Carl Friederich Gauss

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Carl Friederich Gauss
Carl Friederich Gauss
The Prince of Mathematics
Pauca, sed matura
Carl Friederich Gauss
• Born 1777 in Braunschweig
(Brunswick), Germany to Gebhard Dietrich Gauss, a gardener and bricklayer, and Dorothea Gauss, the daughter of a stonecutter
• Supposedly showed extreme genius as a toddler, correcting his father’s arithmetic and teaching himself to read. Maybe.
Carl Friederich Gauss
• The classic Gauss story:
• In his arithmetic class, his teacher, a man named Büttner, told the class to write down all the numbers from 1 to 100 and add them. • Gauss wrote a single number (5050) on his slate, and handed it in. • Gauss figured thusly: …and so on, giving a sum of 101 for each pair taken form the beginning and end of the sequence. There are 50 such pairs, and Carl Friederich Gauss
• To his credit, Büttner was • In 1792 he began study at impressed, and got Gauss a the Brunswick Collegium better arithmetic book. Carolinum, with the help of a stipend from Ferdinand, • Büttner’s assistant, Martin the Duke of Brunswick. Bartels, was also impressed. • Büttner and Bartels helped • Here, Gauss independently discovered Bode's law, the Gauss get into the binomial theorem and the Gymnasium (secondary arithmetic‐ geometric school) in 1788.
mean, as well as the law of quadratic reciprocity and the prime number theorem. Carl Friederich Gauss
• Entered University of Göttingen in 1795, where he became friends with Farkas
Bolyai (with whom he corresponded for years).
• Here, Gauss’ teacher was Abraham Gotthelf
Kästner, whom Gauss didn’t think much of. Carl Friederich Gauss
• Published a book on • He left Göttingen
number theory, without a degree, Disquisitiones
returned to Brunswick, Arithmeticae , in 1801. earned a degree, and at It was largely ignored the request of the Duke, submitted a dissertation for 20 years.
to the University of • Used least‐squares Helmstedt on what is approximation to now known as the predict the location of Fundamental Theorem Ceres, a dwarf planet of Algebra. within the asteroid belt. Carl Friederich Gauss
• Married Johanna Osthoff in 1805. • In a letter to Bolyai: “A wondrously fair madonna
countenance, a mirror of spiritual peace and health, kind, somewhat romantic eyes, a perfect figure and size (that is something) a clear understanding and an intelligent conversation (that is also something), but a quiet, happy, modest, and chaste angelic soul which can harm no one, that is the best.” Carl Friederich Gauss
• From a tear‐stained letter • Gauss’ father, his wife found by his grandson: Johanna, and one of their three children all died in “Lonesome, I sneak about the next four years. the happy people who surround me here. If for • So did Ferdinand, the a few moments they Duke who supported him.
make me forget my • Gauss remarried and had sorrow, it comes back three more children, but with double force….Even life didn’t seem to hold as the bright sky makes me much joy thereafter.
sadder…”
Carl Friederich Gauss
• “O, beseech the Eternal – could he refuse you anything? – only this one thing, that your infinite kindheartedness may always hover and float, living, before me, helping me, poor son of earth that I am, to struggle after you as best I can.”
Carl Friederich Gauss
• Was appointed head astronomer at the observatory in Göttingen in 1807; made astronomical observations up until age 70.
• Helped with a geodesic survey in the 1820’s. • His second wife died after a long illness in 1831. • In 1831 Gauss began a 6‐year collaboration with Wilhelm Weber, probably Gauss’ closest friend outside of Bolyai.
Carl Friederich Gauss
• A second son left for • He became estranged America and eventually from one of his sons, became successful in a who he encouraged not
to go into mathematics St. Louis boot and shoe in order that the family business with his name remain unsullied. brother‐in‐law: Fallenstein and Gauss.
• His son left for America and eventually became a successful businessman. They later reconciled. Description of Gauss as Teacher
• Did not enjoy teaching because he felt his students were generally underprepared.
• Richard Dedekind, one of his final doctoral students, wrote a description of Gauss as a teacher. “... usually he sat in a comfortable attitude, looking down, slightly stooped, with hands folded above his lap. He spoke quite freely, very clearly, simply and plainly: but when he wanted to emphasize a new viewpoint ... then he lifted his head, turned to one of those sitting next to him, and gazed at him with his beautiful, penetrating blue eyes during the emphatic speech. “ Description of Gauss as Teacher
“If he proceeded from an explanation of principles to the development of mathematical formulas, then he got up, and in a stately very upright posture he wrote on a blackboard beside him in his peculiarly beautiful handwriting: he always succeeded through economy and deliberate arrangement in making do with a rather small space. For numerical examples, on whose careful completion he placed special value, he brought along the requisite data on little slips of paper.”
Carl Friederich Gauss
• Died in his sleep in Göttingen in 1855.
After His Death, The King Honored Him
• Georgius V. rex
Hannoverge
Mathematicorum
principi
(George V. King of Hanover
to the Prince of mathematicians)
Gauss’ Mathematical Accomplishments
• At age 19, Gauss demonstrated a method for constructing (with straightedge and compass) a regular heptadecagon (17‐gon). • In fact, he proved that regular n‐gons can only , be constructed when where the ’s are Fermat primes of the form . Gauss’ Mathematical Accomplishments
• At age 19, Gauss proved the law of quadratic reciprocity, which tells the conditions under which is solvable. • The same year, he independently discovered the Prime Number Theorem, which states that
or in other words, that the prime numbers less than x are distributed among the positive integers
roughly like Prime Number Theorem
Graph of Gauss’ Mathematical Accomplishments
• Again at age 19, he proved that every positive integer is representable as a sum of at most three triangular numbers; or as he put it in his .”
notebook, “Heureka! • And to round out his 19th year, he published a result on the number of solutions of polynomials with coefficients in finite fields. • 1796 was a very good year (although many results weren’t published until later – or ever). Gauss’ Mathematical Accomplishments
• For his doctoral dissertation, he proved the Fundamental Theorem of Algebra. This one the first of four proofs he provided through his life, the last more rigorous (by modern standards) than all the others.
• Every polynomial with complex coefficients has at least one complex root. Gauss’ Mathematical Accomplishments
• In his Disquisitiones Arithmeticae, he introduced modular arithmetic, which provided an organizational framework and basis for much of elementary number theory. if and only if . •
•
and Gauss’ Mathematical Accomplishments
• Likely developed the technique of least‐
squares ten years before Legendre published it (Legendre accused him of plagiarism). Used it to help find Ceres again after it disappeared behind the sun.
• Came to understand the logical consistency of non‐Euclidean Geometry as a young man, but never did publish it. We can look at this in some detail.
Gauss on Geometry
• “On the supposition that Euclidean geometry is not valid, it is easy to show that similar figures do not exist; in that case, the angles of an equilateral triangle vary with the side in which I see no absurdity at all. The angle is a function of the side and the sides are functions of the angle, a function which, of course, at the same time involves a constant length. It seems somewhat of a paradox to say that a constant length could be given a priori as it were, but in this again I see nothing inconsistent. Indeed it would be desirable that Euclidean geometry were not valid, for then we should possess a general a priori standard of measure.“ –
Letter to Gerling, 1816
Gauss on Geometry
• "I am convinced more and more that the necessary truth of our geometry cannot be demonstrated, at least not by the human intellect to the human understanding. Perhaps in another world, we may gain other insights into the nature of space which at present are unattainable to us. Until then we must consider geometry as of equal rank not with arithmetic, which is purely a priori, but with mechanics.“ – Letter to Olbers, 1817
Gauss on Geometry
• " There is no doubt that it can be rigorously established that the sum of the angles of a rectilinear triangle cannot exceed 180°. But it is otherwise with the statement that the sum of the angles cannot be less than 180°; this is the real Gordian knot, the rocks which cause the wreck of all.... I have been occupied with the problem over thirty years and I doubt if anyone has given it more serious attention, though I have never published anything concerning it. ”
• (“Over thirty years” puts the start date before 1794, or before age 17.)
Gauss on Geometry
• "The assumption that the angle sum is less than 180°
leads to a peculiar geometry, entirely different from Euclidean, but throughout consistent with itself. I have developed this geometry to my own satisfaction so that I can solve every problem that arises in it with the exception of the determination of a certain constant which cannot be determined a priori. The larger one assumes this constant the more nearly one approaches the Euclidean geometry, an infinitely large value makes the two coincide. The theorems of this geometry seem in part paradoxical, and to the unpracticed absurd; but on a closer and calm reflection it is found that in themselves they contain nothing impossible....”
Gauss on Geometry
• “All my efforts to discover some contradiction, some inconsistency in this Non‐Euclidean geometry have been fruitless, the one thing in it that seems contrary to reason is that space would have to contain a definitely determinate
(though to us unknown) linear magnitude. However, it seems to me that notwithstanding the meaningless word‐
wisdom of the metaphysicians we know really too little, or nothing, concerning the true nature of space to confound what appears unnatural with the absolutely impossible.
Should Non‐Euclidean geometry be true, and this constant bear some relation to magnitudes which come within the domain of terrestrial or celestial measurement, it could be determined a posteriori.“ – Letter to F. A. Taurinus, 1824.
Gauss on Geometry
• “There is also another subject, which with me is nearly forty years old, to which I have again given some thought during leisure hours, I mean the foundations of geometry.... Here, too, I have consolidated many things, and my convictions has, if possible become more firm that geometry cannot be completely established on a priori grounds. In the mean time I shall probably not for a long time yet put my very extended investigations concerning this matter in shape for publication, possibly not while I live, for I fear the cry of the Bœotians which would arise should I express my whole view on this matter.” ‐ Letter to Bessel, 1829.
• Forty year earlier would have been about 1789, with Gauss about 12 years of age. Gauss on Geometry
• The “Bœotians” he refers to are the followers of the philosopher Immanuel Kant, who insisted that “the concept of [Euclidean] space is by no means of empirical origin, but is an inevitable necessity of thought.” • Gauss never did make his work on non‐Euclidean geometry known publicly, partly because he didn’t want to be drawn into debates with the Kantians, and partly because he was a perfectionist and only published completed works of his mathematical results. “Few but ripe” was his personal motto. Gauss’ Mathematical Accomplishments
• Given a point on a surface, find the normal. All the planes containing that normal intersect the surface in curves. The two curves with largest and smallest radius of curvature (R and r, say) always meet at right angles; their curvatures are the principal curvatures.
Gauss’ Mathematical Accomplishments
• The Gaussian curvature is . defined by • It can be positive, negative, or zero.
• “Remarkable Theorem:” Gaussian curvature of a surface can be determined from the measurements of length on the surface itself. Gauss’ Mathematical Accomplishments
• While working in astronomy, Gauss realized that measurement errors produced a bell‐
shaped curve – now called a Gaussian distribution or more popularly a normal distribution. • Made the first systematic investigation into the convergence of a series.
Gauss’ Mathematical Accomplishments
• Published works fill “only” 12 volumes, as compared to Euler’s 70 volumes and Cauchy’s 27 volumes. • “A cathedral is not a cathedral until the last piece of scaffolding is removed.” • “Thou, nature, are my goddess; to thy laws my services are bound.” Carl Friederich Gauss, or Mr. Darcey?