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Thursday, October 30, 2014

Throwback Fact of the Week - Vampire Numbers - 10/30/14

Vampire numbers were first defined by Clifford Pickover in 1994. 

A vampire number is a composite whole number with an even number of digits, n. The vampire number has form v = xy where the digits x and y each have n/2 digits. These digits consist only of the digits in the vampire number, but in any order. Note that x and y are not allowed to both end in trailing zeros. 

1260 is the smallest vampire number.
 v = xy
1260 = 21*60

You'll notice that in this case, the vampire number has 4 digits; x and y each have 2 digits, and those digits consist of a re-arrangement of the digits in the vampire number. 

The digits x and y are known as the fangs of the vampire number. Vampire numbers sometimes have multiple pairs of fangs that fit the rules for forming the vampire number. Such as 125460 = 204 × 615 but it also equals 246 × 510.

An even more interesting type of vampire number would be a vampire number where its fangs are also its prime factors. For example, 117067 = 167 × 701, where 167 and 701 are the prime factors of 117067, making this number a prime vampire number! 

Thursday, October 23, 2014

Throwback Fact of the Week - Benford's Law - 10/23/14

Benford's Law, or more descriptively, the first-digit law, states that in most number lists and real-life collections of data, (such as death rates, baseball statistics, etc.) 1 occurs in the leftmost leading position roughly 30% of the time. It also asserts that larger digits occur less and less frequently, with 9 occurring a mere 4.6% of the time as the first digit.

The significance of this law lies in the fact that it shows that the most common first digit in an arbitrary source of data is not random; if it were random, every digit would be expected to occur first about 11.1% of the time, a 1 out of 9 probability.

The graph below shows probability that each digit occurs as the leading digit. A given set of numbers satisfies Benford's Law if the leading digit d, where d can be the digits 1 through 9, occurs with probability log10(1+1/d).

The law is named after physicist, Dr. Frank Benford, who stated it in 1938. However, it had been previously pointed out by Simon Newcomb in 1881, when he noticed that the pages of logarithms containing the numbers beginning with 1 were much more worn out than other pages.

The law has been used as a method to detect fraud; for example, an accountant could detect a fraudulent tax document if the occurrence of leading digits does not sync up closely with Benford's Law.

Thursday, October 16, 2014

Throwback Fact of the Week - Zero - 10/16/14

Zero. We often overlook the importance of this number and its existence does not seem foreign or bizzare to us... anymore.

The Greeks lacked a symbol and concept for zero. They wrestled with the philosophical implications of how nothing could be something.

The Babylonians developed an excellent sexagesimal number system (base 60) but they lacked the concept of zero. Over time, the Babylonians would develop a system of using a space as a placeholder between digits that functioned similarly to the modern zero. However, this space was not zero; the concept of representing nothing was foreign to them. Without the idea of zero, there was no way to distinguish certain Babylonian numbers, just like how today we wouldn't be able to distinguish the numbers 13, 103, 130 10003 without the use of zero. 

The development of zero as an actual number (not just an empty space) is credited to 7th century Indian mathematics. Around the same time, the concept was being used by the Mayan civilization. However, it was the Indian concept of zero that spread to Arabia, Europe and China. 

Indian mathematics treated zero like any other number, using it in calculations (even in division). It was the Indian mathematician, Bragmagupta, who first laid out a set of rules governing the use of zero. These rules included things like a number subtracted from itself is zero and a number times zero is zero. 

The importance of zero can be summed up by Pierre-Simon Laplace, who said:

"The ingenious method of expressing every possible number using a set of ten symbols (each symbol having a place value and an absolute value) emerged in India. The idea seems so simple nowadays that its significance and profound importance is no longer appreciated ... The importance of this invention is more readily appreciated when one considers that it was beyod the two greatest men of antiquity, Archimedes and Apollonius."

Monday, October 13, 2014

Center of Math NEWs: Greg Moore - Physical Mathematics and the Future

Center of Math NEWs brings you fun and interesting new mathematics news from around the world.

Topics: Physical mathematics, string theory, infinite-dimensional Lie algebras, geometry, analytic number theory, topology,  quantum gravity, string theory, supersymmetry

Physical Mathematics and the Future - Gregory W. Moore: paper (PDF)
Show slides: here 

 Runtime 16:02

Strings 2014: go

Thursday, October 9, 2014

Throwback Fact of the Week - Viviani's Theorem - 10/09/14

Viviani's Theorem can be broken into 2 steps:

  1. Choose a point anywhere inside of an equilateral triangle
  2. Draw perpendicular lines from the point to each of the 3 sides of the triangle
The theorem states that the sum of the lengths of these lines is equal to the height of the triangle. Using the image below, the theorem states that x + y + z = h, no matter where inside the triangle you place point P. 

The theorem was proven by Vincenzo Viviani around the year 1659. The proof can be derived easily from the formula for the area of a triangle (Area = .5bh, where b is the base of the triangle and h is the height). To quote the typical math textbook, "this proof is left as an exercise for the reader."

This theorem can be generalized to any regular n-sided polygon. For the case of an n-sided polygon, the sum of the perpendicular distances from an interior point to each of the n sides is equal to n times the length of the apothem of the polygon (Recall that an apothem of a regular polygon is a perpendicular line segment from the center of the polygon to the midpoint of one of its sides).

Thursday, October 2, 2014

Throwback Fact of the Week - Tower of Hanoi - 10/02/14

The Tower of Hanoi is an intriguing puzzle game made popular by French mathematician, Edouard Lucas, in 1883. 

The puzzle consists of disks of different sizes that can be moved onto any of the 3 pegs. At the start, the disks are all stacked on the leftmost peg in order of size, with the smallest at the top. The goal of the puzzle is to move the entire stack to the rightmost peg, while obeying these 3 simple rules:
  1. Only one disk at a time may be moved to another peg 
  2. You can only move the top disk from a stack of disks
  3. A larger disk cannot be placed on top of a smaller disk
The minimum number of moves necessary to solve the puzzle with n disks is 2n - 1. There are simple algorithms and strategies for solving this puzzle in the minimal number of steps. 

Check out the video below to see the game being played optimally starting with 5 disks (2^5-1 = 31 moves). 

Video Credit: Mohammad Al-Khanfar

Wednesday, October 1, 2014

New Center of Math Animation!

     These past few weeks have been busy at the center; with our followers rapidly growing, we decided to make a short video to thank everyone for their support! After days of toiling with cameras, lighting, and crazy setups, we finally finished our own whiteboard animation. Come check out our Youtube channel to see our new video!

Previously mentioned ridiculous setup...