{"id":44,"date":"2016-07-02T07:59:10","date_gmt":"2016-07-02T07:59:10","guid":{"rendered":"http:\/\/pi2e.ch\/blog\/?p=44"},"modified":"2024-02-17T09:04:11","modified_gmt":"2024-02-17T09:04:11","slug":"god-loves-the-odd-number","status":"publish","type":"post","link":"https:\/\/pi2e.ch\/blog\/2016\/07\/02\/god-loves-the-odd-number\/","title":{"rendered":"God Loves the Odd Number"},"content":{"rendered":"

In the late 17th century, Gottfried Wilhelm Leibniz<\/a> discovered a beautiful series expansion of \u03c0\/4 which only involves odd numbers. The identity can be derived from the derivative of the Arctangent \\( \\huge \\frac{d}{dx}\\arctan(x) = (1+x^2)^{-1}\\). Based on this we have
\n$$\\Large \\begin{align} \\frac{\\pi}{4} = \\arctan(1) &= \\int_0^1 \\frac{1}{1+x^2}\u00a0 \\\\
\n&=\\int_0^1 1-x^2+x^4-x^6+\\ldots\\\\
\n&=1-\\frac{1}{3}+\\frac{1}{5}-\\frac{1}{7}+\\ldots\\end{align}$$
\nwhere we have used the formula for the sum of a geometric series. Leibniz comments this result with a citation from Virgil<\/a>:<\/p>\n

Numero DEUS impare gaudet (God loves the odd number).<\/p><\/blockquote>\n

Leibniz, although living more than 300 years ago, made important contributions to computer science. He invented a mechanical calculator<\/a> and worked on the binary numeral systems. He used numbers, especially the digits 0 and 1, to illustrate <\/a>the Christian concept of a creation ex nihilo:<\/p>\n

Wie denn dergestalt die Zahlen gleichsam als in einem Spiegel die Sch\u00f6pfung oder den ursprung der dinge aus Gott und sonst nichts darstellen, und also dienen solches denen heyden unbekande geheimniss aus dem Licht der vernunft einiger massen zu ercl\u00e4ren und anzudeuten, wie unrecht die heidnische philosophi, die materi als einen mitursprung, Gott gleichsam an die Seite gesezet.<\/p><\/blockquote>\n

With this sentence, Leibniz touches on the still debated question, whether a person (“God”) or matter and energy is the ultimate reality. He argues <\/a>for the former using the principle of sufficient reason:<\/p>\n

Maintenant il faut s\u2019\u00e9lever \u00e0 la m\u00e9taphysique, en nous servant du grand principe, peu employ\u00e9 commun\u00e9ment, qui porte que rien ne se fait sans raison suffisante, c\u2019est-\u00e0-dire que rien n\u2019arrive sans qu\u2019il soit possible \u00e0 celui qui conna\u00eetrait assez les choses de rendre une raison qui suffise pour d\u00e9terminer pourquoi il en est ainsi, et non pas autrement. Ce principe pos\u00e9, la premi\u00e8re question qu\u2019on a droit de faire sera\u00a0: pourquoi il y a plut\u00f4t quelque chose que rien\u00a0?<\/p><\/blockquote>\n

In spite of contrary claims, there is no convincing solution, which shows how something can come into being out of nothing. For many proposed solutions, “nothing” turns out not to be literally nothing, but for example a quantum vacuum as in the case of Lawrence Krauss. In another prominent case, the Hartle-Hawking cosmology, the universe turns out to be an eternally existing de Sitter space<\/a>. A de Sitter space is a space-time with a constant positive curvature. This leads to the unusual fact, that the ratio of a circle’s circumference to its diameter is smaller than \u03c0.<\/p>\n

This finally brings us back to \u03c0. Of course, you could use the Leibniz series to compute \u03c0 numerically, but the convergence is really slow. Let’s give it a try:<\/p>\n