

\DeclareMathVersion{sans}
\SetSymbolFont{operators}{sans}{OT1}{cmbr}{m}{n}
\SetSymbolFont{letters}{sans}{OML}{cmbrm}{m}{it}
\SetSymbolFont{symbols}{sans}{OMS}{cmbrs}{m}{n}
\SetMathAlphabet{\mathit}{sans}{OT1}{cmbr}{m}{sl}
\SetMathAlphabet{\mathbf}{sans}{OT1}{cmbr}{bx}{n}
\SetMathAlphabet{\mathtt}{sans}{OT1}{cmtl}{m}{n}
\SetSymbolFont{largesymbols}{sans}{OMX}{iwona}{m}{n}

\DeclareMathVersion{boldsans}
\SetSymbolFont{operators}{boldsans}{OT1}{cmbr}{b}{n}
\SetSymbolFont{letters}{boldsans}{OML}{cmbrm}{b}{it}
\SetSymbolFont{symbols}{boldsans}{OMS}{cmbrs}{b}{n}
\SetMathAlphabet{\mathit}{boldsans}{OT1}{cmbr}{b}{sl}
\SetMathAlphabet{\mathbf}{boldsans}{OT1}{cmbr}{bx}{n}
\SetMathAlphabet{\mathtt}{boldsans}{OT1}{cmtl}{b}{n}
\SetSymbolFont{largesymbols}{boldsans}{OMX}{iwona}{bx}{n}

\newif\IfInSansMode
\let\oldsf\sffamily
\renewcommand*{\sffamily}{\oldsf\mathversion{sans}\InSansModetrue}
\let\oldmd\mdseries
\renewcommand*{\mdseries}{\oldmd\IfInSansMode\mathversion{sans}\fi\relax}
\let\oldbf\bfseries
\renewcommand*{\bfseries}{\oldbf\IfInSansMode\mathversion{boldsans}\else%
   \mathversion{bold}\fi\relax}
\let\oldnorm\normalfont
\renewcommand*{\normalfont}{\oldnorm\InSansModefalse\mathversion{normal}}
\let\oldrm\rmfamily
\renewcommand*{\rmfamily}{\oldrm\InSansModefalse\mathversion{normal}}

\usepackage[textwidth=9cm]{geometry}

\begin{document}
\section{Inline-$m\alpha \tau h$}
Normal: A $Ax\alpha+\chi b$

\sffamily Sans: A $Ax\alpha+\chi b$

\bfseries Bold-Sans: A $ Ax\alpha+\chi b$

\normalfont Normal: A $ Ax\alpha+\chi b$

\bfseries  Bold: A $ Ax\alpha+\chi b$

\normalfont Normal: A $ Ax\alpha+\chi b$

\section{Display-$\mu\alpha\tau\eta$}
Text in roman family
\begin{equation}
     \sqrt{(a_1+a_2+\gamma)^2}=\sum_{i=1}^2 a_i + \gamma
\end{equation}

\sffamily
Text in sans family
\begin{equation}
    \sqrt{(a_1+a_2+\gamma)^2}=\sum_{i=1}^2 a_i + \gamma
\end{equation}

\sffamily\bfseries
Text in sans family, bold version
\begin{equation}
    \sqrt{(a_1+a_2+\gamma)^2}=\sum_{i=1}^2 a_i + \gamma
\end{equation}

\rmfamily\bfseries
Text in roman family, bold version
\begin{equation}
    \sqrt{(a_1+a_2+\gamma)^2}=\sum_{i=1}^2 a_i + \gamma
\end{equation}
\end{document}

For comparison, this is how one can switch from sans to serif math and back in ConTeXt.

\def\N{{\bf N}}
\def\R{{\bf R}}
\def\Q{{\bf Q}}
\def\Z{{\bf Z}}
\def\C{{\bf C}}

\startbuffer
Some random text: $\R, \Q, \N, \Z, \C$ are sets.
\stopbuffer

% To format the examples
\setuphead[subject][style=bold, after=]

\starttext

\subject{Default} \getbuffer

\subject{Sans math}

\switchtobodyfont[ss] 
\appendtoks \ss \to \everymathematics
\getbuffer

\subject{Serif math}
\switchtobodyfont[rm]
\appendtoks \rm \to \everymathematics
\getbuffer


\stoptext

You can also use group {...} to limit the effect of font changes.