Throwback Fact: The Birthday Problem

image: www.flickr.com

In 1939, mathematician Richard von Mises proposed a problem. The answer was so counter-intuitive to many people, including mathematicians across the globe, that it continues to be one of the most discussed math problems ever. It is very difficult for people to understand, not because the mathematics behind it are hard, but just because the answer is so different from what anyone would expect. That problem is stated as follows:

Suppose you are in a large room that random people generally enter. How many people must be in the room before the probability that some share a birthday becomes at least 50 percent?

Counting Systems: Ancient Babylon

One of the most fascinating parts of math, for me, is the history behind it. Math is one of those things that shows up in every time period, no matter what civilization was in power or what languages were spoken. As a miniseries, I'd like to share some ancient counting systems and facts about math at that time. First up is the math of the Ancient Babylonians.

The Region: Babylonia was name of the cultural region in the fertile crescent, stretching from the Persian Gulf to modern-day Baghdad and west to the Mediterranean Sea. Their history dates back as far as 4000 BCE with the Sumerians, though very little is known about the Sumerian culture.

The Babylonian civilization rose and fell for millenia, and their culture can be classified into several distinct periods. The mathematical records that have survived come from two different periods: the First Babylonian Dynasty period (1800s - 1500s BCE) and the Selucid period (400 - 0 BCE). Very few documents fall between those two periods. The main focus of researchers are documents from the First Babylonian Dynasty, but the second period is interesting because it overlaps with great mathematicians from Ancient Greece.

One particularly famous Babylonian was Hammurabi, the king of Babylon for much of the 18th century BCE. Hammurabi's Code is the oldest known form of constitution for a government, and the stone on which it is written is the longest surviving work from the Old Babylonian period.

The Numerals: The scholars of Babylonia used cuneiform numerals to write their numbers. They pressed the wedge-shaped end of a reed into soft clay, and hardened it into stone tablets. The numbers are written from left to right like modern Latin numerals, but that is where the similarities end.

Problem of the Week

Happy Tuesday from the Center of Math team! Our Problem of the Week is coming to you one day later than usual because we celebrated Memorial Day yesterday. Here's the problem, which we posted on our Facebook, Twitter, and Google+ accounts just a little while ago:

As usual, I've provided my solution to this geometry problem below...

3 Memorial Day Mathematicians

image: www.nj.com

For the Center of Math, along with our United States fanbase, today is a public holiday. Many Americans celebrate with a backyard barbeque, or a day at the beach, but it's important to remember what Memorial Day is really for!

Of course, we're celebrating here by honoring a few math-minded people who have served in the United States armed forces. While there are surely many mathematicians who have served their countries all around the world (Alan Turing comes to mind), here are three Americans that you may not have heard of...

Throwback Fact: The Origins of Pi

We've mentioned pi quite a few times on the blog (our pi day post in particular). However, we've never discussed the reasoning behind the symbol.

Pi, of course, has existed in approximations since the time of the Babylonians (who got to 28/8 = 3.125) and the Ancient Egyptians (who found (16/9)^2 = 3.1605 on the Rhind Papyrus). Approximations have become better and better over time, but it will never end. Pi is transcendental- it is irrational, and cannot be found as the result of an algebraic function. We can continue to calculate more and more digits, but the number of trailing digits is an uncountable infinity, and therefore we will never know exactly what pi is.

Problem of the Week

 Happy Monday from the Center of Math team! Here's our newest Problem of the Week, which we posted to our social media accounts just a little while ago:

Click to expand!

A geometry and trig problem! We haven't seen one of these in a while. There are several ways to solve this problem, and I'll show just one of them below...

Throwback Fact: Quaternions

A digital representaion of a Julia set quaternion

In 1843, Irish mathematician Sir William Hamilton was walking across Dublin's Brougham Bridge when inspiration struck him. He had been working on finding a method to multiply together points in space (three dimensions) instead of just points on a plane (two dimensions.) On his walk with his wife that morning in october, he realized that while he could not "multiply triples," he could multiply points in four dimensions. He carved this basic rule for multiplication into the bridge:

i² = j² = k² = ijk = -1

Hamilton named a quadruple of this type a quaternion and he devoted the rest of his career to their study. The simplest definition of such a number is a four-dimensional number. Quaternions are still studied today, and present a number of uses in modern mathematics. According to The Math Book by Clifford Pickover, they've been used to describe the dynamics of motion in three dimensions, and have been applied to fields including virtual reality computer graphics, game programming, robots, bioinformatics, and flight software of spacecrafts.

The plaque on Brougham Bridge in Dublin, Ireland

The invention, or discovery, of quaternions represents a moment of great ingenuity in math history. You can actually still visit Brougham (Broom) Bridge today, and see the plaque commemorating Hamilton's discovery. There is even an annual commemorative walk following Hamilton and his wife's path on that fateful morning. Perhaps we should have added this location to our list of math destinations!

One more fun fact about quaternions is that they generated such polarizing responses from the math community. Scottish physicist William Thomson(1824 - 1907) considered them an "unmixed evil to those who have touched them in any way," while mathematician Oliver Heaviside thought of their invention as a feat of human ingenuity. Curiously, one mathematician (who gained his fame from a harshly different circumatance) who studied quaternions was Theodore Kaczynski, the "Unabomber."

4 Examples of Math in Art

We at the Center of Math have been inspired in unforeseen ways by this beautiful weather. Our artistic sides are out and about! So, we've compiled this list of greatly varied pieces of art that are influenced by mathematics.

http://www.mcescher.com/gallery/mathematical/study-for-stars/

Study for Stars (Escher)- M. C. Escher was a Dutch artist born in 1898, and is most known for his drawings and woodcuts of mazes and seemingly-impossible objects. Escher’s work lives on today as one of the inspirations for the burgeoning field of graphic design. 

A Center of Math Milestone

Thank you, thank you, thank you.

The Center of Math has surpassed 25,000 followers on Twitter!

As a small textbook publisher, social media means everything to our company. We use Twitter (as well as Facebook, Youtube, Google+, Instagram...) to get our name out there. We put ourselves out there with around two posts per day to create brand awareness, and to get our videos, textbooks, and math resources within reach of students all over the world. Reaching 25K followers is a huge milestone on our path to becoming the go-to mathematics resource on the internet.

Here are some fun facts about our Twitter account:

Problem of the Week

Happy Monday from the Center of Math! I hope you've all been looking forward to the Problem of the Week as much as I looked forward to writing it. The problem today is rather interesting; take a look just below:

Click on the picture to expand!

 So we have a lot of variables to work with and not many solid numbers. This is the perfect ground to look for patterns. See my solution below, AND a solution by one of our Twitter followers...

Mother's Day Special: Variation of the Simple Marriage Problem

One of my favorite things to do at the Center is create video content for fun math topics. The intern here before me did two "Holiday Specials" around Thanksgiving and Christmas, and I decided to continue his work! One of my first math videos was the Valentine's Day Special: Monty Hall problem. That one was a lot of fun.

This Sunday is Mother's Day (hi Mom!), so I took a math topic that I think is fun and themed it to fit the holiday. The Stable Marriage Problem is the famous mathematical scenario that we chose. Instead of a small village that has 10 women and 10 men who need to be paired in matrimony, we used mothers and the presents that their children picked out for Mother's Day.

Throwback Fact: Perfect Numbers

Tori's handwriting demonstrates a few properties of the first two perfect numbers

We've mentioned the cult of Pythagoras and his followers a few times in the past. And we know that their mathematical findings did not stop at the Pythagorean theorem. One mathematical idea that fascinated the group was perfect numbers.

A perfect number is defined as a positive integer that is equal to the sum of its positive divisors, excluding itself. The definition appeared as early as Euclid's Elements, and only four of these numbers had been discovered before the Renaissance. The discovery process was so slow because these numbers are inredibly rare- the first is 6. It's a relatively small number, and makes the viewer think that the numbers may common. The next perfect number is 28, and then 496. A Greek mathematician (circa 100 CE) named Nicomachus is credited with the discovery of the fourth perfect number: 8128. Then, more than a millenium passed before an unknown mathematician recorded the fifth perfect number, 33,550,336, for the first time.

Cinco de Mayo Mathematician Highlight

After doing a little research, I’ve discovered that Cinco de Mayo is a holiday celebrated possibly more in the United States than in Mexico, its country of origin. Cinco de Mayo began as the celebration of a battle- it commemorates the Mexican army’s victory over France at the Battle of Puebla in 1862. This victory was particularly important in the Franco-Mexican war because it was an impossible feat as the Mexicans won against a much better equipped army.

In the United States, it is often utilized by Mexican Americans as a celebration of Mexican culture, including festivals and parades in some cities. In typical Center of Math fashion, we will use this opportunity to feature a prominent Mexican mathematician, Fray Diego Rodriguez!

5 Movies that Every Math Person Should See

We love movies. We love math. But our favorite thing is when those two worlds are combined. In the list below, we've done our best to curate a collection of math movies that everyone will love.

Pi – Any list of math movies has to begin with Darren Aronofsky’s directorial debut from 1998, Pi. Starring Sean Gullette as mathematician and number theorist Max Cohen, Pi tells a story of one man’s descent towards insanity, and the dark side that comes with pursuing a lifelong passion. Max is hell-bent on finding a way to “crack the code” of nature through the use of mathematics, and will stop at nothing until he can accomplish his goal. Max’s quest takes him from the stock market and financial world to studying Gematria, an ancient Jewish tradition in which letters are given numerical values. This film sticks with you for a long time after you’ve seen it, and really does make viewers wonder if there is a code to crack the world.

Problem of the Week: Star Wars

Happy Monday, and a good May the 4th to any Star Wars fans in our audience! I discovered this problem while browsing the internet a few months ago, and I'm glad some simple googling led me to find it again. While this is not a particularly challenging Problem of the Week for our more advanced college and beyond followers, it's still a fun review of algebra!

I've transcribed the word problem that I found in this article. Isn't it great that such a young math student created this problem? I posted it on our Facebook, Twitter, and Google+ pages, and I've included my solution below.

Click to make the picture larger!

 This word problem is just that: wordy. Beneath the Star Wars jargon is an algebra problem similar to one you could find in the SAT math section, or on an algebra test. See my solution below...