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 in Euclid’s Elements, an encyclopedia of all known mathematics around 300 B.C. But how do you actually prove the Pythagorean theorem?
Fractions and Circles
Published on 2010-02-06
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
Published on 2010-01-09
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 millennium problems presented by the Clay Mathematics Institute.
The Stern-Brocot Tree of Fractions
Published on 2009-12-04
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,
Book review: The Pleasures of Counting
Published on 2009-11-15
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 variations, and The pleasures of thought.
Continued Fractions and Continuants
Published on 2009-11-10
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}}}}
Computing the Greatest Common Divisor
Published on 2009-10-29
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.
Remembering Trigonometric Addition Formulas
Published on 2009-09-23
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}
\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.
Release: SputArithmetic 0.1
Published on 2009-09-19
This is the first release of the SputArithmetic library. You can download the library or read about it.
Useful Properties of the Floor and Ceil Functions
Published on 2009-09-09
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.
Useful Properties of the Floor and Ceil Functions continued »
