the official magazine for Scientific Malaysian

by Chin Pin Wong

To celebrate the 10th issue of the Scientific Malaysian Magazine, I was invited to write an article about the number 10 in view of my mathematical background. Unfortunately, I must confess that the number 10 rarely ever pops up in the mathematics I do. Thanks to this invitation, however, I have learnt that there is a lot to be discovered by following the trail of the number 10, even if the results are sometimes unexpected.

1. The base 10 system that most modern societies use is known as the Hindu-Arabic system. A popular but much debated claim is that the system originates from the fact that we have 10 fingers [1]. It is generally agreed that the main advantage of using this base is that we are accustomed to using it. In some cases, it turns out to be more useful to use other bases. For example, in mathematics, base e (Euler’s number) often occurs, while computer systems use binary numbers i.e., base 2.

Fun Fact: The Mayan civilisation is known to have used a base 20 system while the Babylonians used a base 60 system (see reference [2]) for more details).

2. The number 10 is an even number.

Fun Fact: Humans subconsciously assign genders to numbers. People see even numbers as feminine while odd numbers are seen as masculine. Two psychologists at Northwestern University, Galen Bodenhausen and James Wilkie, conducted a series of experiments and found that foreign names and faces of babies tagged with an odd number were more likely to be classified as masculine while those tagged with an even number were more likely to be classified as feminine. For more details, see reference [3].

3. 10 is the sum of the first 3 prime numbers. A prime number is a positive integer greater than 1 that has no positive integer divisors other than 1 and itself. Examples of primes are 2, 17, 541 and 179,424,673. Prime numbers are an important area of research in mathematics. One of the most important results related to prime numbers is the Fundamental Theorem of Arithmetic which states that every positive integer can be uniquely written as a product of primes.

Fun Fact: This unique factorisation of primes plays an important role in public key cryptography which is the basis of computer security. One of the main reasons why we use this for encryption is because it is incredibly difficult to find the prime factorisation of very large numbers, even with the aid of computers. For more details, see reference [4].

4. 10 is the 4th triangular number. Triangular numbers are numbers that can be represented as a triangle of discrete points as below:

Generally, we study n-gon numbers, for example square numbers and pentagonal numbers.

Fun Fact: Fermat’s Polygonal Number Theorem states that any positive integer is a sum of at most 3 triangular numbers, 4 square numbers, 5 pentagonal numbers and n n-polygonal numbers [5].

5. 10 is also a tetrahedral number. The n-th tetrahedral number is the sum of the first n triangular numbers. 10 is the 3rd tetrahedral number as 10 = 1 + 3 + 6. The numbers correspond to placing discrete points to form the shape of a tetrahedron just like we did above for the triangular numbers.

Fun Fact: There are only 5 numbers which are both triangular and tetrahedral namely 1, 10, 120, 1,540 and 7,140 [6].

6. 10 occurs in the 6th row of Pascal’s triangle. Pascal’s triangle is a number triangle whose rows consist of the binomial coefficient. The triangle can be built as the following: Begin with 1 at the peak and build the following rows to have one more number than the one above to form a triangle. Every number (except the peak) is the sum of the two numbers above it. The first few rows of Pascal’s triangle are as below:

Apart from the binomial numbers, Pascal has many other interesting patterns e.g., the second diagonal contains the natural numbers, the third diagonal contains the triangular numbers and the fourth the tetrahedral numbers.

The sum of the shallow diagonals (see the next figure) form the Fibonacci series. For more interesting patterns which one can get from Pascal’s triangle, see [7].

Fun fact: Although Pascal’s triangle is named after Blaise Pascal, a French mathematician in the 17th century, it had been described 500 years earlier by a Chinese mathematician, Yanghui [8].

7. The SI prefix denoting a factor of 10 is deca– or deka–. Fun Fact: A 10-sided regular polygon, i.e. a decagon, can be constructed using a straightedge and compass. Straightedge and compass problems date from Greek antiquity and are still of great interest today as they have links to much deeper mathematics. For a solution to how one might construct a decagon using a straightedge and compass, have a look at the following video https://www.youtube.com/watch?v=KGOp_NBXcUw

8. One can construct a polygon inside a triangle which has an area that is 1/10th of the area of the triangle. Take any triangle and divide each side into 3 equal parts. Connect each dividing point to the opposite corner using straight lines. The area of the polygon that forms from the intersection of these lines in the centre of the triangle is  1/10th of the area of the triangle (see figure below):

Fun Fact: This result is known as Marion’s Theorem as it was proven by Marion Walters, a university professor. There is a generalisation of this result called Morgan’s Theorem which was proven by a high school student, Ryan Morgan [9].

9. 10 can be written as the sum of two squares i.e., 10 = 12 + 32. The question of which integers, n, can be represented as the sum of two squares is a very classical problem. There are many variants of this problem: such as the study of the Pythagorean equation, namely the study of which squares, “z2” can be written as the sum of two squares, i.e. z2 = x2 + y2.

Fun Fact: One of the most well-known problems of this type is Fermat’s Last Theorem, which states that the equation zn = xn + yn has no integer solutions for n > 2 and x, y, z ≠ 0. Pierre de Fermat was a well-known French mathematician who scribbled down this theorem in the margin of a book in the 17th century and claimed that he had found proof but did not write the proof down as well [10]. It took 300 years before anyone found actual proof, and that proof uses many modern mathematical concepts. You can find the (non-technical) story of Fermat’s Last Theorem in the book Fermat’s Last Theorem by Simon Singh.

10. Last but not least, the Scientific Malaysian Magazine celebrates its 10th issue here. Let us wish it a googol (10100) more fantastic issues!

Fun Fact: The search engine Google comes from a misspelling of the word ‘googol’. The founders were looking for a word which would represent the indexing of huge amounts of data and was about to use the term ‘googol’ but was misspelt when registering the name of their domain [11].

## References:

[1] Ifrah, Georges, The universal history of numbers: From prehistory to the invention of the computer, Translated from the 1994 French original by David Bellos, E. F. Harding, Sophie Wood and Ian Monk, John Wiley & Sons, Inc., New York, 2000.
[2] http://bit.ly/1K7xUc3
[3] http://bit.ly/1Eg9b4v
[4] http://bit.ly/1Fg4FDK
[5] http://bit.ly/1JfGmbm
[6] http://bit.ly/1ILrCh3
[7] http://bit.ly/1uFqg00
[8] http://bit.ly/1DRCdpq
[9] http://bit.ly/1ySnqfg
[10] http://bit.ly/1fbYlNm
[11] http://stanford.io/1HYfTNS