Recent Posts by Ian Lucas

Concerning Leap Years

Who Introduced the Leap Year? Julius Ceasar, believe it or not. The Roman calendar had got seriously out of date with the solar calendar, even though Roman officials were supposed to introduce an extra month every so often. So Caesar consulted his top astronomers who worked out that the solar year—the time it takes for the Earth to do exactly one rotation around the sun—was 365.25 days. So, by maintaining a calendar with 365 days, and adding one extra day every ...

Drawing Graphs on the TI-84

A previous post on solving equations with the TI-84 has proved very popular, so here are some tips and tricks when using the GDC for drawing graphs and associated functionality. Pre-set scales It's easy enough to type in an equation and then press the graph button. However, the most important thing to do which helps make sense of any graph is to set up the correct window: the minimum and maximum value of the displayed coordinates, and the associated scale marks. The ...

What’s the difference between an expression and an equation?

Well, I’ll answer that in a moment. More generally, you should be familiar with algebraic terminology since relevant words are used in exam questions—you need to know exactly what they mean. Let’s start at the beginning and build up… Constants and variables A constant is a fixed number, such as 2. A variable is a letter which can take on different values, such as $x$. To complicate things, letters can sometimes be used to stand for constants! For example, I can define ...

Muhabbat Sharapova, Mathematician – and Hero of Uzbekistan

To be named as a "Hero of Uzbekistan" is to be awarded the highest national honour in that country. Recently, Muhabbat Sharapova, one of the country's top mathematicians, received the award - not for her mathematical prowess, but because she has turned down job offers from many leading universities and institutions in favour of remaining a maths teacher in her small home town. In an interview for the BBC's 100 women initiative, she says: "Mathematics is the mother of all sciences." ...

More puzzles

It's sometimes hard to define whether a puzzle can be classified as "mathematical" or not. They certainly are if they require mathematical techniques or knowledge to solve them. Today's puzzles don't, but I would still class them as mathematical: firstly, because they use numbers or geometrical space or logic; and secondly because they tickle the mathematical bits of your brain and so, in a very small way, hone your skills. 1. The two jugs Solve this in under 30 seconds - excellent. ...

Understanding Averages

You will have learnt that there are three types of average: the mean, the median, and the mode. Let's look at them closely - why we need all three, what you can do with them, and when to use each one. The Mean The mean is calculated by adding all the values and dividing by the number of values. Using sigma notation, where the sigma sign means "the sum of", we can write: $m = {{\sum x } \over n}$. The best way ...

Catch up on useful Mathematics blogs…

I have been writing these blogs for IB Mathematics students for a year and a half now: this is my 36th. They divide into three types: a)    General interest and recreation. These blogs look at how mathematics impinges on everyday life and aim to give you a wider appreciation of its relevance. Examples are the probability behind backgammon; Olympic records; some thoughts about zero; a series of blogs about great mathematicians; mathematical puzzles. b)    Help with specific areas of mathematics, ...

A democratic paradox

Democracy depends on people voting for their leaders, but there are many different voting systems in use, all of which aim for a fair result: first past the post; single transferable vote; alternative vote plus are three examples. All have advantages, all have disadvantages. Many popular voting methods use a system which allows voters to select candidates in order of preference. Suppose there are three candidates, A, B and C. You would expect that if more voters preferred A to B, ...

Coincidence or conspiracy?

Consider these well known historical connections: TITANIC:  In 1898, a British author wrote a story about a luxury liner, 800ft long, which was travelling at full speed when it hit an iceberg in the North Atlantic, and sank with the loss of 2500 people. The loss of life was so great because the ship didn't carry enough lifeboats. Its name - The Titan!     D-DAY:    In May 1944 a schoolteacher, Leonard Dawe, who composed the daily crossword for a London newspaper, included ...

Practical applications: Vectors

Are vectors just a mathematical idea, or are they of any practical use? Every time you fly anywhere, it's because of vectors that you end up in the right place! Here's why. What is a vector? A vector represents any quantity that has both magnitude and direction. So, vectors can be used to represent displacement (ie change of position), velocity, force - but nor distance, speed or energy. For example, if the distance between two towns along a winding road is 120km, ...

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