Mathematics Weblog

Some textbooks misuse infinity

Monday 18 October 2004 at 3:43 pm | In Articles | Post Comment

It’s happened again! $\infty$ used in a textbook (unnamed to protect the guilty) as if it were a real number instead of an idea. In a discussion of the formula for the acute angle $\theta$ between two lines

$\theta=\tan^{-1}\left |\dfrac{m_1-m_2}{1+m_1m_2}\right |$

the following appears:

Putting gives an angle $\tan^{-1}(\infty)=90^{\circ}$ , confirming the condition for the lines to be perpendicular

This is of course complete nonsense. As I’ve said before $\tan(90^{\circ})$ doesn’t exist and $\tan^{-1}(x)$ is only defined on $\mathbb{R}$ ie for $-\infty<x <\infty$
The textbook was written by the examiners (which is one reason why we use it); this worries me even more.
I suppose this is better than one well-known textbook back in the eighties which solved the equation t(t-3)=t^2-4 by putting $\frac{1}{m}=t$ then ‘showing’ $t=\infty$ or $t=\frac{4}{3}$ . This seems to show that all linear equations are quadratics in disguise; or cubics, quartics – who knows where this nonsense leads 😕
See also Division by zero shock!