### A Beautiful Equation

Euler gave us this jewel which is regarded by many as the most beautiful equation in mathematics; it has both elegance and simplicity. It is comprised of five fundamental numbers and the operations of addition, [more…]

### Commutativity

A binary operation is said to commute if the order of the operands does not alter the result. In the real numbers, , the following commutative laws hold : for all for all Although these [more…]

### Going Round in Circles

Or : Multiplying by . The beginning of complex numbers is often attributed to Cardano (1501 – 1576).  Complex numbers provide a solution to the equation . They are an extension of the number system [more…]

### The Basel Problem

One sometimes gets very interesting results when one turns divergent sums upside down. The Basel problem was asked by Pietro Mengoli in . He asked what is the exact summation of the reciprocals of the [more…]

### Hark! The Herald Angels Sing

Painted in about 1470/75, it is in the National Gallery. One of my favourite carols: Merry Christmas!   OldTrout 2018

### Pistachio Ice Cream

Ingredients : 8oz (225g) unsalted, shelled pistachio nuts 1 pint (570ml) double cream 1 pint (570ml) single cream 4oz (110g) golden caster sugar 8 large egg yolks 1 1/2 tablespoons of custard powder   Method [more…]

### The Möbius Function

The Möbius function is denoted by , where is the Greek letter mu. Depending on the factorisation of , the function takes the values . The last says that if is even, then and if [more…]

### The Fields Medal

“Rise above oneself and grasp the world” These are awarded every four years for advancement in the field of mathematics.  There is an age limit of 40 on the recipients.  It is a recognition of [more…]

### Euler’s Totient Function

Euler’s totient function counts the number of positive integers up to that are relatively prime to , where is considered to be relatively prime to all .  These numbers are called the totatives of . [more…]