Gauss also made important contributions to number theory with his 1801 book Disquisitiones Arithmeticae ( Latin, Arithmetical Investigations ), which, among things, introduced the symbol ≡ for congruence and used it in a clean presentation of modular arithmetic, contained the first two proofs of the law of quadratic reciprocity, developed the theories of binary and ternary quadratic forms, stated the class number problem for them, and showed that a regular heptadecagon ( 17-sided polygon ) can be constructed with straightedge and compass.

Though Gauss had up to that point been financially supported by his stipend from the Duke, he doubted the security of this arrangement, and also did not believe pure mathematics to be important enough to deserve support.

Gauss proved the method under the assumption of normally distributed errors ( see Gauss – Markov theorem ; see also Gaussian ).

Gauss also claimed to have discovered the possibility of non-Euclidean geometries but never published it.

Gauss also upheld religious tolerance, believing it wrong to disturb others who were at peace with their own beliefs.

See also the letter from Robert Gauss to Felix Klein on 3 September 1912.

Germany has also issued three postage stamps honoring Gauss.

He was a contemporary of Carl Gauss, also a mathematician and physicist.

Gauss – Jordan elimination, an extension of this algorithm, reduces the matrix further to diagonal form, which is also known as reduced row echelon form.

Its major part resembles the field of a bar magnet (" dipole field ") inclined by about 10 ° to the rotation axis of Earth, but more complex parts (" higher harmonics ") also exist, as first shown by Carl Friedrich Gauss.

Other standard iterative methods for matrix equation solutions can also be used, for example the Gauss – Seidel method, where updated values for each patch are used in the calculation as soon as they are computed, rather than all being updated synchronously at the end of each sweep.

Gauss has also found some of Vega's errors in the calculations in the range of numbers, of which there are more than a million.

He also has a South African spider named after him, Araneus drygalskii ( Strand, 1909 ), based on material collected on the Gauss expedition.

Gauss conjectured that this condition was also necessary, but he offered no proof of this fact, which was provided by Pierre Wantzel in 1837.

The Riemann – Roch theorem can also be seen as a generalization of Gauss – Bonnet.

In fact, although Gauss also conjectured that there are infinitely many primes such that the ring of integers of is a PID, to this day we do not even know whether there are infinitely many number fields ( of arbitrary degree ) such that is a PID!

X ( f ( t )) is also a Gauss – Markov process

In mathematics, the error function ( also called the Gauss error function ) is a special function ( non-elementary ) of sigmoid shape which occurs in probability, statistics and partial differential equations.

Starting with Gauss ' law for electricity ( also one of Maxwell's equations ) in differential form, we have:

Another interpretation of the metric tensor, also considered by Gauss, is that it provides a way in which to compute the length of tangent vectors to the surface, as well as the angle between two tangent vectors.

Humboldt also secured a recommendation letter from Gauss, who upon reading his memoir on Fermat's theorem wrote with an unusual amount of praise that " Dirichlet showed excellent talent ".

Riemann later named this approach the Dirichlet principle, although he knew it had also been used by Gauss and by Lord Kelvin.

Bolzano also gave the first purely analytic proof of the fundamental theorem of algebra, which had originally been proven by Gauss from geometrical considerations.

Gauss usually declined to present the intuition behind his often very elegant proofs — he preferred them to appear " out of thin air " and erased all traces of how he discovered them.

This method ( and the general idea of an FFT ) was popularized by a publication of J. W. Cooley and J. W. Tukey in 1965, but it was later discovered ( Heideman & Burrus, 1984 ) that those two authors had independently re-invented an algorithm known to Carl Friedrich Gauss around 1805 ( and subsequently rediscovered several times in limited forms ).

Several other experiments followed, with André-Marie Ampère, who in 1820 discovered that the magnetic field circulating in a closed-path was related to the current flowing through the perimeter of the path ; Carl Friedrich Gauss ; Jean-Baptiste Biot and Félix Savart, both of which in 1820 came up with the Biot-Savart Law giving an equation for the magnetic field from a current-carrying wire ; Michael Faraday, who in 1831 found that a time-varying magnetic flux through a loop of wire induced a voltage, and others finding further links between magnetism and electricity.

Gauss discovered that the law of biquadratic reciprocity and its supplements were more easily stated and proved as statements about " whole complex numbers " ( i. e. the Gaussian integers ) than they are as statements about ordinary whole numbers ( i. e. the integers ).

as discovered by Gauss.

* Theorema Egregium − The " remarkable theorem " discovered by Gauss which showed there is an intrinsic notion of curvature for surfaces.

Five years later, on March 29, 1807, he discovered the asteroid Vesta, which he allowed Carl Friedrich Gauss to name.

Legendre and Gauss both applied the method to the problem of determining, from astronomical observations, the orbits of bodies about the Sun ( mostly comets, but also later the then newly discovered minor planets ).

He also discovered the presence of new constraints which he suggested to be interpreted as the equivalent of Gauss constraint of Yang Mills field theories.

It was discovered by the British National Antarctic Expedition, 1901 – 04, which named this feature after Professor Carl Friedrich Gauss, a German mathematician and astronomer.

The constant is named after Carl Friedrich Gauss, who on May 30, 1799 discovered that

In his 1799 doctorate in absentia, A new proof of the theorem that every integral rational algebraic function of one variable can be resolved into real factors of the first or second degree, Gauss proved the fundamental theorem of algebra which states that every non-constant single-variable polynomial with complex coefficients has at least one complex root.

* July 10 – Carl Friedrich Gauss discovers that every positive integer is representable as a sum of at most 3 triangular numbers.

Property ( 3 ) means that every Gauss – Markov process can be synthesized from the standard Wiener process ( SWP ).

The Gauss map provides a mapping from every point on a curve or a surface to a corresponding point on a unit sphere

For every Borel subset E of I, we also define the Gauss – Kuzmin measure of E

Gauss had considered canceling the Christmas tour, first because of financial considerations, and then because of alumni criticism, which " in nearly every case … came as the result of the excessive drinking on the part of a few of your men.

Joseph Louis Lagrange proved the square case in 1770, which states that every positive number can be represented as a sum of four squares, for example, .. Gauss proved the triangular case in 1796, commemorating the occasion by writing in his diary the line " ΕΥΡΗΚΑ!

Gauss proved that for every value D, there are only finitely many classes of binary quadratic forms with discriminant D. Their number is the class number of discriminant D. He described an algorithm, called reduction, for constructing a canonical representative in each class, the reduced form, whose coefficients are the smallest in a suitable sense.

Already Gauss had shown that, in fact, every quadratic field is contained in a larger cyclotomic field.