## Archive for August, 2017

### Saros 145/22: The Great American Eclipse

Next Monday, the shadow of the Moon will bring a two-minute spell of darkness as it sweeps across the United States along a path from Oregon to South Carolina. The eclipse is one of a series known as Saros 145. [TM121 or search for “thatsmaths” at irishtimes.com].

Saros series 145 recurring every 18 years, 10 days and 8 hours.
[Image from www.GreatAmericanEclipse.com ]

### Fractions of Fractions of Fractions

Numbers can be expressed in several different ways. We are familiar with whole numbers, fractions and decimals. But there is a wide range of other forms, and we examine one of them in this article. Every rational number ${x}$ can be expanded as a continued fraction:

$\displaystyle x = a_0 + \cfrac{1}{ a_1 + \cfrac{1}{ a_2 + \cfrac{1}{ a_3 + \dotsb + \cfrac{1}{a_n} } }} = [ a_0 ; a_1 , a_2 , a_3 , \dots , a_n ]$

where all ${a_n}$ are integers, all positive except perhaps ${a_0}$. If ${a_n=1}$ we add it to ${a_{n-1}}$; then the expansion is unique.

### It’s as Easy as Pi

Every circle has the property that the distance around it is just over three times the distance across. This has been known since the earliest times  [see TM120 or search for “thatsmaths” at irishtimes.com].

The constant ratio of the circumference to the diameter, denoted by the Greek letter pi, is familiar to every school-child. You might expect to find a proof in Euclid’s Elements of Geometry, he could not prove it, and he made no mention of the ratio (see last week’s post).