If 10^-3 m is 1 mm, then 10^-6 m is 2 mm, isn’t it?

This is a common question that students have but are ashamed to ask.

As per your logic, $1mm={10}^{-3}m$. 

Multiplying both sides by 2 to maintain equality we get

$2\times 1mm=2\times {10}^{-3}m$ and not ${10}^{-6}mm$.

Please recall the property of exponents which is as follows....

$a^m\times a^n=a^{\left(m+n\right)}$ therefore, $a^{-m}\times a^{-n}=a^{\left(-m-n\right)}$

Applying this property in our example, 

${10}^{-6}={10}^{-3}\times {10}^{-3}$=${10}^{\left(-3-3\right)}={10}^{-6}$

So,

$\begin{array}{l}{10}^{-6}m={10}^{-6}\times 1m\\ ={10}^{-6}\times {10}^{+3}mm\left[::1m=1000mm\right]\end{array}$

${10}^{-6}m={10}^{\left(-6+3\right)}mm$

$\begin{array}{l}{10}^{-6}m={10}^{-3}mm\\ =\frac{1}{{10}^3}mm=\frac{1}{1000}mm\end{array}$