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10 Cool Mathematics Inventions

Many people find math to be mind boggling, useless, boring, and/or mental abuse to humans. Math can be quite the opposite if you forget about these preconceived notions and draw your own conclusions from personal experience. The world is made up of math and it brings meaning to our world. You see it and use it everyday. Yep! The house you live in, restaurants, the malls and stores you visit, so on and so forth were all built by the construction segment of industry employment which includes construction laborers, worksite supervisors, brickmasons, stonemasons, carpet/tile installers, cement masons, equipment operators, drywall installers, electricians, glaziers, insulation workers, painters, plumbers, plasters, rebar workers, roofers, and sheet metal workers whom had to have knowledge of math. Therefore, math is actually quite beautiful isn't it?

1. Pythagoras Theorem

Pythagoras: a2 + b2 = c2

His full name is Pythagoras of Samos and he is considered "the father of numbers."  He is best known for the widely used formula known as the Pythagorean Theorem. So it is only right to name his own invention after him :)

What is it?

The Pythagorean theorem deals with the lengths of the sides of a right triangle. The theorem states that: The sum of the squares of the lengths of the legs of a right triangle ('a' and 'b' in the triangle shown below) is equal to the square of the length of the hypotenuse ('c').

When would I use the Pythagorean theorem?

Dr. Math provides an excellent real life example.

2. Euler's Formula

It appears that Leonard Euler (1707-1783) was the first person to notice the fact that for convex 3-dimensional polyhedra V - E + F = 2.

Euler mentioned his result in a letter to Christian Goldbach (of Goldbach's Conjecture fame) in 1750. He later published two papers in which he described what he had done in more detail and attempted to give a proof of his new discovery.

V- E + F = 2

where

V = number of vertices
E = number of edges
F = number of faces

Buckyball

60 - 90 + 32 = 2

3. The 4-Color Theorem

The four-color theorem states that any map in a plane can be colored using four-colors in such a way that regions sharing a common boundary (other than a single point) do not share the same color. This problem is sometimes also called Guthrie's problem after F. Guthrie, who first conjectured the theorem in 1852. The conjecture was then communicated to de Morgan and thence into the general community. In 1878, Cayley wrote the first paper on the conjecture.

Fallacious proofs were given independently by Kempe (1879) and Tait (1880). Kempe's proof was accepted for a decade until Heawood showed an error using a map with 18 faces (although a map with nine faces suffices to show the fallacy). The Heawood conjecture provided a very general assertion for map coloring, showing that in a genus 0 space (including the sphereor plane), four colors suffice. Ringel and Youngs (1968) proved that for genus , the upper bound provided by the Heawood conjecture also give the necessary number of colors, with the exception of the Klein bottle (for which the Heawood formula gives seven, but the correct bound is six).

Six colors can be proven to suffice for the  case, and this number can easily be reduced to five, but reducing the number of colors all the way to four proved very difficult. This result was finally obtained by Appel and Haken (1977), who constructed a computer-assisted proof that four colors were sufficient. However, because part of the proof consisted of an exhaustive analysis of many discrete cases by a computer, some mathematicians do not accept it. However, no flaws have yet been found, so the proof appears valid. A shorter, independent proof was constructed by Robertson et al. (1996; Thomas 1998).

In December 2004, G. Gonthier of Microsoft Research in Cambridge, England (working with B. Werner of INRIA in France) announced that they had verified the Robertson et al. proof by formulating the problem in the equational logic program Coq and confirming the validity of each of its steps (Devlin 2005, Knight 2005).

J. Ferro (pers. comm., Nov. 8, 2005) has debunked a number of purported "short" proofs of the four-color theorem.

Summary. William Playfair, engineer, political economist and scoundrel, was the most important developer of statistical graphics. In the two centuries since, there has been no appreciable improvement on his basic designs.

William Playfair was born at Liff in Scotland during the Enlightenment - a Golden Age in the arts, sciences, industry and commerce. He died in London, in 1823, after an eventful life, though unmarked at the time by any apparently significant or memorable contribution. He made little impression in his native land, and his impact was only slightly greater in England and France. Yet he is responsible for inventions familiar and useful to us all: he was the first to devise and publish all of the common statistical graphs - the pie chart, the bar chart, and the statistical line graph. He invented a universal language useful to science and commerce alike and though his contemporaries failed to grasp the significance, Playfair had no doubt that he had forever changed the way we would look at data. Over a span of thirty six years he published many books and pamphlets containing statistical charts and though this work was received with indifference, even hostility, he never faltered in his conviction that he had found the best way to display empirical data. However, it took almost a century after his death before his invention was fully accepted.

5. World War II Cryptography

By World War II, mechanical and electromechanical cipher machines were in wide use, although—where such machines were impractical—manual systems continued in use. Great advances were made in both cipher design andcryptanalysis, all in secrecy. Information about this period has begun to be declassified as the official British 50-year secrecy period has come to an end, as US archives have slowly opened, and as assorted memoirs and articles have appeared.

The Germans made heavy use, in several variants, of an electromechanical rotor machine known as Enigma. Mathematician Marian Rejewski, at Poland's Cipher Bureau, in December 1932 deduced the detailed structure of the German Army Enigma, using mathematics and limited documentation supplied by Captain Gustave Bertrand of French military intelligence. This was the greatest breakthrough in cryptanalysis in a thousand years and more, according to historian David Kahn. Rejewski and his mathematical Cipher Bureau colleagues, Jerzy Różycki andHenryk Zygalski, continued reading Enigma and keeping pace with the evolution of the German Army machine's components and encipherment procedures. As the Poles' resources became strained by the changes being introduced by the Germans, and as war loomed, the Cipher Bureau, on the Polish General Staff's instructions, on 25 July 1939, at Warsaw, initiated French and British intelligence representatives into the secrets of Enigma decryption.

Soon after the Invasion of Poland by Germany on 1 September 1939, key Cipher Bureau personnel were evacuated southeastward; on 17 September, as the Soviet Union attacked Poland from the East, they crossed into Romania. From there they reached Paris, France; at PC Bruno, near Paris, they continued breaking Enigma, collaborating with British cryptologists at Bletchley Park as the British got up to speed on breaking Enigma. In due course, the British cryptographers - whose ranks included many chess masters and mathematics dons such as Gordon WelchmanMax Newman, and Alan Turing (the conceptual founder of modern computing) - substantially advanced the scale and technology of Enigma decryption.

German code breaking in World War II also had some success, most importantly by breaking the Naval Cipher No. 3. This enabled them to track and sink Atlantic convoys. It was only Ultra intelligence that finally persuaded the admiralty to change their codes in June 1943. This is surprising given the success of the British Room 40 code breakers in the previous world war.

At the end of the War, on 19 April 1945, Britain's top military officers were told that they could never reveal that the German Enigma cipher had been broken because it would give the defeated enemy the chance to say they "were not well and fairly beaten"

text from wikipedia

6.First Mechanical Calculator

Schickards's Calculator

A 1623 letter from William Schickard to astronomer Johannes Kepler is the only surviving record of Schickard’s calculator.

Schickard combined Napier’s Bones, for multiplication and division, with a toothed-wheel system to add and subtract. It is the earliest known mechanical four-function calculator.

Schickard built two calculators around 1623. One, for his astronomer friend Johannes Kepler, was destroyed by fire. The other is, as far as we know, lost. We know about them only from Schickard’s handwritten letters, which contain sketches of what he had built:

A sketch of Schickard's calculator

The Pascaline

Blaise Pascal was a philosopher, mathematician…and good son. In the 1640s, he invented for his father, a tax collector, a machine that could add.

Pascal built about 50 Pascalines, making it the first adding machine produced in even modest quantity—though he only sold about 15.

7.Euler's Equation

When you first look at Euler's equation it seems like simple math: e^(i*pi)+1=0. What makes Euler's equation so beautiful is how he was able to unify math's most important numbers (e, i, pi, 0, and 1) into an equation.

8.Format's Last Theorem

Fermat's last theorem is a theorem first proposed by Fermat in the form of a note scribbled in the margin of his copy of the ancient Greek text Arithmetica by Diophantus. The scribbled note was discovered posthumously, and the original is now lost. However, a copy was preserved in a book published by Fermat's son. In the note, Fermat claimed to have discovered a proof that theDiophantine equation  has no integer solutions for  and .

The full text of Fermat's statement, written in Latin, reads "Cubum autem in duos cubos, aut quadrato-quadratum in duos quadrato-quadratos, et generaliter nullam in infinitum ultra quadratum potestatem in duos eiusdem nominis fas est dividere cuius rei demonstrationem mirabilem sane detexi. Hanc marginis exiguitas non caperet" (Nagell 1951, p. 252). In translation, "It is impossible for a cube to be the sum of two cubes, a fourth power to be the sum of two fourth powers, or in general for any number that is a power greater than the second to be the sum of two like powers. I have discovered a truly marvelous demonstration of this proposition that this margin is too narrow to contain."

As a result of Fermat's marginal note, the proposition that the Diophantine equation

 (1)

where , and  are integers, has no nonzero solutions for  has come to be known as Fermat's Last Theorem. It was called a "theorem" on the strength of Fermat's statement, despite the fact that no other mathematician was able to prove it for hundreds of years.

9.Euclid's Postulates

1. A straight line segment can be drawn joining any two points.

2. Any straight line segment can be extended indefinitely in a straight line.

3. Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center.

4. All right angles are congruent.

5. If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough. This postulate is equivalent to what is known as the parallel postulate.

Euclid's fifth postulate cannot be proven as a theorem, although this was attempted by many people. Euclid himself used only the first four postulates ("absolute geometry") for the first 28 propositions of the Elements, but was forced to invoke the parallel postulate on the 29th. In 1823, Janos Bolyai and Nicolai Lobachevsky independently realized that entirely self-consistent "non-Euclidean geometries" could be created in which the parallel postulate did not hold. (Gauss had also discovered but suppressed the existence of non-Euclidean geometries.)

10.Game Theory

Image from Fakescience.tumblr.com

Game theory is a branch of mathematics that deals with the analysis of games (i.e., situations involving parties with conflicting interests). In addition to the mathematical elegance and complete "solution" which is possible for simple games, the principles of game theory also find applications to complicated games such as cards, checkers, and chess, as well as real-world problems as diverse as economics, property division, politics, and warfare.

Game theory has two distinct branches: combinatorial game theory and classical game theory.

Combinatorial game theory covers two-player games of perfect knowledge such as gochess, or checkers. Notably, combinatorial games have no chance element, and players take turns.

In classical game theory, players move, bet, or strategize simultaneously. Both hidden information and chance elements are frequent features in this branch of game theory, which is also a branch of economics.

• Game theory under the microscope: These articles take an in-depth look at game theory, including equations and calculations.
• Game theory in the news: These articles report on news about game theory and its applications, giving you an overview of what can be done with it and how.
• Game theory in the real world: is any theory designed to calculate the maximum efficacy of a players strategy. It is also believed to be applicable to real world situations.

Resources:

https://people.math.osu.edu/fiedorowicz.1/math655/Euler.html

Howard Wainer. "William Playfair" (version 3). StatProb: The Encyclopedia Sponsored by Statistics and Probability Societies. Freely available at http://statprob.com/encyclopedia/WilliamPLAYFAIR.html

Weisstein, Eric W. "Fermat's Last Theorem." From MathWorld--A Wolfram Web Resource.http://mathworld.wolfram.com/FermatsLastTheorem.html

Weisstein, Eric W. "Four-Color Theorem." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Four-ColorTheorem.html

Weisstein, Eric W. "Game Theory." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/GameTheory.html

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Notes

Figure This Challenge #56

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Created by Wanda Collins May 10, 2015 at 1:56pm. Last updated by Wanda Collins May 10, 2015.

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Math Limerick

Question: Why is this a mathematical limerick?

( (12 + 144 + 20 + 3 Sqrt[4]) / 7 ) + 5*11 = 92 + 0 .

A dozen, a gross, and a score,
plus three times the square root of four, divided by seven, plus five times eleven, is nine squared and not a bit more.

---Jon Saxton (math textbook author)

Presentation Suggestions:
Challenge students to invent their own math limerick!

The Math Behind the Fact:
It is fun to mix mathematics with poetry.

Resources:

Su, Francis E., et al. "Math Limerick." Math Fun Facts.

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