# Georg Cantor, Father of Sets

Today (March 3) is the birthday of Georg Cantor (1845-1918). Cantor was a German mathematician, best known as the inventor of set theory. Set theory is the branch of mathematics that studies collections of objects, known as sets. Cantor established the importance of one-to-one correspondence in sets and is responsible for the idea that “some infinities are bigger than others”. For example, the counting numbers (1, 2, 3, 4,…), known as natural numbers, is an infinite set. The real numbers is also an infinite set, but this set includes the natural numbers, as well as, negative integers, fractions, and irrational numbers, such as pi.  Each number in the set of natural numbers corresponds to the natural numbers within the set of real numbers, but the set of real numbers includes many other numbers. Therefore, even though both sets are infinite, the set of real numbers is larger.

If you watched the documentary about fractals that I shared a while ago, you may recognize that Cantor Set, illustrated below. It was one of the earliest fractals, known at the time as “monsters”. The Cantor Set is formed by taking a line, removing the middle third, then removing the middle third of the remaining lines, and so on. It would seem that eventually there would be nothing left, but there is always a set of remaining points, just ever smaller and smaller.

# Happy 450th Birthday Galileo!

Today (February 15, 2014) is the 450th anniversary of the birth of Galileo Galilei. Galileo made many contributions to science including describing the motion of pendulums and falling objects, his improvements to the telescope, his astronomical discoveries (including 4 of Jupiter’s moons), and his support of heliocentrism (the astronomical model wherein the planets move around the sun, described by Copernicus).

Until Galileo, scientists typically conducted qualitative studies, meaning they relied on descriptions of characteristics rather than on measured quantities or values. One of Galileo’s most important contributions to science was that he used mathematics to describe his scientific observations. He was one of the first thinkers to state that the laws of nature are mathematical. He also understood the relationship between mathematics and physics. He understood that the parabola is a conic section , a mathematical function where the y-value is a function of the square of the x-value (for example, y=x2), and the trajectory of a falling object. If you throw ball, it follows a parabolic path.

Galileo mostly applied the standard mathematics of the day to his scientific pursuits, but he did produce the Galileo’s paradox, which shows that there are as many perfect squares as there are whole numbers, even though most numbers are not perfect squares.

Galileo also proved that objects fall at the same rate, regardless of weight. Before that, people believed that heavier objects fell faster than lighter objects. Of course, we know that sometimes lighter objects fall more slowly, but this is due to air resistance. On the Apollo 15 mission, Commander David Scott showed that in the absence of air, a feather falls at the same rate as a hammer.

Legend has it, Galileo performed the falling object experiment by dropping objects off the Tower of Pisa. In the video below, Galen Weitkamp, professor of mathematics at Western Illinois University, explains some of the math describing falling objects.

# Benoit Mandelbrot

Benoit Mandelbrot and Mandelbrot Set

Today (November 20) is the birthday of Benoit Mandelbrot (1924-2010). Mandelbrot is known for his work in fractal geometry, specifically hIs “theory of roughness” and his discovery of the Mandelbrot set. Mandelbrot also coined the term “fractal”.

Mandelbrot was born in Poland, but his family fled to France to escape Nazi persecution. He studied mathematics at university in Paris, and then came to the U.S. to pursue a master’s degree in aeronatics at CalTech.  Mandelbrot returned to Europe for a time, but he then took a job at IBM in New York, where he worked for 35 years.

Other mathematicians had studied the mathematical sets we now know as fractals, but Mandelbrot had the advantage of access to IBM computers. Fractal sets are created by feeding a value into an equation, and then taking that result and feeding it back into the equation.  Up until the advent of computers, calculations had to be done by hand, limiting the number of iterations that were possible. Using computers, Mandelbrot was the first to do millions of iterations, creating his now famous Mandelbrot set.

Nova did a great show on fractals and Mandelbrot, which you may have seen in a previous post. But, if you missed it, hear it is again.

# August Ferdinand Möbius

November 17 is the birthday of August Ferdinand Möbius a German mathematician and astronomer, who lived from 1790 to 1868. Möbius, of course, lends his name to the Möbius strip (although it was also independently discovered by  Johann Benedict Listing around the same time). Other mathematical concepts sporting Möbius’ name include  the  Möbius configurationMöbius transformations, the Möbius transform, the Möbius function, and the Möbius inversion formula.

Möbius is most known for his work in mathematics, but he also published work in astronomy concerning the occultation of planets and the motion of celestial objects.

One of Möbius’ teachers was Carl Friedrich Gauss, whom some consider one of the greatest mathematicians of all time.Mobius served as chair of astronomy and higher mechanics at the University of Leipzig.

One of the first posts on the blog was about the Möbius strip. If you missed it check it out here: Möbius Strip: One-Sided Wonder. Also, check out More Möbius.

# Tomorrow is Ada Lovelace Day

Ada Lovelace is a international celebration aiming to raise the profile of women in science, technology, engineering and maths by encouraging people around the world to talk about the women whose work they admire

Ada Lovelace was an English mathematician and writer known for her work on Charles Babbage’s Analytical Engine, an early mechanical computer. She wrote what is recognized as the first algorithm intended to be processed by a machine, and, is often described as the world’s first computer programmer.

There are Ada Lovelace Day events worldwide. Check here to see if there is an event in your area.

# Andrew Wiles and Fermat’s Last Theorem

In 1995, Andrew Wiles proved Fermat’s Last Theorem, a problem that had stumped mathematicians for over 350 years.

Andrew Wiles (Left) and Pierre de Fermat (Right)

Never heard of Fermat’s Last Theorem? Well, to explain, let’s start with the Pythagorean Theorem. As you may know, the Pythagorean Theorem states that for right angle triangles, a2+b2=c2, where c  is the length of the hypotenuse, and a and b are the lengths of the other 2 sides. We know that there are a number of positive whole number solutions to the Pythagorean Theorem: 3, 4, and 5; 6, 8, and 10; etc.

In 1637, Pierre de Fermat, a French lawyer and amateur mathematician, conjectured that there are no positive whole number solutions for an+bn=cn for values of n>2. So, in other words, there are no positive whole number solutions for a3+b3=c3, or a4+b4=c4, etc. He claimed to have a proof, but did not have room in the margin of the book (Arithmetica) where he wrote this conjecture. For over 350 years, mathematicians tried to discover Fermat’s proof but were unsuccessful. Wiles finally proved the theorem, but he used some modern techniques that would have been unknown to Fermat, which makes ones wonder if Fermat had some other simpler proof that has yet to be discovered.

If you are interested in learning more about Fermat’s Last Theorem, there is a great book and a documentary (both by Simon Singh) that you should check out.

The book, Fermat’s Enigma, provides a detailed and interesting account of Andrew Wiles quest to solve Fermat’s Last Theorem, including explanations of the math appropriate for non-mathematicians. Singh gives us the entire story of Fermat’s Last Theorem starting from Fermat himself and covering the mathematicians along the way that attempted to solve the proof or provided important breakthroughs used by Wiles in his successful proof. Simon Singh wrote the book after producing a documentary about Wiles’ journey for the BBC. The documentary, Fermat’s Last Theorem does not provide as much detail, or course, but I love how, through the interviews with Wiles, you really get a sense of how much of himself he pored into this problem. This problem, which he first learned about from a library book at the age of 10, truly was his life’s work. You can hear the emotion in his voice when he describes how important it was to him and how he felt when he finally completed the proof. The documentary was also produced in the U.S. as The Proof by NOVA, but it isn’t available online. You can see the BBC version in full on YouTube (embedded below).

# Gauss: Great Mathematician or the Greatest Mathematician?

Today (April 30) is the birthday of Carl Friedrich Gauss (1777-1855), a German mathematician, considered by some to be the greatest mathematician in history. Gauss influenced many fields, including number theory, algebra, differential geometry, geophysics, electrostatics, astronomy, and optics.

Gauss was born to poor working-class parents in Brunswick, Germany. His mother, who was illiterate, never recorded his birthday, remembering only that he had been born on a Wednesday, eight days before the Feast of the Ascension. To solve the puzzle of his birthdate, Gauss had to derive methods to compute a date in past and future years.

Gauss’ genius was evident at an early age, and there are several anecdotes about his precocity. According to one legend, at age 3 he corrected, in his head, an error his father had made while calculating finances. According to another famous story, one of his primary school teachers gave him the task of adding the numbers 1 to 100, as a punishment for misbehavior. The teacher was astonished when Gauss produced the answer a few seconds later. Gauss realized that he could add the numbers as pairs from opposite ends of the list (1+100, 2+99, 3+98, etc.), and that each of these pairs equals 101. Since there are 50 of these pairs, the addition of all numbers from 1 to 100 must equal 50 x 101 or 5,050.

Some of Gauss’ contributions:

• He proved the fundamental theorem of algebra, which states that every polynomial has a root of the form a+bi. (Although his proof had a gap that was filled in 1920 by Alexander Ostrowski.)
• In 1801, he proved the fundamental theorem of arithmetic,  which states that every natural number can be represented as the product of primes in only one way.
• At age 24 (in 1801), Gauss published Disquisitiones Arithmeticae, considered one of the most brilliant achievements in mathematics. In this publication, Gauss systematized the study of number theory (properties of the integers), proved that every number is the sum of at most three triangular numbers, and developed the algebra of congruences.
• Also in 1801, G Piazzi, an Italian astronomer, discovered Ceres, a dwarf planet. Piazzi had only been able to observe 9 degrees of its orbit before it disappeared behind the Sun. When Ceres was rediscovered later that year, it was almost exactly where Gauss had predicted. Gauss made the prediction based on his least squares method of approximation. Gauss published a theory of the motion of planetoids disturbed by large planets in 1809. His work was such an improvement over the cumbersome mathematics of 18th century orbital prediction that his work remains a cornerstone of astronomical computation.

Things named in honor of Gauss include:

• Degaussing, the process of eliminating a magnetic field
• The CGS unit for magnetic field was named gauss
• The crater Gauss on the Moon
• Asteroid 1001 Gaussia
• The ship Gauss, used in the Gauss expedition to the Antarctic
• Gaussberg, an extinct volcano discovered by the above mentioned expedition
• Gauss Tower, an observation tower in Dransfeld, Germany

Sources:

Carl Friedrich Gauss. Wikipedia. Retrieved April 30, 2013, from http://en.wikipedia.org/wiki/Carl_Friedrich_Gauss

Gauss, Karl Friedrich. Scienceworld.Wolfram.com. Retrieved April 30, 2013, from http://scienceworld.wolfram.com/biography/Gauss.html

Johann Carl Friedrich Gauss. Retrieved April 30, 2013, from http://www-groups.dcs.st-and.ac.uk/history/Biographies/Gauss.html