Divisors

Sunday 30 January 2005 at 8:25 pm | In Articles | 2 Comments

Most people, when asked how many divisors the number 60 has (including 1 and 60), would struggle to do so without listing them all. Yet once you know that the prime factorisation of 60 is 60=2^2\times3\times5 you can immediately say that the number of divisors is 3 \times 2 \times 2=12. In other words, you take each index, add 1 then multiply them together.

This is easy to see if you list the divisors as

    \2^0\times3^0\times5^0\ 2^1\times3^0\times5^0\ 2^2\times3^0\times5^0\ 2^0\times3^1\times5^0\ 2^1\times3^1\times5^0\ \cdots\ 2^2\times3^1\times5^1\

It’s then not difficult to prove the general result:

    If n=p_1^{\alpha_1}.p_2^{\alpha_2}.\cdots.p_k^{\alpha_k} then the number of divisors of n is d(n)=\prod\limits^k_{i=1}\left(\alpha_i + 1}\right)

Quadratic Formula Song

Monday 24 January 2005 at 2:39 pm | In Articles | 1 Comment

Got the blues about remembering \dfrac{-b \pm \sqrt{b^2-4ac}}{2a} then listen to The Quadratic Formula Song at Calculus Music.

Tom Lehrer is the best known singing mathematician (zillions of pages on the net). Art Garfunkel (half of Simon & Garfunkel) will be known to older readers; he has an MA in mathematics and started a PhD but his music career came first.

A Million Random Digits With 100,000 Normal Deviates

Sunday 23 January 2005 at 11:52 am | In Articles | Post Comment

Mathematical humour is always welcome, particularly as it is rare. Who would have thought that a book of a million random numbers could inspire amusing reviews at Amazon.com?

Thanks to Isabel’s math blog.

Mathematical 404 Page

Sunday 16 January 2005 at 9:18 pm | In Articles | Post Comment

Gooseania spotted this mathematical 404 error page. Any others out there?

Four Colour Theorem Latest

Monday 10 January 2005 at 9:46 pm | In Articles | 2 Comments

Listen to a short interview with Keith Devlin about the latest verification of the computer proof of the Four Colour Theorem at Solving the ‘Four-Color Problem’ of Map Making (near the bottom of the page). Then read more about it on Devlin’s Angle.

Thanks to Mathforge.net for alerting me to this.

Spherical Trigonometry

Saturday 8 January 2005 at 9:42 pm | In Articles | 8 Comments

The idea that the angles of a triangle add up to 180° is so well engrained that it comes as a shock to some students that it isn’t true in other geometries. These geometries don’t have to be obscure or abstract since the angles of a spherical triangle drawn on the earth’s surface (assuming it is a sphere) always add up to more than 180° by an amount proportional to its area.

It’s a shame that like much else, spherical trigonometry has long since disappeared from the A level syllabus. The 2-dimensional sine rule

    \displaystyle \frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}

is (or ought to be 8-)) well-known but how many are aware of the 3-dimensional version

    \displaystyle \frac{\sin a}{\sin A}=\frac{\sin b}{\sin B}=\frac{\sin c}{\sin C}

or the fact that the great circle distance (the shortest distance) n in nautical miles between points with latitude & longitude \left(\alpha_1,\beta_1\right),\left(\alpha_2,\beta_2\right) is given by

    \cos n = \sin \alpha_1 \sin \alpha_2 + \cos \alpha_1 \cos \alpha_2 \cos \left(\beta_1-\beta_2\right)

where n is measured in minutes? You could use sites such as Surface Distance Between Two Points of Latitude and Longitude but it’s not the same as sitting down and proving the general result.

Without this introduction to three dimensions how is one going to start visualising geometry in four or more dimensions? Of course, reading Flatland would be an excellent start. Phoenix-Library has some excellent versions of this book in a number of online formats.

This is perhaps a suitable place to publicise my all time favourite puzzle:

    A hunter leaves his house one morning and walks one mile due south. He then walks one mile due west and shoots a bear, before walking a mile due north back to his house. What colour is the bear?

Powered by WordPress with Pool theme design by Borja Fernandez.
Entries and comments feeds. Valid XHTML and CSS. ^Top^