Wednesday, December 31, 2014

New Years Day Warm Up.

Here is a warm up for tomorrow.  There are some things you need to know about 2015 before it begins:

The year 2015 will have 13 full moons.
The year 2015 will have 3 Friday the 13ths.
2015: Time travelers Marty McFly and Doc Brown arrive from the year 1985. (From the movie “Back to the Future Part II”)

2015 has 8 divisors (1, 5, 13, 31, 65, 155, 403, and 2015), whose sum is σ = 2688.  The sum of its proper divisors is 673.  Its totient is φ = 1440.  The sum of its prime factors is 49.  The product of its (nonzero) digits is 10, while the sum is 8.
It is a sphenic number (or 3-almost prime), since it is the product of 3 distinct primes: 2015 = 5 * 13 * 31.
2015 is a composite, deficient, evil, odd and square-free number.
It is a Duffinian number.
It is a zygodrome in base 12.
It is a junction number, because it is equal to n + sod(n) for n = 1993 and 2011.
It is a congruent number.
It is a polite number, since it can be written in 7 ways as a sum of consecutive naturals, for example: 2015 = 50 + 51 + ... + 79 + 80.
It is an arithmetic number, because the mean of its divisors is an integer number (336).
22015 is an apocalyptic number.
2015 is a wasteful number, since it uses less digits than its factorization.
2015 can be expressed as the sum of 10 different powers of 2: 2015 = 20 + 21 + 22 + 23 + 24 + 26 + 27 + 28 + 29 + 210.
2015 is a trapezoidal number (the difference of two triangular numbers): 2015 equals the 63rd triangular number minus the 1st triangular number.
2015 is a 31-smooth and a 5-rough number.
2015 is a palindromic and Cyclops number when written in Base 2: 111110111112.
2015 is written MMXV in Roman numerals.
2015 is an undulating number when written in base 8: 37378.
2015 can be expressed as the difference of two squares in four different ways: 2015 = 482 - 172 = 842 - 712 = 2042 - 1992 = 10082 - 10072.
2015 is one of 5 number located between the sexy prime pair of 2011 and 2017.
2015 is a Lucas-Carmichael number.
The string 2015 occurs at position 19038 (counting from the first digit after the decimal point.  The “3.” is not counted.).  This string occurs 20,090 times in the first 200 million digits of Pi.

Have a Happy and Safe New Year.


Thursday, December 18, 2014

Professor Keith Devlin on Youtube

Dr. Keith Devlin is a mathematics professor at Stanford University in California.  He has authored many books in the area of mathematics (and Maths relationship to other subjects).  Some of course are related to college level mathematics, but others are what I call books on popular mathematics, books for the general public or for the more educated public, but not for a mathematics course.  He has been pretty successful will selling the books, so that reflects well on how well he is able to explain mathematical concepts to the un-initiated or mathematics enthusiast.
I discovered today that he has a series of five videos, each about 2 hours long, discussing a brief history of mathematics, titled “Mathematics: Making the Invisible Visible”.  It is from a continuing education course that he taught in 2012.  They are available free on Youtube, and I highly recommend then for more serious high schoolers, college students, and adults.  The websites are provided below:
The first video talks about the development of ancient mathematics (counting and numbers).
The second video talks about the Fibonacci sequence and the Golden Ratio, especially about which popular beliefs are true, and which are false.
The third talks about the development of Algebra and how it is different from arithmetic.
The fourth talks about the development of Calculus and how it has become so effective a tool for us to use.
The fifth video focuses on how human beings acquired the ability to do mathematics.
A quick search of can show you the books he has written.  If you are trying to prepare for Fibonacci Day next month (11/23/2014 – 1, 1, 2, 3, …) I can recommend “The Man of Numbers” which discusses Leonardo Fibonacci the mathematical revolution that he kicked off.
Dr. Devlin also teaches a MOOC course for Corsera (online, non-credit, and FREE) that teaches mathematical thinking.  See  for details.
He authors a monthly article for the Mathematical Association of America (MAA) called “Devlin’s Angle”.  See:  And he has a website at:
Don't forget - check out the Youtube videos - I think you will enjoy them.


Wednesday, December 17, 2014

Language and Mathematics

If you write out the names of the natural numbers in English (ONE, TWO, THREE, FOUR, etc.) you will be surprised to to learn that:
ONE contains the first O, the first N, and the first E.
TWO contains the second O.
THREE contains the third E.
ELEVEN contains the 11th E.
TWENTY-FOUR contains the 24th T.

TWENTY-NINE contains the 29th N.
THIRTY-ONE contains the 31st N.
ONE HUNDRED NINE contains the 109th N.
ONE HUNDRED NINETY-NINE contains the 199th D.
TWO HUNDRED FIFTY-ONE contains the 251st O.
FOUR HUNDRED FIFTY-FOUR contains the 454th U.
FIVE HUNDRED FIFTY-NINE contains the 559th I.
1,174 contains the 1,174th O.
1,716 contains the 1,716th S.
5,557 contains the 5,557th F.
6,957 contains the 6,957th F.
15,756 contains the 15,756th F.
17,155 contains the 17,155th F.
24,999 contains the 24,999th Y.
43,569 contains the 43,569th F.
735,759 contains the 735,759th V.
1,105,807 contains the 1,105,807th V.
1,107,785 contains the 1,107,785th V.
1,584,504 contains the 1,584,504th V.
1,707,941 contains the 1,707,941st V.
1,921,567 contains the 1,921,567th L.



Saturday, December 13, 2014

One of Peggy Sue's Other Numbers

Remember that sweet Italian girl I told you about a month ago.  She's back.  It seems that the coolness of the Tyrrhenian Sea and the heat of Vesuvius creates a temperature difference powerful enough to drive a person's mind into new territories.

The last time we saw a fraction whose decimal expansion produced the 13 times table.  Now we have a fraction whose decimal expansion produces powers of 13.

Powers of 13 beginning with 13^0 through 13^20, in 24 digit chunks:

1 / 999,999,999,999,999,999,999,987 =