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 m_1m_2=-1 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!

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