Sputsoft

C++ Templates and Usual Arithmetic Conversions

sput — Sat, 28 Aug 2010 09:56:11 +0000

If you add a short int and a char in C++, what is the resulting type? What if you subtract a long int from an unsigned int? The answers actually depend on the compiler and the target architecture (int or unsigned in the first case and long int or unsigned long int in the second). [...]

Bitwise Operators and Negative Numbers

sput — Sat, 24 Jul 2010 17:55:05 +0000

When representing integers using a fixed number of bits, negative numbers are typically represented using two’s complement. If using n bit numbers, the two’s complement of a number x with 0 ≤ x < 2n is (-x) mod 2n = 2n - x. But what do you do if you want to work with unbounded/multiple-precision [...]

Release: Sputsoft Numbers 0.2 (formerly SputArithmetic)

sput — Sat, 19 Jun 2010 07:57:00 +0000

I am very excited about this release. The library has been redesigned and almost everything has been rewritten. Even the name has changed, it is now called Sputsoft Numbers (instead of SputArithmetic). Features include: A generic, portable backend that should make compilation possible on practically every modern C++ compiler and architecture. Compilation against a GMP [...]

Book Review: The Book of Numbers

sput — Sun, 23 May 2010 14:46:18 +0000

The Book of Numbers is a wonderful book about, well, numbers. And lots of them. From ancient ways of writing numbers to Gaussian integers to surreal numbers. The authors are some tough mathematicians, too. John H. Conway is Professor of Mathematics at Princeton University, an authority in game theory and group theory, and the inventor [...]

Arithmetic by Geometry

sput — Sat, 24 Apr 2010 15:29:08 +0000

Today real numbers are most often represented by applying (elementary) functions to (decimal) integers. Throughout history, though, arithmetic and propositions involving (positive) real numbers were often considered from a purely geometrical point of view. Real numbers were identified by the length of some line segment and, e.g., the product of two numbers was identified by [...]

Line-line Intersection in the Plane

sput — Tue, 30 Mar 2010 21:12:42 +0000

How do you calculate the point where two lines in the plane intersect? It is not very hard to do, but the formula can look quite complicated, depending on how you write it up. This article is a reminder that it can be expressed in a simple manner. We start out by not restricting ourselves [...]

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. [...]

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,

\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 [...]