Recent Posts by Ian Lucas

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, ...

Some websites to prevent mathematical cobwebs

If you have a long break this (Northern hemisphere) summer, it's important to keep your brain ticking over mathematically so that you don't have to start up from cold when you go back to school next term. I've been on the lookout for some recreational mathematical activity on the web: even just a few minutes each day will be beneficial. I've included a partial screenshot from each site. Pattern recognition Enlarge any one of these 240 patterns. You have to work out ...

Great Mathematicians 4 – Euclid

Given that Euclid's influence on mathematics, geometry in particular, has never diminished over two thousand years, it is extraordinary that we know so little about his life. He was born around 300BC, and was amongst the first teachers at the great university of Alexandria, founded by Ptolemy I, but it is likely that he studied mathematics in Athens with some of Plato's students. Euclid wrote around a dozen influential mathematical books, but it is his 13 volume treatise The Elements ...

A chicken pecking problem

The following problem was posed in the 2017 Raytheon MathsCounts national competition: In a barn, 100 chicks sit peacefully in a circle. Suddenly, each chick randomly pecks the chick immediately to its left or right. What is the expected number of unpecked chicks? The competition is aimed at 13 to 15 year olds, and they have just 45 seconds to answer - the first with the correct solution wins. In this case, a 13 year old Texan boy came up with the answer in ...

Proving Pythagoras’ Theorem

Hopefully everyone reading this can state Pythagoras' Theorem by heart - but, just in case: "Given a right-angled triangle, the square of the length of the hypotenuse is the sum of the squares of the two shorter sides."  Stated more simply, and with the aid of a diagram:   c2 = a2 + b2     But how can this result be proved? It's no good drawing lots of right-angled triangles and measuring their sides - firstly, because it's impossible to measure accurately; and secondly, showing that something is true ...

Understanding significant figures

I'll start with a question - to how many significant figures is the number 500? What does "significant" mean? Consider the number 315 - which digit can you change which would have the least effect on the size of the number? Clearly the 5: increase or decrease it by 1, and 315 changes by 1. But move the 3 up or down 1, and the number will change by 100. So, the 3 is more "significant" than the 5 - and the further ...

A Human Binary Counter

At the heart of every computer, however complex, are binary numbers - that is, numbers formed only of 0's and 1's. This is because it is easy to represent just two digits electronically: a switch can be on or off; a current can flow one way or the other; a pulse can change a stored digit from one state to the other. How does the binary system work?  When we count up to 9 in the decimal system, we've run out ...

Last Call for Revision

I have just returned from teaching on an OSC revision course at the Anglo-American school in Moscow, and was impressed by the focus of the students there towards revision, and how well they were able to sustain that focus over a fairly gruelling week. By the time this blog is published I hope to have seen some of you at the Oxford revision courses as well; my aim, and that of all the teachers on OSC revision courses, is to ...

Puzzle time again

Last year I posed three mathematical puzzles in a blog. This time there's just one, but don't just read it and then look at the answer - it's a really good puzzle to spend some time trying to solve; most people give up, and yet when you see the solution, it's surprisingly straightforward. And please note: there are no catches, no tricks, no sneaky wordplay; the puzzle is exactly as I present it. Four people are being chased by a dragon. ...

Understanding logarithms

A popular topic on OSC revision courses is logarithms. I usually start by asking who knows what a logarithm is, to be met by most of the class staring at their fingernails! To fully comprehend the laws of logarithms, and how to use them, it really helps to start with a clear understanding of the basics: and as long as you can cope with indices (powers), logs follow closely behind. A logarithm is really just a power. If I asked you: ...

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