Visualizing the Pythagorean Theorem

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?

Visualizing the Pythagorean Theorem continued »

Fractions and Circles

Conway and Guy: The Book of Numbers

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.

Fractions and Circles continued »

Book Review: Prime Obsession

Derbyshire: Prime Obsession

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.

Book Review: Prime Obsession continued »

The Stern-Brocot Tree of Fractions

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,

The Stern-Brocot Tree of Fractions continued »

Book review: The Pleasures of Counting

Korner: The Pleasures of Counting

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.

Book review: The Pleasures of Counting continued »

Continued Fractions and Continuants

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}}}}

Continued Fractions and Continuants continued »

Computing the Greatest Common Divisor

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.

Computing the Greatest Common Divisor continued »

Remembering Trigonometric Addition Formulas

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.

Remembering Trigonometric Addition Formulas continued »

Release: SputArithmetic 0.1

This is the first release of the SputArithmetic library. You can download the library or read about it.

Release: SputArithmetic 0.1 continued »

Useful Properties of the Floor and Ceil Functions

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 »