Some Math Magazines

Quanta is really a science magazine but has good articles of interest to math fans. Mathematical Digest is a magazine that has been produced for nearly 50 years now by Professor John Webb of the University of Cape Town. I grew up on these in high school and along with Martin Gardner's columns they fuelled my love of math. I was privileged to be instructed by Prof. Webb as an undergraduate math student as well; he was regularly awarded honors as a distinguished teacher and these were well-deserved.

Finally, there is Manifold, a student magazine of the University of Warwick, which is where the well-known math professor and writer Dr Ian Stewart hails from. I haven't looked at these yet but expect them to be interesting.

If you know of other sources of such magazines let me know in the comments!

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A Christmas Carroll

On Christmas day, 1877, Lewis Carroll, author of the Alice books, entertained two bored young girls by inventing a new game that he called Word Links: given two words, change one word to the other by changing a single letter at a time with the intermediate steps all being valid words themselves. For example, to change "cold" to "warm", one can use the steps "cord", "card", "ward". Carroll later popularised this form of puzzle in a series of articles in Vanity Fair magazine, changing the name to Doublets - from the "double, double, toil and trouble" witches' incantation in Shakespeare's Macbeth.

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The Gym Locker Paradox

It happens so often - you go to your section of the gym change room, and there is just one other person there, and they are using the locker next to yours and blocking the way. I noticed this a long time ago, and noticed that others noticed it too, referring to Murphy’s law, or “just my luck”, or some such explanation. Why does it happen so often?

I call this the Gym Locker Paradox. It is related to a much more well known paradox, the Birthday Paradox, which is commonly formulated as a puzzle: how many people do you need to have in a room together before the odds are better than even that two share the same birthday?

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The Calculus of Democracy

Earlier this year I went to a “guys movie night” at a friends house. Before the movie we sat around drinking Croatian Schnapps and eating Chinese food, and the conversation turned to politics. There was general agreement that our various political systems were highly suspect and not very democratic, but Ray, the Texan lawyer with the Croatian wife who had provided the schnapps, explained the Australian system of instant runoffs and asserted that this was at last a fair system.

For those not familiar with this system, let me explain: voters get to rank the candidates in order of preference. The votes for everyone’s first preference are counted, and the candidate who gets the least votes is eliminated. That candidate is also struck from the voters choices and the process is repeated until there is a candidate that has more than 50% of the vote.

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Fibonacci ties it all together

In the past few posts I have discussed a diverse set of topics, from number representation to recurrence relations, Pythagorean triples.

All of these topics are related, and this is particularly illustrated when we consider the work of Leonardo of Pisa, better known as Fibonacci (from the Latin “de filiis Bonacci”, or “of the family of Bonacci”). Fibonacci was born in 1175, in Pisa, Italy, shortly after the start of the construction of the famous leaning tower. The city of Pisa was a maritime republic which had its own colonies, including Bugia (in modern day Algeria), where Leonardo’s father moved in 1192 to be a clerk in the customs house. It was here that Fibonacci became acquainted with the Hindu numerals and zero, and where his Muslim teacher introduced him to the the great book Al-Jabr wal-Muqabalah. Fibonacci subsequently travelled broadly, where he learnt from mathematicians of different cultures. Around 1200 he returned to Pisa and began work on his masterpiece Liber Abaci, which was published in 1202, and begins with these words:

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The Horn of Gabriel

And the seventh angel sounded [his horn]; and there were great voices in heaven, saying, The kingdoms of this world are become the kingdoms of our Lord, and of his Christ; and he shall reign for ever and ever.” (Revelations 11.15, King James Edition)

In Christian and Islamic folklore, it is the angel Gabriel who is considered to be the seventh angel, who announced the coming of judgement day. An angel with such a huge responsibility clearly needs an instrument worthy of the task, and indeed there is one: Gabriel’s horn, also known as Torricelli’s trumpet, after its discoverer, Evangelista Torricelli, a student of Galileo. Gabriel’s horn has infinite surface area but finite volume, and is described the rotating the curve \(y=\frac{1}{x}\) for \(x \geq 1\) around the \(x\) axis.

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More on Pythagoras

Pythagoras is known for two great contributions to mathematics – he established the need for formal proofs instead of just conjecture and rules of thumb, and he established the existence of the irrationals. In popular culture of course, Pythagoras is more well known for the Pythagorean theorem – that the square of the hypotenuse of a right angled triangle is the sum of the squares of the other two sides – but this was one of the oldest known results in mathematics and in fact predates Pythagoras by as much as a thousand years. Nonetheless, Pythagoras provided a formal proof, and the result led him to ask whether there were rational numbers that worked in the case where the hypotenuse had the length two. That is, what was the fractional representation of the square root of two? Pythagoras managed to show there was none, leading to the discovery of the irrational numbers.

His proof was quite simple. Assume that \(\sqrt{2}\) can be expressed as a fraction \(\frac{a}{b}\) of two whole numbers \(a\) and \(b\). Assume these are the two smallest such whole numbers – that is, they have no common divisor allowing the fraction to be further reduced. Then:

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Much Ado about Nothing

Many people have heard that the number zero was introduced later than the other digits. Upon hearing this it seems fantastical - how could people have managed even elementary arithmetic without zero? However, it is not the concept of “nothing” that was missing, but rather the use of a zero in other places when representing or recording numbers. For example, in the number 101, zero is used as a separator, and it is this concept that was lacking in early number systems.

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Primal Soup

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Number theory really began with Euclid, around 300BC, in books 7 through 9 of his masterwork, The Elements. It is here that we find the original definitions of odd and even numbers, prime and composite numbers, perfect numbers (numbers which are the sum of their factors, e.g. 6 = 3 + 2 + 1), and more. But the greatest achievements of all were his proofs that composite numbers are the products of primes, that this factorization is unique, and that there are an infinity of primes. Most introductory algebra or number theory classes cover these three great proofs, but they are worth revisiting for those who may have forgotten.

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Archimedes counts the sand

In a previous post I described the Babylonian/Sumerian sexagesimal (base 60) counting system. Unlike this system, most cultures adopted a base 10 counting system due to the natural inclination to count with the fingers (leading to the term digits). Less common were quinary (base 5) systems, but vigesimal (base 20) were not unusual - for example, this was widespread in native American culture, including (with a novel variation) the Mayans. Duodecimal (base 12) has also played a role; we still have remnants in the groupings of dozens and gross.

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