Showing category "Mathematics" (Show all posts)

Solving The Wire Filling Problem

Posted by on Saturday, January 7, 2023, In : Mathematics

As we start into a new year, and start to anticipate all of the amazing new discoveries and scientific theories that will come with it, I was trying to think of a good subject for the first article of 2023. I considered writing reviews of leading edge artificial intelligence or quantum computing techniques, or perhaps a discussion of some of the latest theories in astrophysics or exotic particle physics. In the end though, I decided that coming out of the holidays we should go with a bit of l...
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Fourier Transforms

Posted by on Tuesday, June 25, 2019, In : Mathematics

I have spent the last few weeks pondering what I should write about for my first article back. Should I go into something exotic in astrophysics such as magnetic monopoles or topological defects? Should I go into complicated mathematical research and paradoxes? Should I go into the bizarre world of quantum mechanics and its foundations? Nothing seemed quite right.

So I decided to write about Fourier transforms. That's right, after thinking about some of the strangest concepts in physics and ma...
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Surreal Numbers

Posted by on Friday, February 23, 2018, In : Mathematics

I am always amazed at how simple some of the most interesting ideas and research in modern mathematics truly is. There are problems in mathematics that can be explained to a small child and yet the greatest minds of the past centuries have been unable to solve. Mathematics is one of the few fields of study where anyone can understand topics that the leading experts are still trying to solve. One such topic is the surreal numbers.

Everyone remembers as a child learning the integers, or counting...
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Simpson's Paradox Visualized

Posted by on Monday, July 10, 2017, In : Mathematics

A while ago I wrote an article about an interesting statistical phenomenon known as Simpson's Paradox. According to Simpson's Paradox, a company can have discriminatory hiring policies in spite of each of its individual departments being completely fair. A new medical treatment can work better than existing methods for both the young and the old, and yet it gives worse results when you don't know the age of the patient. And it can make a single data set produce opposite and contradictory resu...
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The Axiom Of Choice

Posted by on Thursday, October 13, 2016, In : Mathematics

Mathematics is the pinnacle of logic and science. Most people assume that mathematics is based on solid foundations of logic, and that everything is well understood. Some sciences contain controversies and differing opinions, but surely mathematics is pure and definitive. At the very least the basics that are taught in school must be based indisputable.

And yet deep in the heart of mathematics lies a problem, known as the axiom of choice.

Suppose that you have a collection of boxes, each conta...
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Berkson's Paradox

Posted by on Tuesday, September 13, 2016, In : Mathematics

A couple of years ago I wrote an article explaining one of the most interesting phenomena in statistics, Simpson's Paradox. For those who have forgotten (or just didn't read it), Simpson's paradox occurs when a single set of data is analyzed by two different, equally valid methods, and the results of the two methods give opposite conclusions. The reason for this is that often a data set contains information that is not included in the analysis but is important to the results, such as drug tes...
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Nine Kinds Of Unbounded Primes

Posted by on Thursday, June 16, 2016, In : Mathematics

For those who have been enjoying my recent series on unsolved mathematical problems, we have another entry today. This time featuring the properties of prime numbers. (And for those who prefer my physics and astronomy articles, I promise that I will try to write a few more of those as well in the coming weeks).

As most (and hopefully all) of you know, a prime number is a number which cannot be written as the product of two other numbers. It can only be properly divided by the number 1, and by ...
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Perfect Numbers

Posted by on Tuesday, June 7, 2016, In : Mathematics

Continuing with this week's theme of simple yet unproven mathematical problems, today I thought I would discuss perfect numbers. Although they appear to be simply counting numbers, they have some very unique properties and mysteries.

First though I must apologize for part of yesterday's article. As I wrote in the introduction, my aim was to provide three simple problems in number theory. And while readers did enjoy both the Erdos-Strauss conjecture and Pillai's Conjecture, it would seem that m...
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Three Problems In Number Theory

Posted by on Monday, June 6, 2016, In : Mathematics

Last week I wrote an article outlining a few simple yet unsolved problems in mathematics. At the time I had thought it would be a one off posting, rather than the start of a series of articles. However based on the positive responses I have received and the numerous inspiring mathematical discussions it has provoked, it is clear that there is a demand in society for more of these type of mathematical articles.

And so here we go again...

The last article that I wrote focused on colourings and on...
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Beyond Four Colouring

Posted by on Sunday, May 29, 2016, In : Mathematics

A couple of years ago I wrote an article on the famous four colouring theorem. (For those who missed it, the basic idea is that any map drawn on a flat plane will be able to be coloured such that no two neighbouring regions have the same colour, using only four colours). Although this problem has been solved, with a proof being completed nearly forty years ago now, it is a controversial problem in that the only known proof of this theorem is a brute force calculation by a computer, rather tha...
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Card Shuffling Problem

Posted by on Tuesday, November 3, 2015, In : Mathematics

A few days ago I was watching an old re-run of the excellent British television program, QI, and the host of the show made an interesting claim regarding a normal deck of playing cards. He took a new deck, which is ordered by number and suit, and gave it a few shuffles. Afterwards he claimed that no deck of cards had ever before been in that order, as the number of possible orderings was so large that statistically it was (almost) impossible that two randomly shuffled decks could ever be in t...
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Three Prime Problems...

Posted by on Wednesday, March 25, 2015, In : Mathematics

Last week I wrote about the state of modern physics, and how - in my opinion - it was now so complex and filled with subtle technicalities that an untrained amateur would have no chance of producing some great new theory or result. However it is interesting to note that although theoretical physics and mathematics are very closely related, the field of mathematics is still ripe for amateurs to produce important results. Even primary school students can understand mathematics problems that the...
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The Three Game Paradox

Posted by on Saturday, November 15, 2014, In : Mathematics

In my continuing series on curious mathematical paradoxes, I have reviewed several interesting and counter-intuitive phenomena in both mathematics and statistics. Recently a couple of readers of my blog asked me about the Three Game Paradox. I must admit that I don't see this as a true paradox, but rather simple statistics, but it is still interesting and as such I present it here for your pleasure...

Suppose that you have just joined a chess club, and you are really a novice player. It seems ...
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The Rare Test Paradox

Posted by on Tuesday, June 3, 2014, In : Mathematics

Due to the positive comments I have received from my last two articles on mathematical and statistical paradoxes, I have written another one. However where the first two were interesting mathematical puzzles with minimal real world applications, this one is nastier because many people have suffered from its results. It is commonly known as the Rare Test Paradox

Suppose that you have an excellent medical test for some rare disease. Just for the sake of argument, suppose that this test is so ac...

A Million Dollar Problem

Posted by on Wednesday, April 9, 2014, In : Mathematics

With tax season here once again, I thought it would be a good time to present an interesting mathematical problem that could be worth one million dollars to whoever solves it. It is simple to present, but has stumped mathematicians for decades.

Let me begin with a review of logic circuits (although it can also be considered as mathematical operations with no connection to electronics). You begin with a set of "inputs" that can have a value of true or false. You can have as many or as few as yo...
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Simpson's Paradox

Posted by on Friday, March 7, 2014, In : Mathematics

During an online tutoring session I was hosting recently, the subject of Simpson's Paradox came up, and I realized that this beautiful example of counter intuitive statistics is not that well known yet, and even those who have heard of it have great trouble understanding it and resolving it in their own minds. And so I thought I would write a simple explanation that helped my students in the past to visualize how it works.

First an explanation of Simpson's Paradox. In general, this refers to a...
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Dr. Christopher Bird