Any sketch of a cubic function will show the curve crossing the axis at least once. If it has exactly one real root then it must have two complex conjugate roots as well. While the real root is easily seen as the intercept, the complex roots are not so obvious. This blog however illustrates a simple technique that enables an estimate to be made on them. Given a scaled sketch, nothing more than a ruler and a sharp pencil is required to obtain the estimate.
In many departments of the public service, a fortnightly salary is determined as ‘twelve parts per three hundred and thirteen’ of an employee’s gross annual income. Just why this and its ramifications is discussed in this short blog. We think that it’s worth reading because the fraction may well be playing havoc with your earnings.
There are several types of Diophantine equation – equations requiring integer solutions. Although very little is known with certainty about Diophantus, the study of Diophantine equations has been important in number theory. We set out the standard way of solving the linear Diophantine equation ax + by = k.
Every 20 years or so Jupiter laps Saturn, appearing together in the same zodiacal region of the sky (technically with the same right ascension). It’s called a Great Conjunction and the next one occurs on 21 December 2020. That’s not too far away. Jupiter is chasing down Saturn and will catch it on that day, and the reflected light of the two gas giants will be some spectacle. This blog develops a useful formula for working out lapping times, the time required for a faster object to lap a slower object. The blog also follows the efforts of Johannes Kepler in his attempt to make sense of the heavenly phenomena.
Puzzles similar to this one are often used to motivate the study of ideas about numbers and divisibility going back to the time of Euclid. However, a straightforward approach is possible, avoiding complications of the Ancient Greek kind. Before long, we may find that some deeper mathematical ideas are needed but we can begin intuitively.
The reason why the Gregorian Calendar was adopted in place of the Julian Calendar was because, quite simply, the Earth doesn’t really take 365 and a quarter days to get around the Sun. It’s a tad short of that figure, and problems began to appear with the alignment of the vernal equinox. This story traces a little of the history, and then introduces Christian Zeller, the mathematician who is understood to be the first to find a formula for calculating the elapsed days of the year.
Lewis Carroll was really Charles Dodgson, an Oxford Scholar and a brilliant mathematician to boot! One of his party tricks was to be able to determine the day of the week for any day of the Gregorian calendar. John Conway, an American mathematician also noticed something quite odd with new Calendar and invented the Doomsday. Both are worth learning.
This problem is perhaps one of the most well known mathematical puzzles of the last hundred years. According to Wikipedia, a short story writer Ben Williams modified an older problem and included it in a story in the October 9, 1926 issue of The Saturday Evening Post. The magazine was swamped by interested readers wanting a solution. Much later Martin Gardner (1914-2010), the American popular mathematics writer, featured the problem in his April 1958 Mathematical Games column in the magazine Scientific American. Gardner once told his son that it was his favourite puzzle.
The Monarchy is an excel spreadsheet tracing the English Monarchy from William the Conqueror to the present day Elizabeth the second. The various houses are identified and annotations of historical events are included as well.
Recurrence relationships arose in relation to the cake numbers and in the cannonball-stacking problem.
We give two techniques for finding an explicit function given a linear recurrence relation.