Haec dedicatio est causa celebrantis Caroli Frederici Gauss, Principis Mathematicorum, hac die pridie kalenda Maia, MMVIII.
Gauss fuit unus trium mathematicorum maximorum qui mirantem prudentiam mathematica habuit. Natus MDCCLXXVII, Brunswick, Germania, ingenium atque tantum tres anni aetate demonstravit. Dum suus pater inceptum mathematica studente recusavit ingeniumque mathematicae ingoravit, ipse studio mathematica cum auxilio matri et Carolo Guilielmo Ferdinando praevaluit. In mathematica, omne area Gauss fuit dexter. Etiamsi hoc, arithmeticam reginam mathematicae nominavit; mathematicam, reginam scientiam, et fortasse maximam scriptarum operarum Disquisitionem Arithmeticae scripsit. Non modo ipse fuit mirabilis mathematicus, sed etiam mirabilis astronomus, ac rectorem observatorii Gottingene factus est. Fortasse in astronomia maximum factum fuit computatio ambitus Cereris. Fuit nihil mathematica in illo die quod Gauss non facere potuit. Quicquid aequales comperiverunt, ipse iam comperiverant--et plus. Multa haec in mathematica ephemere quod etiam iam difficile lectu tentus sunt. Tantum fuit studium mathematicae ut recognoscens idem factum ceteris ipsi pauca interfuit famaque fortuna non coegerunt. Gauss fuit dexter, non modo in mathematica, sed etiam in lingua. Enim non modo Germanam linguamque Latinam linguam sed etiam multam linguam Europa didicit, et cum sexaginta duo anni aetate fuerit, Russiam linguam conatus est, ac paucis annis, omnino hanc linguam perdidicit, ipseque bene legit ac scripsit. Talis fuerunt facta tanti viri ut Gauss. Et tamen, alicui ut dextro ut ipsi, cuius scivit paucum edidit. Sua sententia fuit "pauca sed matura", et arbitratus est quid esse levis aequalesque quid potuerunt uti non edidit. Etiam cum scivit ceteros suas sententias non intellexerit non edidit. Gauss mortuus est ante diem septem kalenda Martia, MDCCCLV, Gottingene, Germania. Postea Principem Mathematicorum nominatus est.
Tamen, etiamsi magnum ingenium Caroli Frederici Gauss in mathematica, ac honorque reverentia quas ipse meruit, vita asperior aut tristior ipsi non esse potuit. Nam sua uxor, quam quam aliquas in orbe terrarum amavit modo quinque annis post conubium vixit, tum mense post tertium infantem natum mortuus est, fotusque Carolus Guilielmus Ferdinandus saeve Napoleone occisus est, paucosque amicos quod magnum ingenium, tantum super ceterorum solitudinemque hostesque quod facta ceterorum non recognovit habuit, melancholiamque ac hypochondriam ac alios morbos senectute passus est. Mathematica fuit tantum suum solaciumque medicamen. Et tamen, nihil horum facientem magnum virum impedivit. Nam ipse fuit magnus, non modo in mathematica ac ingenio, sed etiam in more. Humanusque lenis, placidusque mollis fuit, certaminaque pugnas non amavit, religiosamque intolerantiam non praedicavit, tamen, potens in animo ac ingenio.
Id est magna cum honore quo anniversaria dedicatio duocentesimus tricesimus primus Carolo Frederico Gauss celebramus, viro cuius moresque ingenium sunt dignus celebrantis. Ita huic occasioni, hanc effigiem Caroli Frederici Gauss cum uno eius noto dicto, "Si ceteri veri mathematica meditentur ut penitus ac perpetuo meditatus sim, mea inventa faciant." ac "Regina scientiae est mathematica, ac regina mathematicae est arithmetica." traxi.
This dedication is to commemorate Carl Friedrich Gauss, Prince of Mathematicians, on this day of April 30, 2008.
Gauss was one of the three greatest mathematicians, who had an amazing insight in mathematics. Born in 1777, Brunswick Germany, he displayed his talent as soon as he was only three. While his father resented his attempts in studying mathematics and ignored his genius in mathematics, he prevailed in his study for mathematics with the help of his mother and Duke Carl Wilhelm Ferdinand. Gauss was well versed in every field in mathematics. Despite this, he dubbed arithmetics queen of mathematics; mathematics was queen of science, and he wrote the Disquisitiones Arithmeticae, perhaps the greatest of his written works. Not only was he an extraordinary mathematician, but also an extraordinary astronomer, and became the director of the observatory in Gttingen. Perhaps his greatest achievement in astronomy was his calculation of the orbit of Ceres. There was nothing in mathematics in his day which Gauss could not do. Whatever his contemporaries discovered, he had already discovered--and more. Many of these were kept in his mathematics journal which even today are difficult to read. So great was his devotion to mathematics that he cared little to recognize the same achievements of others, and fame and fortune did not drive him. Gauss was skilled, not only in mathematics, but also in languages. For he learned not only German and Latin, but also many European languages, and when he was 62, he tried Russian, and in a few years, had mastered the language completely, and read and wrote it well. Such were the achievements of so great a man as Gauss. And yet, for someone as skilled as himself, he published little of what he wrote. His motto was "few but ripe", and did not publish what he thought to be trivial and what his contemporaries could use. When he knew that others would not understand his ideas, he also did not publish. Gauss died on February 23, 1855 in Gttingen, Germany. Afterwards, he was dubbed the Prince of Mathematicians.
And yet, despite the great genius of Carl Friedrich Gauss in mathematics, and the honor and respect he deserved, life could not have been more harsh or more sad for himself. For his wife, whom he loved more than anyone else in the world lived only five years after marriage, then died a month after his third child was born, and the cherished Carl Wilhelm Ferdinand was brutally killed by Napoleon, and he had few friends because of his great genius, so much above that of others and isolation, and made enemies because he did not recognize the achievements of others, and suffered melancholia, hypochondria and other illnesses in his later life. Mathematics was his only solace and drug. And yet, none of these things prevented him from becoming a great man. For he was great, not only in mathematics and genius, but also in character. He was kind and gentle, calm and sensitive, and did not like conflicts, and did not preach religious intolerance, yet, he was strong in spirit and intellect.
It is with great honor that we commemorate the 231st anniversary dedication to Carl Friedrich Gauss, a man whose character and genius are worthy of commemorating. So, for this occasion, I have drawn these pictures of Carl Friedrich Gauss with one of his well known quotes, "If others would reflect on mathematical truths as deeply and continuously as I have, they would make my discoveries." and "Mathematics is the Queen of Sciences, and arithmetic, the Queen of Mathematics."http://img.photobucket.com/albums/v405/btl/Gauss_sketch.jpghttp://img.photobucket.com/albums/v405/btl/Gauss_J_color.jpg
Here also is a Gauss Widget
for those using a mac (OS X Tiger or higher).
And also commemorating this event, are some cool math problems I've encountered on exams, from other people, and on my own, which I've successfully solved on my own. You'll find some of these easy, and some of them hard, although I included the easier ones because they were still interesting. I may choose to post the solutions later. I've divided them into three sections, the Pidgeonhole Principle, Logic and Proof Related (you'll find the logic stuff the easiest).
For those not familiar, the pidgeonhole principle is the idea that if you are given n sets and m number of elements, where m>n, then there is at least one set which contains at least m/n elements rounded to the nearest integer (up).
1) Socks in a Drawer
A drawer contains 6 black socks, 8 white socks, and 10 blue socks. One night you need to get some socks but the lights go out.
Without being able to look inside the drawer, what is the smallest amount of socks that you need to take from the drawer to guarantee that you have picked up a pair of identically colored socks?
What is the smallest number of socks you need to take from the drawer to guarantee that you have picked up 5 socks of one color?
What is the smallest number of socks you need to take from the drawer to guarantee that you have picked up 5 blue socks?
1) Five people, Five Occupations, and Two Genders
Five people with family names Dow, Elliot, Finley, Grant and Hanley have exactly one of the following occupations: appraiser, broker, cook, painter and singer.
Use the following information to determine each person's sex and occupation.
-Three of these five people are men.
-The broker and the appraiser are father and son.
-The singer, the appraiser and Grant like Italian food.
-Dow and Hanley are playing in a ladies golf tournament.
-The singer told Finley that he likes hockey.
-The cook owes Hanley $50.
2) The Prisoners in The Tower
An elderly queen, her daughter, and little son, weighing 195 pounds, 105 pounds and 90 pounds respectively, were kept prisoners at the top of a high tower. The only communication with the ground below was a rope passing over a pulley with a basket at each end, and so arranged that when one basket rested on the ground the other was opposite the window. Naturally, if the one were more heavily loaded than the other, the heavier would descend; but if the excess on either side was more than 15 pounds, the descent became so rapid as to be dangerous, and from the position of the rope the captives could not check it with their hands. The only thing in the tower to help them was a cannon ball, weighting 75 pounds. They, not withstanding, contrived to escape.
How did they manage it?
3) A Traditional River Crossing Problem
Three men, traveling with their wives, came to a river that they wished to cross. There was but one boat, and only two could cross at one time. Since the husbands were jealous, no woman could be with a man unless her own husband was present. In what manner did they get across the river?
Bonus: Try this problem with four or more couples.
4) Handshakes at Dinner
Four couples attended a dinner. At the beginning of the dinner, they shook hands (a couple does not shake hands with each other, and one does not shake hands with oneself). When the host asked how many people shook hands, there were seven different answers. If you know how many people shook hands with the host, how many people shook hands with the host's wife?
1) Laplace Equation
Verify that if u(x,y) and v(x,y) are two functions that satisfy the Laplace equation, ∇²f = f_xx + f_yy = 0 and ∇u∙∇v = 0, then the function uv(x,y) = u(x, y)v(x, y) satisfies the Laplace equation.
2) Ceva's Theorem + Area of a Triangle in a Triangle
Consider a triangle ABC. Take points A₁, B₁, C₁ such that A₁ ∈ BC and BA₁: A₁C = 2:1, B₁ ∈ CA and CB₁: B₁A = 2:1, C₁ ∈ AB and AC₁: C₁B = 2:1. Consider the triangle which sides belong to the lines AA₁, BB₁, CC₁. Prove that its area is equal to 1/7 of the area of the triangle ABC.
3) Number Theory
Suppose, for some positive integer, n, that 2n + 1 is prime. Prove that 2n + 1 | M_n or 2n + 1 | M_n + 2, where M_n = 2ⁿ - 1.
(Hint: use M_n(M_n +2).)
4) The Last Digit of Every Fermat Number
Prove that the last digit of every Fermat number, 2^2^k + 1 is 7, where k ≥ 2.
5) Divisibility Tests
Prove the following:
A number is divisible by 2 if its last digit is divisible by 2 (is even).
A number is divisible by 3 if the sum of its digits are divisible by 3.
A number is divisible by 4 if its last two digits are divisible by 4.
A number is divisible by 5 if its last digit is 0 or 5.
A number is divisible by 7 if the difference of the sum of its digits except the last one and the last digit doubled is divisible by 7.
A number is divisible by 8 if its last three digits are divisible by 8.
A number is divisible by 9 if the sum of its digits are divisible by 9.
A number is divisible by 11 if the difference of the sum of its alternating digits is divisible by 11.
Bonus: Find a divisibility test for 2^n and 5^n and prove it.
6) Right Triangle within a Circle
Construct a triangle inside of a circle with one side the diameter of the circle, and one vertex along the circle itself. Prove that any such triangle you construct must be a right triangle.
7) 1 = 0.999...
Prove that 1 = 0.999...
8) The nth Residues of a Prime
Prove that the nth residues for a prime, p occur the same number of times, for 1 ≤ b ≤ p-1 in a ≡ b^n mod p.
9) The nth Residue, m - 1 of m
Prove that for the nth residue of m, such that a^n ≡ b mod m, (m-1)^n ≡ (-1)^n mod m.
10) Geometry in a Square
Construct a square, ABCD. Now construct a bisector of ∠BAC that meets on BC at a point F. Now construct a line from a point E on AC to F, where AE = AB. Prove that FE is perpendicular to AC and that BF = FE = EC.
11) Menelaus Problem
Prove that in any non-isosceles triangle ABC the three points of intersection of the bisectors of its external angles with the opposite sides belong to one line. Hint: If P is the point of intersection of the bisector of the external angle A with the extension of the side BC, then PC:PB = AC:AB. Prove it (using similar arguments to our proof of a similar statement for the interior bisector).
12) 6p Congruence
If a is an integer and p is prime, and p > 3, show that a^p ≡ a mod 6p.
13) Angles in a Circle
Suppose there is a circle with two line segments, each intersecting the circle at two different locations, (denote these points, A and B) and which intersect each other at another point on a circle, C. Prove that if there are two other line segments passing through the same two points as the first two segments and which meet at a different point on the circle, D, the angles ACB and ADB are the same.
Bonus: For a regular n-gon inscribed in a circle, draw line segments from one vertex of the n-gon to the other n-1 vertices. Prove that the angles between these lines and the first vertex are all the same.
14) Ratios in a Triangle
Suppose there is a triangle with vertices, A, B and C. Suppose D is the median of AB, and E is the median of DC. Prove the ratio of the sides along AC when it is cut by a line through BE is 2:1.
15) Relation between Pythagorean Triples and their Product
Prove that for any Pythagorean Triple (a^2 + b^2 = c^2; a, b, c are integers), their product abc is divisible by 60.
And also, I will now introduce a mathematical object in R^3: a polyhedron, with regular faces, the octahedron. You can take some paper and make one for real! Here's how:http://img.photobucket.com/albums/v405/btl/Octahedron.jpg