6.1 Smart calculators are too smart

1 Introduction 2 Mathematics is not only about computing 3 Basic concepts 4 Elementary functions 5 Analytical geometry 6 Warning 6.1 Smart calculators are too smart 6.2 Frequently asked questions 7 List of the used symbols Index Bibliography

6.1.1 How come that $\ln(-1)$ or $\sin(\ii)$ are evaluated and do not return an error?

Virtually all elementary functions can be extended almost to the entire set of complex numbers. And indeed, $\ln(-1) = \ii\pi$ and $\sin(\ii) = \ii \sinh(1)$. There is not enough time to study the calculus of complex numbers in BIE-ZMA. However, we will at least mention how to define $e^z$ for any complex number $z$.

Mathematica, for example, works implicitly in „complex mode“. This may be very confusing for an uneducated user.

6.1.2 How come that $\sqrt[3]{-1}$ is evaluated as $\frac{1}{2} + \frac{\sqrt{3}}{2}\ii$ and not as $-1$?

If you are curious you can easily verify that this answer is not wrong:

\begin{equation*} \begin{aligned} \Bigg( \frac{1}{2} + \frac{\sqrt{3}}{2} \ii \Bigg)^{\!\!3} &= \Bigg( \underbrace{\frac{1}{4} + \frac{\sqrt{3}}{2}\ii - \frac{3}{4}}_{-\frac{1}{2} + \frac{\sqrt{3}}{2}\ii} \Bigg) \cdot \Bigg( \frac{1}{2} + \frac{\sqrt{3}}{2} \ii \Bigg) = \\ &= -\frac{3}{4} - \frac{1}{4} = -1. \end{aligned} \end{equation*}

The „problem“ is that in complex numbers there are in total three solutions to the equation

\begin{equation*} z^3 = -1, \quad z\in\CC. \end{equation*}

The one solution that we got is what we call principal solution – the solution with the least „argument“. We repeat again the calculus of complex numbers is not part of the BIE-ZMA course.

6.1.3 Equality in CAS Mathematica

In CAS Mathematica there are several symbols for equality with the following meaning:

Symbol == denotes logical equality (comparison, writing equations).
Symbol = denotes assignment.
Symbol := denotes „delayed evaluation“.

We will demonstrate the various meanings using the example below. The output of this piece of code

a = 4;
b = a;
Print[b]
a = 2;
Print[b]

4
4

On the other hand, the cell containing

a = 4;
b := a;
Print[b]
a = 2;
Print[b]

results in the output

4
2

section

Chapter