Monthly Archives: April 2013

Gauss: Great Mathematician or the Greatest Mathematician?

468px-Carl_Friedrich_GaussToday (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

 

Origami: Art and Math

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Origami is the Japanese art of paper folding. But it is more than an art form; it is a teaching tool and a field of mathematical study.

Origami is a great tool for teaching math.  Origami can increase spatial skills and help students understand many geometric concepts, from shapes and geometric forms, to more complicated concepts like intersecting planes, area, volume, symmetry, and mirroring images. Origami can even be used to teach number theory concepts like fractions and powers of 2. Best of all, origami is fun and creative, and kids love it.

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There are tons of online resources for learning origami and using math as an educational resources, especially on YouTube. Here are videos for origami folding and origami as a teaching tool.

In addition to being a fun tool for teaching math, origami has become its own field of mathematical study. Paper-folding can be used to solve mathematical problems (check out Vi Hart’s video on the origami proof of the Pythagorean Theorem), and math can be used to create incredibly intricate origami designs. In his TED talk, Robert Lang explains the mathematical “laws” behind origami, shows some amazing creations, and talks about how to go from an idea to an origami design using a program called TreeMaker that he offers free on his website. He also discusses some real-world applications that have grown out of the study of origami.

 

Women in Math: Recent Research

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image source: IGNACIOLEO

There have been several studies in the literature in the past few weeks about women in math.

I recently posted about the BYU study that found that while boys tend to do better in the first round of math competitions, girls do just as well or better than the boys in subsequent rounds. But there are a few other studies worth mentioning.

IU study: Feelings of power can diffuse effects of negative stereotypes: In this study, researchers used several tools to make women feel high, low, or neutral in power, and then gave these groups a math exam with instructions that either did or did not reinforce the stereotype that women are not good at math. They found that those who felt high in power performed better in math than those in both the low power and control group, despite stereotype reinforcing instructions.

Could Playing ‘Boys’ Games Help Girls in Science and Math? In this review of 12 studies, researchers found that in general men do better at spatial tasks than women. However, there was more variation found within each gender than between genders. Spatial ability seemed to be more associated with gender-identification than with biological gender. The authors suggest that, because spatial ability is refined through play and recreational activities, girls could improve spatial abilities through participating in activities that are stereotypically considered masculine.

More Career Options May Explain Why Fewer Women Pursue Jobs in Math and Science: In this study, researchers examined test scores and conducted surveys with 1,490 college-bound US students. The students were surveyed in 12th grade and again at 33 years old. The researchers found that among students with high math abilities, those who also had high verbal abilities, a group that contained more women than men, were less likely to have chosen a career in math and science than those with moderate verbal abilities. The authors suggest that having both high math and verbal abilities means having more career options, and that this may partly explain why fewer women enter these fields. “Because they’re good at both, they can consider a wide range of occupations.”

The First Computer is Older Than You Thought

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Antikythera Mechanism Reproduction
Mogi Vicentini

PBS recently aired a fascinating episode of NOVA, titled “Ancient Computer,” which details the discovery and study of a geared machine, dubbed the Antikythera Mechanism. Fragments of this machine, determined to be from around 100 BC, were discovered on the floor of the Mediterranean Sea off the coast of the Greek island of Antikythera in 1900. But, it wasn’t until a century later, that a team of scientists were able to use modern technology to unravel what the machine did and how it worked.

After much study, the scientists determined that this ingenious system of 30 (or more) gears was able to predict solar and lunar eclipses and the movement of the planets known at that time. It could even take into account the elliptical orbits of the moon and planets. This machine is believed to be the first analog computer.

This episode is currently available to view at PBS.org and on YouTube (embedded below).

And just for fun…

3 Points on Triangles

There are several easy demonstrations using cut-outs than can help show some of the properties of  triangles.

1. Area of a Triangle

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The area of a triangle equals 1/2 base times height. This is true for any type of triangle, but easiest to demonstrate for a right triangle. Simply use 2 identical right triangles, flip one upside down and put them together to form a rectangle (shown below) with Area = a x b). The area of the triangle is simply one half of this (1/2 a x b).

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It is slightly more complicated to show this with other triangles because the triangles will need to be rearranged a bit.  For these acute and obtuse triangles, draw a line for the height, which must form right angle with the base. For this example, assume that the longest side on the obtuse triangle is the base. Take one of the triangles and cut along this height line and then arrange the three pieces into a rectangle with Area = a x b (shown below).

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There is a second way to look at the obtuse triangle. You can use one of the shorter sides as the base. But, then you have to cut both triangles, as shown in the picture below.

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To see this demonstration as a computer animations check out these Animated Triangles.

2. Angles of a Triangle

The angles of a triangle always add to 180o (an angle of 180o is simply a straight line). This can be shown by taking three identical triangles and lining up the 3 angles as shown in the picture. You can also take 1 triangle and cut or tear off 2 of the angles and line them up with the third. See how they add up to a straight line? this works for any type of triangle.

TriangleAngles180Pictures

There is an interactive web tool showing angle sums here. There is also a Wolfram demonstration showing how this can be demonstrated with folding (you’ll need the free Wolfram CDF player to view).

3. Pythagorean Theorem

The Pythagorean Theorem applies to only right triangles. It states that the sum of the squares of the 2 sides adjacent to the right angle is equal to the square of the other side.  You may remember the formula a2 + b2 = c2. This can be demonstrated a couple of ways.

For the first demonstration, take 4 identical right triangles, arrange them into a square as shown below, leaving a diagonal square of side c in the center. Draw the outer square on the paper below, then rearrange the triangles to give 2 smaller squares of sides a and b. This shows that the area of the diagonal square (c2)  in the first arrangement is the same as the sum of the 2 squares (a2 +b2) in the second arrangement.

PythDemo1

You can see the same demonstration in the animation below.

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Source: Wikimedia Commons
Author: JohnBlackburne

You can also arrange the triangles to create a square of side c by putting the hypotenuse (the side opposite the right angle) along the outside edge of the square as shown in this animation:

Copyright © 1998 by Davis Associates, Inc.All Rights Reserved

Copyright © 1998 by Davis Associates, Inc.
All Rights Reserved

For the second demonstration, start with two squares of side a and b. (I’ve included the triangles in the picture below, so you can see that the lengths of the sides are the same.) Then draw diagonal lines to create right triangles of sides a and b – the easiest way to do this is to lay the triangles down on the squares as shown and draw a line along the triangle edge. Next, cut along these diagonal lines and rearrange the pieces to create a single square of side c. So, a2 + b2 becomes c2.

PythDemo2

There is also a nice Wolfram Demonstration: An Intuitive Proof of the Pythagorean Theorem

 

Cicadas: Mathematicians of the Insect World

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Source: Wikimedia Commons (Author: Bruce Marlin)

Cicadas are coming to the East Coast this year. This year marks the end of the 17-year life-cycle for this particular population of cicadas. Periodical cicadas (Magicicada septendecim) spend most of their lives underground sucking fluids from roots, but emerge after 13 or 17 years (depending on the population) to mate and then die.

What does this have to do with math?

13 and 17 are both prime numbers, and therefore cannot be divided evenly by any other numbers (besides 1 and themselves). Researchers believe that prime number life-cycles help the periodical cicadas avoid predators and parasites. For example, a cicada with a 17-year cycle and a parasite with a two-year cycle would meet only twice in a century (year 34 and year 68). A 17-year cicada would only meet a 5-year parasite once in a century, year 85 (5 x 17). Also, the prime number cycles keep different populations from meeting and interbreeding. Since the life-cycle is determined by genes, interbreeding between 13-year and 17-year cicada populations would throw off their clockwork cycle.

How do the cicadas know when to emerge? They count, of course. Cicadas feed on the roots of trees that flower every year. Scientists at University of California-Davis were able to make some cicadas hatch a year early by transplanting them onto potted trees and forcing the trees to flower twice in one year.

Read more at:

Science News for Kids: Prime Time for Cicadas

The Baltimore Sun: Mathematicians explore cicada’s mysterious link with primes