{"id":143,"date":"2013-02-13T20:26:34","date_gmt":"2013-02-13T20:26:34","guid":{"rendered":"http:\/\/florian-roemer.de\/blog\/?p=143"},"modified":"2013-02-13T20:31:42","modified_gmt":"2013-02-13T20:31:42","slug":"a-funny-number","status":"publish","type":"post","link":"https:\/\/florian-roemer.de\/blog\/a-funny-number\/","title":{"rendered":"A funny number"},"content":{"rendered":"

I was giving tutorials for an undergraduate course in Signals and Systems theory. Since students came from quite varying backgrounds I had to start very simple – my first tutorial used to be just on complex numbers. This left some of the students bored, saying they’ve seen it all. To keep them thinking I usually asked them to compute a complex number for me: Let $j$ be the imaginary unit, what is the value of $j^j$?<\/p>\n

This question is not as trivial as it may seem at first sight. It requires a generalization of the exponentiation $a^b$ from $a, b \\in \\mathbb{R}$ to $a, b \\in \\mathbb{C}$. There are rigorous ways of doing this which I do not want to discuss in detail here. Let us just assume that we found a generalization of $a^b$ to complex numbers which satisfies the laws of powers, in particular the law $a^b = {\\rm e}^{b \\cdot \\ln(a)}$ where $\\ln(x)$ is the natural (base-e) logarithm. Then, we can rewrite $j^j$ as $j^j = {\\rm e}^{j \\cdot \\ln(j)}$. What is $\\ln(j)$ though? Well, $j$ can be written as ${\\rm e}^{j \\cdot \\pi\/2}$, so $\\ln(j)$ should be $j \\cdot \\pi\/2$, right? This finally brings us to $j^j = {\\rm e}^{j \\cdot j \\cdot \\pi\/2} = {\\rm e}^{-\\pi\/2}$.<\/p>\n

Some of my students would actually obtain this result. They would usually be surprised to get a real (and sort of strange number) out of such an operation, but always they would be very proud to have the answer. The tech-savvy ones would even check their result <\/a>with whatever internet-able device they carried and be extra sure to have it right.<\/p>\n

However, my (admittedly a bit discouraging) reply would be that the answer is, despite being correct, incomplete. Actually infinitely incomplete. There are (infinitely) more “correct” answers. How so? Well, $j$ can be written as ${\\rm e}^{j \\cdot \\pi\/2}$ but it can also be written as ${\\rm e}^{j \\cdot (\\pi\/2 + 2\\pi)}$ or ${\\rm e}^{j \\cdot (\\pi\/2 – 2\\pi)}$, and so on. We can add any integer multiple of $2\\pi$ to the phase due to the periodicity of complex numbers with respect to their phase.<\/p>\n

For the (natural) logarithm of a complex number this means that it is in general ambiguous: $\\ln(j) = j \\cdot \\pi\/2 + 2\\cdot k \\cdot \\pi$ for any $k \\in \\mathbb{Z}$. To avoid the confusion it is common to write ${\\rm Ln}(x) = \\ln(x) + 2\\cdot k \\cdot \\pi$ where $\\ln(x)$ is the (unique) principle value of the logarithm which we obtain by choosing the principle phase of $x$.<\/p>\n

In that sense, the full answer would be $j^j = {\\rm e}^{-\\pi\/2} \\cdot {\\rm e}^{-2 \\cdot k \\cdot \\pi}$, where $k=0$ corresponds to the principle value of $j^j$ given by ${\\rm e}^{-\\pi\/2}$.<\/p>\n

Infinitely many solutions, all of them are real, and all of them connect ${\\rm e}$ and $\\pi$. Pretty cool, right?<\/p>\n","protected":false},"excerpt":{"rendered":"

I was giving tutorials for an undergraduate course in Signals and Systems theory. Since students came from quite varying backgrounds I had to start very simple – my first tutorial used to be just on complex numbers. This left some of the students bored, saying they’ve seen it all. To keep them thinking I usually […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":[],"categories":[10],"tags":[],"_links":{"self":[{"href":"https:\/\/florian-roemer.de\/blog\/wp-json\/wp\/v2\/posts\/143"}],"collection":[{"href":"https:\/\/florian-roemer.de\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/florian-roemer.de\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/florian-roemer.de\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/florian-roemer.de\/blog\/wp-json\/wp\/v2\/comments?post=143"}],"version-history":[{"count":12,"href":"https:\/\/florian-roemer.de\/blog\/wp-json\/wp\/v2\/posts\/143\/revisions"}],"predecessor-version":[{"id":155,"href":"https:\/\/florian-roemer.de\/blog\/wp-json\/wp\/v2\/posts\/143\/revisions\/155"}],"wp:attachment":[{"href":"https:\/\/florian-roemer.de\/blog\/wp-json\/wp\/v2\/media?parent=143"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/florian-roemer.de\/blog\/wp-json\/wp\/v2\/categories?post=143"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/florian-roemer.de\/blog\/wp-json\/wp\/v2\/tags?post=143"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}