Math Goodies

This page is where I stash a few favorite (or fun) equations for easy reference.

Starting with this mathematical limerick:

\(\int_{1}^{\sqrt[3]{3}} {z}^{2} {dz} \times \cos \left( \frac{3 \times \pi}{9} \right) = \ln \sqrt[3]{e}\)

Translation:

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    Integral z-squared dz,
    From one to the cube root of three.
        Times the cosine,
        Of three pi over nine,
    Equals log of the cube root of e.

Best of all, the math works:

\(\displaystyle\cos\left(\frac{\pi}{3}\right)\!\times\!\left[\frac{1}{3}{z}^{3}\right]_{1}^{\sqrt[3]{3}}\!\!=\!{0.5}\!\times\!\left(\!\!\frac{\left(\sqrt[3]{3}\right)^{3}}{3}\!-\!\!\frac{{1}^{3}}{3}\!\right)\!={0.5}\!\times\!\left(\!\frac{3}{3}\!-\!\frac{1}{3}\!\right)\!=\frac{1}{3}\)

And “log of the cube root of e” is also 1/3.

A personal favorite is Euler’s Identity:

\(\displaystyle{e}^{{i}\pi}-1=0\)

And Euler’s Equation:

\(\displaystyle\eta\,{e}^{{i}\pi\theta}=\eta\,\left(\cos\theta,\;{i}\sin\theta\right)\)

In general, I love the many faces of the complex numbers:

\(\displaystyle{z}=({a}+{bi})=[{x},\,{y}]=({r},\,\theta)={r}(\cos\theta+{i}\sin\theta)={re}^{{i}\pi\theta}=\begin{bmatrix}{a}&{-b}\\{b}&{a}\end{bmatrix}\)

They can even be expressed as matrices!

continuing…

Logos con Carne

Math Goodies