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Thursday, September 25, 2014

Throwback Fact of the Week - Collatz Conjecture - 9/25/14

Collatz Conjecture

This conjecture is named after German mathematician, Lothar Collatz, who first proposed the problem in 1937. 

Start with any positive integer  a_0 , then apply the following recursion:

 a_n={1/2a_(n-1)   for a_(n-1) even; 3a_(n-1)+1   for a_(n-1) odd
The Collatz conjecture states the following: regardless of which positive integer is chosen initially, this sequence of numbers will always eventually reach 1.

For example, starting with 10 (which is even, so divide by 2) you get the following sequence of numbers (5,16,8,4,2,1). Or starting with 7 you get (22,11,34,17,52,26,13,40,20,10,5,16,8,4,2,1). Note that upon reaching 1 you then get (4,2,1,4,2,1...) indefinitely.  

Despite the apparent simplicity with which the problem is stated, there has been no proof that every positive integer will eventually reach 1 using this recursion. While there has been no proof that it works for every integer, it has been checked to work for every number up to 5.764×1018

The Collatz Conjecture has fascinated mathematicians for decades. Mathematician Paul Erdos even commented on the complexity of the problem by saying "Mathematics is not yet ready for such problems."

Monday, September 22, 2014

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Thursday, September 18, 2014

Throwback Fact of the Week - HP-35: The First Scientific Pocket Calculator - 9/18/14

HP-35: The First Scientific Pocket Calculator

In 1972, the Hewlett-Packard company (commonly known as HP) introduced the first ever scientific pocket calculator, called the HP-35. The term scientific meant that it was able to calculate trigonometric and exponential functions. The HP-35 was the first ever calculator to do these calculations; prior calculators only performed addition, subtraction, multiplication and division.

Before the introduction of the HP-35, trigonometric and exponential calculations were done using a slide rule, which were only accurate to a few significant figures. The fact that you may have never even heard of or seen a slide rule before is because the introduction of this calculator made slide rules obsolete.

An HP-35 Scientific Pocket Calculator
Image Credit: Seth Morabito 

The name HP-35 came from the 35 keys on the device. It was only anticipated to sell about 10,000 units in the first year. However, HP greatly underestimated the demand for such a device and ended up selling over 100,000 in the first year.

Thursday, September 11, 2014

Throwback Fact of the Week - Euler's Identity - 9/11/14

Euler's Identity

Euler's Identity is considered to be one of the most amazing and beautiful mathematical relationships ever discovered. The statement of the equality is as follows:

e^{i \pi} + 1 = 0

You may wonder why this equality is looked upon with such awe and admiration.

The identity seamlessly links some of the most important mathematical symbols: the number 0 (the additive identity), the number 1 (the multiplicative identity), the transcendental numbers π and e (Euler's number), and i (the imaginary unit). If this wasn't enough, it also makes use of three of the basic arithmetic operations: addition, multiplication and exponentiation.

The derivation of the identity follows from Euler's formula, which states:

e^{ix} = \cos x +  i\sin x \,\!

Evaluating this formula at x = π yields the identity. 

Kasner and Newman note, "We can only reproduce the equation and not stop to inquire into its implications. It appeals equally to the mystic, the scientist, and the mathematician."

Thursday, September 4, 2014

Throwback Fact of the Week - Wheat on a Chessboard - 9/4/14

Wheat on a Chessboard

This 800-year-old problem in mathematics is notable because of the fluency at which it illustrates the startling nature of geometric growth.

The legend goes that the Indian King, Shirham, asked his Grand Vizier, Sissa ben Dahir, what reward he would like for inventing the game of chess. The Grand Vizier asked for one grain of wheat for the first square of the chessboard, two grains for the second square, four grains for the third, eight grains for the fourth square, and so on for the sixty-four total squares. This appeared to the king to be a small and foolish request, so he granted the grand vizier's wish.

Little did the king know just how many grains of wheat would accumulate by the 64th square. This number would be the sum 1+2+22+23+…+263 = 264-1. This comes out to be 18,446,744,073,709,551,615 grains of wheat (over 18 quintillion!). Clearly the king would run out of grain long before this total. This is enough wheat to cover the entire earth several inches deep!

Watch the video below to get an illustration of this legend and the absurd amount of wheat!

Video Credit: Jill Britton