SputSoft

Visualizing the Pythagorean Theorem

sput — Sun, 14 Feb 2010 12:13:18 +0000

Most people are familiar with the Pythagorean theorem: In a right-angled triangle the square of the hypotenuse is equal to the sum of the squares of the other two sides. As the name of the theorem implies, it is attributed to Pythagoras, a Greek mathematician who lived around 500 B.C. The theorem is also included [...]

Fractions and Circles

sput — Sat, 06 Feb 2010 10:45:35 +0000

Fractions produced by mediants have some very interesting properties. We saw some of them in connection with the Stern-Brocot tree. This articles explores a more curious property, relating fractions to circles in the plane. It was discovered in 1938 by Lester R. Ford and is also mentioned in Conway and Guy’s The Book of Numbers. Let [...]

Book Review: Prime Obsession

sput — Sat, 09 Jan 2010 19:15:23 +0000

Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics is a book about the Riemann Hypothesis, posed by Bernhard Riemann in 1859. As the book title says, it is one of the greatest unsettled mathematical conjectures remaining today. It is among David Hilbert’s list of twenty-three mathematical problems and one of the seven [...]

The Stern-Brocot Tree of Fractions

sput — Fri, 04 Dec 2009 20:21:54 +0000

Consider two fractions \frac{m_1}{n_1} and \frac{m_2}{n_2} with positive numerators and denominators. The fraction \frac{m_1+m_2}{n_1+n_2} is called the mediant of \frac{m_1}{n_1} and \frac{m_2}{n_2}. It is straightforward to show that the mediant is placed numerically between the original fractions, (1) \frac{m_1}{n_1} < \frac{m_2}{n_2} \quad \Rightarrow \quad \frac{m_1}{n_1} < \frac{m_1+m_2}{n_1+n_2} < \frac{m_2}{n_2}. Consider now the following simple procedure. Start with the [...]

Book review: The Pleasures of Counting

sput — Sun, 15 Nov 2009 13:48:16 +0000

The Pleasures of Counting is a book about people working with mathematics and challenges they have faced. The book has 544 pages with a total of 19 chapters and 3 appendices. It contains a lot of material and is split into five parts: The uses of abstraction, Meditations on measurement, The pleasures of computation, Enigma [...]

Continued Fractions and Continuants

sput — Tue, 10 Nov 2009 15:27:42 +0000

We will be considering continued fractions of the form a_0 + \displaystyle\frac{1}{a_1 + \displaystyle\frac{1}{\ddots + \displaystyle\frac{1}{a_{n-1} + \displaystyle\frac{1}{a_n}}}} where the a_k’s are real numbers called the partial quotients. Continued fractions can be greatly generalized, where both the “numerators” (here all equal to one) and the partial quotients can be more general mathematical objects. Most common, however, are [...]

Computing the Greatest Common Divisor

sput — Thu, 29 Oct 2009 17:18:49 +0000

The greatest common divisor of two integers is the largest positive integer that divides them both. This article considers two algorithms for computing \hbox{gcd}(u,v), the greatest common divisor of u and v. Some key properties of \hbox{gcd} are: \hbox{gcd}(u,0) = |u|. \hbox{gcd}(u,v) = \hbox{gcd}(-u,v). \hbox{gcd}(u,v) = \hbox{gcd}(v,u). \hbox{gcd}(u,v) = \hbox{gcd}(u,v+n u) for any integer n. \hbox{gcd}(u,v) = d \cdot \hbox{gcd}(u/d,v/d) if [...]

Remembering Trigonometric Addition Formulas

sput — Wed, 23 Sep 2009 13:13:57 +0000

The addition formulas for sine and cosine look like this: \begin{aligned} \cos(\alpha + \beta) &= \cos \alpha \cos \beta – \sin \alpha \sin \beta, \\ \sin(\alpha + \beta) &= \sin \alpha \cos \beta + \cos \alpha \sin \beta. \\ \end{aligned} I can never remember them. One solution is of course to look them up in a book or search the internet. [...]

Release: SputArithmetic 0.1

sput — Sat, 19 Sep 2009 16:48:35 +0000

This is the first release of the SputArithmetic library. You can download the library or read about it. From the project page: SputArithmetic is a portable library written in C++ for doing multiple-precision arithmetic. The underlying theory and design as well as release announcements can be found among the multi-precision articles. Note that the library was created mainly to support/test [...]

Useful Properties of the Floor and Ceil Functions

sput — Wed, 09 Sep 2009 13:36:13 +0000

This articles explores some basic properties of the integer functions commonly known as floor and ceil. Most of the statements may seem trivial or obvious, but I, for one, have a tendency to forget just how exact you can be when it comes to expressions/equations where floor or ceil functions appear. First, the definitions: \begin{aligned} \lfloor x \rfloor [...]