Piano Sonata No. 17 in D Minor, “The Tempest”, Allegretto, Third Movement. Ludwig van Beethoven, 1801, played by Glenn Gould.

Carl Friedrich !

Gauss!!

!

The Prince of Mathematics

Other Famous German RomanticsComposers: !• Ludwig van Beethoven !• Franz Liszt!• Johannes Brahms!• Franz Schubert!!

Johann van Goethe’s Faust!The Brothers Grimm!!Scientists: !• William Herschel!• Alexander von Humboldt!• Jean-Baptiste Lamarck!• Wilhelm Weber

Romanticism

• Reaction and Response to Enlightenment!!• “Reason is the enumeration of qualities already

known; imagination is the perception of the value of those qualities, both separately and as a whole. Reason respects the differences, and imagination the similitudes of things. Reason is to imagination as the instrument to the agent, as the body to the spirit, as the shadow to the substance.” – Percy Shelley, English Romantic Poet, in A Defense of Poetry, 1821

Wanderer above the Sea of Fog (German: Der Wanderer über dem Nebelmeer), 1818, Caspar David Friedrich

Themes

• Individualism!• Imagination!• Nature!• Preference for the

values and language of the common people over the elite

Gauss: a Romantic Genius• Work in both pure mathematics and applied mathematics

– the natural world as a path to mathematical discoveries!– Rough timeline of work in Applied Mathematics: !• 1800s and 1810s: Astronomy !• 1820s: Geodesics!• 1830s: Physics!

• Rose from humble origins!• Creativity and imagination despite depression, feelings of

being misunderstood by society

Gauss: Personal Life

• Born in Brunswick, Germany in 1777 to poor, working-class parents!

• Recognized as genius early!• Studied at Gottingen University

until 1799 under the sponsorship of the Duke of Brunswick!

• Marries Johanna Ostoff in 1805; she dies after the death of their second child in 1809; Gauss remarried, but often felt misunderstood by family despite their support!

• Depression!• Dies in Gottingen in 1855 after a

long incredibly productive life The Gottingen University and its library in 1815!

Early Contributions to Pure Mathematics!

!!!!

• Groundbreaking mathematical discovery while still in school: the construction of a regular 17-gon by ruler and compass - the most major advance in this field since the time of Greek mathematics. !

• Demystified complex numbers by interpreting them geometrically and rigorously.!

• Fundamental Theorem of Algebra: a polynomial equation of degree n has n complex roots (when counted with multiplicities).

What can be constructed using straight edge and compass?

Around the same time when mathematicians have discovered that not every algebraic equation can be solved (in radicals), they have also learned how to prove that not everything can be constructed using straightedge and compass.

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Regular pentagon• Construction in the Elements

is based on first constructing the golden ratio !!

• Later simplified constructions (like the one on the right) were found by craftsmen in the Islamic world

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Classical challenges:

• Double the cube!• Trisect an angle!• Inscribe regular n-gon!• Square the circle

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Doubling the cube

!!!!!!

Is it possible to solve the equation x3=2 using straight edge and compass?

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Angle trisection• From the homework:!!!

• To trisect a given angle, we need to solve a cubic equation !!using straight edge and compass (c is a constant)

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Regular n-gon• A Swiss mathematician Jobst Bürgi

(1552-1632) (who is also credited with discovery of logarithms along with Napier) have discovered that if x is the side of an inscribed regular (2n+1)-gon then y=x2 is a root of an explicit equation of degree n:!

• 2n+1=5: y2-5y+5=0!• 2n+1=7: y3-7y2+14y-7=0!• …..

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• Solving this cubic equation using the Ferro-Tartaglia method gives a very strange answer:!!!!!!

• Many mathematicians believed that the heptagon can not be constructed using straight edge and compass, including Kepler in Harmony of the World, although his reasons were mostly metaphysical 14

Constructible numbers• In modern terminology, a number is constructible if it can

be computed recursively starting with integers and using only arithmetic operations and extraction of square roots.!

• Notice that square roots can indeed be constructed:

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Wantzel’s Theorem• Pierre Wantzel proved in 1837 (although the result was

probably known to Gauss before him) that if x is a root of an irreducible polynomial with rational coefficients of degree n>2 then x cannot be constructed.!

• Here a polynomial !!!

• is called irreducible if it cannot be factored into two polynomials of smaller degrees with rational coefficients.

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• The following polynomials are irreducible:!• x3-2!

• !!

• y3-7y2+14y-7!• So the cube can not be doubled, the angle of

60 degrees can not be trisected and the regular heptagon can not be constructed!!

• Why are these polynomials irreducible?

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Enter Gauss:

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So the heptadecagon can be constructed!

Gauss-Wantzel Theorem

• Let p be an odd prime number. Then a regular p-gon can be constructed if and only if p is a Fermat prime, i.e. can be written as !!

(with q prime)!• p=3, 5, 17, 257, 65537, 4294967297,

18446744073709551617, …19

• Not all numbers are algebraic (roots of polynomial equations with rational coefficients)

• Liouville (1851): First example of a transcendental number: !

• Hermite (1873) e is transcendental • Lindemann (1882) π is transcendental • In particular, squaring the circle is impossible!

Transcendental numbers

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Disquisitiones Arithmeticae!

!• Gauss publishes Disquisit iones

Arithmeticae in the summer of 1801.!• Considered the cornerstone of number

theory: Gauss brought the work of his predecessors together with his own original work into a systematic framework, filled in gaps, corrected unsound proofs, and extended the subject in numerous ways.

Modular Arithmetic• Before we discuss Theorema Aureum, a few words about modular

arithmetic as developed by Gauss in Disquisitiones Arithmeticae. !• In this system numbers "wrap around" upon reaching a certain value -

the modulus.

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• For example, arithmetic modulo 12 is a familiar “clock arithmetic” where there are only 12 possible hours.!

• 15 is the same as 3, 10 is the same as -2!• Instead of saying “the same”, we say

congruent and write 15 ≡ 3 mod 12!• 9 + 4 ≡ 1 mod 12!• 4 x 7 ≡ 4 mod 12!• 2 - 5 ≡ 9 mod 12

Quadratic residues

• A number p is called a quadratic residue modulo a number q if p is a perfect square modulo q.!!

• !• 0, 1, 2 and 4 are quadratic residues modulo 7!• 3, 5, and 6 are not quadratic residues modulo 7!• If q is any odd prime number then, excepting 0, exactly

half of the remainders modulo q are quadratic residues. !• What are quadratic residues modulo 5?

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x 0 1 2 3 4 5 6

x2 0 1 4 2 2 4 1

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25

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• Theorema Aureum (The law of quadratic reciprocity). Suppose p and q are odd prime numbers. Then p is a quadratic residue modulo q if, and only if, q is a quadratic residue modulo p. There is one exception: if p and q are both congruent to 3 modulo 4 then p is a quadratic residue modulo q if, and only if, q is not a quadratic residue modulo p. !

• For example, is 7 a quadratic residue modulo 619? !• Both are prime numbers congruent to 3 modulo 4. So the behavior of 7

modulo 619 is opposite to the behavior of 619 modulo 7.!• 619=88*7+3 ≡ 3 mod 7 and 3 is not a quadratic residue modulo 7.

Therefore, 7 is a quadratic residue modulo 619! !• So is 5 a quadratic residue modulo 961748941?

Theorema Aureum

The law of quadratic reciprocity was discovered experimentally by Legendre, Lagrange, and Euler. Neither of them could prove it!!!Here is how Gauss wrote about his discovery: !!“In arithmetic the most elegant theorems frequently arise experimentally as the result of a more or less unexpected stroke of good fortune, while their proofs lie so deeply embedded in the darkness that they elude all attempts and defeat the sharpest inquiries … I discovered this theorem independently in 1795 at a time when I was totally ignorant of what had been achieved in higher arithmetic, and consequently had not the slightest aid from the literature on the subject. For a whole year this theorem tormented me and absorbed my greatest efforts until at last I obtained a proof.”!!This proof was published by Gauss in 1801 in Disquisitiones and was followed by seven others in an interval of 17 years.

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Formulation using Legendre symbol

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• The law of quadratic reciprocity: if p and q are odd primes then!!!!

• How to compute the Legendre symbol if a is not prime?

When is -1 a quadratic residue?

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When is 2 a quadratic residue?

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Multiplicativity• The last ingredient that we need is multiplicativity of the

Legendre symbol: if p is prime then!!!!

• Let’s try some examples:!• Is 22 a quadratic residue modulo 65537?!• Is 252 a quadratic residue modulo 257?!• Does a quadratic equation x2+x-4=0 has a solution modulo 67?

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“To me Gauss is like the highest peak amidst our Bavarian mountains as it appears to a spectator from the north. From the east the gradually ascending foothills culminate in the one gigantic colossus, which falls away steeply into the lowlands of a new formation, into which its spurs project for many for many miles and in which the water that streams from it begets new life.” !- Felix Klein, Development of mathematics in the Nineteenth Century

Further reading• D. Kehlmann, Measuring the World!• F. Klein, Development of Mathematics in the 19th Century!• The MacTutor History of Mathematics, http://www-

history.mcs.st-and.ac.uk/!• C.F. Gauss, Disquisitiones Arithmeticae

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