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Carl Friedrich Gauss contribution to mathematics

Gauss contribution to mathematics

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Johann Carl Friedrich Gauss (German: Gauß (About this soundlisten);[Latin: Carolus Fridericus Gauss; (30 April 1777 – 23 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and sciences. Sometimes referred to as the Princeps mathematicorum (Latin for "the foremost of mathematicians") and "the greatest mathematician since antiquity", Gauss had an exceptional influence in many fields of mathematics and science, and is ranked among history's most influential mathematicians.

Johann Carl Friedrich Gauss was born on 30 April 1777 in Brunswick (Braunschweig), in the Duchy of Brunswick-Wolfenbüttel (now part of Lower Saxony, Germany), to poor, working-class parents. His mother was illiterate and never recorded the date of his birth, remembering only that he had been born on a Wednesday, eight days before the Feast of the Ascension (which occurs 39 days after Easter). Gauss later solved this puzzle about his birthdate in the context of finding the date of Easter, deriving methods to compute the date in both past and future years.He was christened and confirmed in a church near the school he attended as a child. Gauss was a child prodigy. In his memorial on Gauss, Wolfgang Sartorius von Waltershausen says that when Gauss was barely three years old he corrected a math error his father made; and that when he was seven, he confidently solved an arithmetic series problem faster than anyone else in his class of 100 students.[9] Many versions of this story have been retold since that time with various details regarding what the series was – the most frequent being the classical problem of adding all the integers from 1 to 100. There are many other anecdotes about his precocity while a toddler, and he made his first groundbreaking mathematical discoveries while still a teenager. He completed his magnum opus, Disquisitiones Arithmeticae, in 1798, at the age of 21—though it was not published until 1801. This work was fundamental in consolidating number theory as a discipline and has shaped the field to the present day. Gauss's intellectual abilities attracted the attention of the Duke of Brunswick, who sent him to the Collegium Carolinum (now Braunschweig University of Technology), which he attended from 1792 to 1795, and to the University of Göttingen from 1795 to 1798. While at university, Gauss independently rediscovered several important theorems. His breakthrough occurred in 1796 when he showed that a regular polygon can be constructed by compass and straightedge if the number of its sides is the product of distinct Fermat primes and a power of 2. This was a major discovery in an important field of mathematics; construction problems had occupied mathematicians since the days of the Ancient Greeks, and the discovery ultimately led Gauss to choose mathematics instead of philology as a career. Gauss was so pleased with this result that he requested that a regular heptadecagon be inscribed on his tombstone. The stonemason declined, stating that the difficult construction would essentially look like a circle.
The year 1796 was productive for both Gauss and number theory. He discovered a construction of the heptadecagon on 30 March. He further advanced modular arithmetic, greatly simplifying manipulations in number theory. On 8 April he became the first to prove the quadratic reciprocity law. This remarkably general law allows mathematicians to determine the solvability of any quadratic equation in modular arithmetic. The prime number theorem, conjectured on 31 May, gives a good understanding of how the prime numbers are distributed among the integers. Gauss also discovered that every positive integer is representable as a sum of at most three triangular numbers on 10 July and then jotted down in his diary the note: "ΕΥΡΗΚΑ! num = Δ + Δ' + Δ". On 1 October he published a result on the number of solutions of polynomials with coefficients in finite fields, which 150 years later led to the Weil conjectures.
Gauss remained mentally active into his old age, even while suffering from gout and general unhappiness. For example, at the age of 62, he taught himself Russian. In 1840, Gauss published his influential Dioptrische Untersuchungen, in which he gave the first systematic analysis on the formation of images under a paraxial approximation (Gaussian optics). Among his results, Gauss showed that under a paraxial approximation an optical system can be characterized by its cardinal points and he derived the Gaussian lens formula. In 1845, he became an associated member of the Royal Institute of the Netherlands; when that became the Royal Netherlands Academy of Arts and Sciences in 1851, he joined as a foreign member.
Gauss contribution to mathematics
In 1854, Gauss selected the topic for Bernhard Riemann's inaugural lecture "Über die Hypothesen, welche der Geometrie zu Grunde liegen" (About the hypotheses that underlie Geometry). On the way home from Riemann's lecture, Weber reported that Gauss was full of praise and excitement.
On 23 February 1855, Gauss died of a heart attack in Göttingen (then Kingdom of Hanover and now Lower Saxony); he is interred in the Albani Cemetery there. Two people gave eulogies at his funeral: Gauss's son-in-law Heinrich Ewald, and Wolfgang Sartorius von Waltershausen, who was Gauss's close friend and biographer. Gauss's brain was preserved and was studied by Rudolf Wagner, who found its mass to be slightly above average, at 1,492 grams, and the cerebral area equal to 219,588 square millimeters (340.362 square inches). Highly developed convolutions were also found, which in the early 20th century were suggested as the explanation of his genius.
Gauss was a Lutheran Protestant, a member of the St. Albans Evangelical Lutheran church in Göttingen. Potential evidence that Gauss believed in God comes from his response after solving a problem that had previously defeated him: "Finally, two days ago, I succeeded—not on account of my hard efforts, but by the grace of the Lord." One of his biographers, G. Waldo Dunnington, described Gauss's religious views as follows:
Gauss contribution to mathematics
For him science was the means of exposing the immortal nucleus of the human soul. In the days of his full strength, it furnished him recreation and, by the prospects which it opened up to him, gave consolation. Toward the end of his life, it brought him confidence. Gauss's God was not a cold and distant figment of metaphysics, nor a distorted caricature of embittered theology. To man is not vouchsafed that fullness of knowledge which would warrant his arrogantly holding that his blurred vision is the full light and that there can be none other which might report the truth as does his. For Gauss, not he who mumbles his creed, but he who lives it, is accepted. He believed that a life worthily spent here on earth is the best, the only, preparation for heaven. Religion is not a question of literature, but of life. God's revelation is continuous, not contained in tablets of stone or sacred parchment. A book is inspired when it inspires. The unshakeable idea of personal continuance after death, the firm belief in a last regulator of things, in an eternal, just, omniscient, omnipotent God, formed the basis of his religious life, which harmonized completely with his scientific research.
Apart from his correspondence, there are not many known details about Gauss's personal creed. Many biographers of Gauss disagree about his religious stance, with Bühler and others considering him a deist with very unorthodox views, while Dunnington (though admitting that Gauss did not believe literally in all Christian dogmas and that it is unknown what he believed on most doctrinal and confessional questions) points out that he was, at least, a nominal Lutheran. In connection to this, there is a record of a conversation between Rudolf Wagner and Gauss, in which they discussed William Whewell's book Of the Plurality of Worlds. In this work, Whewell had discarded the possibility of existing life in other planets, on the basis of theological arguments, but this was a position with which both Wagner and Gauss disagreed. Later Wagner explained that he did not fully believe in the Bible, though he confessed that he "envied" those who were able to easily believe. This later led them to discuss the topic of faith, and in some other religious remarks, Gauss said that he had been more influenced by theologians like Lutheran minister Paul Gerhardt than by Moses. Other religious influences included Wilhelm Braubach, Johann Peter Süssmilch, and the New Testament.
Dunnington further elaborates on Gauss's religious views by writing:
Gauss's religious consciousness was based on an insatiable thirst for truth and a deep feeling of justice extending to intellectual as well as material goods. He conceived spiritual life in the whole universe as a great system of law penetrated by eternal truth, and from this source he gained the firm confidence that death does not end all.
Gauss contribution to mathematics
Gauss declared he firmly believed in the afterlife, and saw spirituality as something essentially important for human beings. He was quoted stating: "The world would be nonsense, the whole creation an absurdity without immortality," and for this statement he was severely criticized by the atheist Eugen Dühring who judged him as a narrow superstitious man. Though he was not a church-goer, Gauss strongly upheld religious tolerance, believing "that one is not justified in disturbing another's religious belief, in which they find consolation for earthly sorrows in time of trouble." When his son Eugene announced that he wanted to become a Christian missionary, Gauss approved of this, saying that regardless of the problems within religious organizations, missionary work was "a highly honorable" task.
On 9 October 1805,[45] Gauss married Johanna Osthoff (1780–1809), and had two sons and a daughter with her. Johanna died on 11 October 1809, and her most recent child, Louis, died the following year. Gauss plunged into a depression from which he never fully recovered. He then married Minna Waldeck (1788–1831) on 4 August 1810, and had three more children. Gauss was never quite the same without his first wife, and he, just like his father, grew to dominate his children. Minna Waldeck died on 12 September 1831.
Gauss had six children. With Johanna (1780–1809), his children were Joseph (1806–1873), Wilhelmina (1808–1846) and Louis (1809–1810). With Minna Waldeck he also had three children: Eugene (1811–1896), Wilhelm (1813–1879) and Therese (1816–1864). Eugene shared a good measure of Gauss's talent in languages and computation. After his second wife's death in 1831 Therese took over the household and cared for Gauss for the rest of his life. His mother lived in his house from 1817 until her death in 1839.

Gauss contribution to mathematics
Gauss eventually had conflicts with his sons. He did not want any of his sons to enter mathematics or science for "fear of lowering the family name", as he believed none of them would surpass his own achievements. Gauss wanted Eugene to become a lawyer, but Eugene wanted to study languages. They had an argument over a party Eugene held, which Gauss refused to pay for. The son left in anger and, in about 1832, emigrated to the United States, where he was quite successful. While working for the American Fur Company in the Midwest, he learned the Sioux language. Later, he moved to Missouri and became a successful businessman. Wilhelm also moved to America in 1837 and settled in Missouri, starting as a farmer and later becoming wealthy in the shoe business in St. Louis. It took many years for Eugene's success to counteract his reputation among Gauss's friends and colleagues. See also the letter from Robert Gauss to Felix Klein on 3 September 1912.

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In the same year, Italian astronomer Giuseppe Piazzi discovered the dwarf planet Ceres. Piazzi could only track Ceres for somewhat more than a month, following it for three degrees across the night sky. Then it disappeared temporarily behind the glare of the Sun. Several months later, when Ceres should have reappeared, Piazzi could not locate it: the mathematical tools of the time were not able to extrapolate a position from such a scant amount of data—three degrees represent less than 1% of the total orbit. Gauss heard about the problem and tackled it. After three months of intense work, he predicted a position for Ceres in December 1801—just about a year after its first sighting—and this turned out to be accurate within a half-degree when it was rediscovered by Franz Xaver von Zach on 31 December at Gotha, and one day later by Heinrich Olbers in Bremen.
Gauss's method involved determining a conic section in space, given one focus (the Sun) and the conic's intersection with three given lines (lines of sight from the Earth, which is itself moving on an ellipse, to the planet) and given the time it takes the planet to traverse the arcs determined by these lines (from which the lengths of the arcs can be calculated by Kepler's Second Law). This problem leads to an equation of the eighth degree, of which one solution, the Earth's orbit, is known. The solution sought is then separated from the remaining six based on physical conditions. In this work, Gauss used comprehensive approximation methods which he created for that purpose.
One such method was the fast Fourier transform. While this method is traditionally attributed to a 1965 paper by J.W. Cooley and J.W. Tukey, Gauss developed it as a trigonometric interpolation method. His paper, Theoria Interpolationis Methodo Nova Tractata, was only published posthumously in Volume 3 of his collected works. This paper predates the first presentation by Joseph Fourier on the subject in 1807.
Zach noted that "without the intelligent work and calculations of Doctor Gauss we might not have found Ceres again". 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. Thus he sought a position in astronomy, and in 1807 was appointed Professor of Astronomy and Director of the astronomical observatory in Göttingen, a post he held for the remainder of his life.
Suppose R is a set of all sets that are not members of themselves. Now is this set R is a member of itself or not? If it is a member of itself then it is one of the sets which are not members of themseves. So it is not member of itself. And if this set R is not member of itself then by definition it is a member of itself. So in either way we come to a contradiction!

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Four Gaussian distributions

The discovery of Ceres led Gauss to his work on a theory of the motion of planetoids disturbed by large planets, eventually published in 1809 as Theoria motus corporum coelestium in sectionibus conicis solem ambientum (Theory of motion of the celestial bodies moving in conic sections around the Sun). In the process, he so streamlined the cumbersome mathematics of 18th-century orbital prediction that his work remains a cornerstone of astronomical computation. It introduced the Gaussian gravitational constant, and contained an influential treatment of the method of least squares, a procedure used in all sciences to this day to minimize the impact of measurement error.
Gauss contribution to mathematics
Gauss proved the method under the assumption of normally distributed errors (see Gauss–Markov theorem; see also Gaussian). The method had been described earlier by Adrien-Marie Legendre in 1805, but Gauss claimed that he had been using it since 1794 or 1795. In the history of statistics, this disagreement is called the "priority dispute over the discovery of the method of least squares."

Similarly probability distribution of two or more random varibles can be established :
Gauss contribution to mathematics
This is the bell curve in two dimensions. This can be generalized to higher dimensional case also.

Non-Euclidean Geometry

Gauss also claimed to have discovered the possibility of non-Euclidean geometries but never published it. This discovery was a major paradigm shift in mathematics, as it freed mathematicians from the mistaken belief that Euclid's axioms were the only way to make geometry consistent and non-contradictory. Research on these geometries led to, among other things, Einstein's theory of general relativity, which describes the universe as non-Euclidean. His friend Farkas Wolfgang Bolyai with whom Gauss had sworn "brotherhood and the banner of truth" as a student, had tried in vain for many years to prove the parallel postulate from Euclid's other axioms of geometry. Bolyai's son, János Bolyai, discovered non-Euclidean geometry in 1829; his work was published in 1832. After seeing it, Gauss wrote to Farkas Bolyai: "To praise it would amount to praising myself. For the entire content of the work ... coincides almost exactly with my own meditations which have occupied my mind for the past thirty or thirty-five years." This unproved statement put a strain on his relationship with Bolyai who thought that Gauss was "stealing" his idea.
Suppose R is a set of all sets that are not members of themselves. Now is this set R is a member of itself or not? If it is a member of itself then it is one of the sets which are not members of themseves. So it is not member of itself. And if this set R is not member of itself then by definition it is a member of itself. So in either way we come to a contradiction!
Letters from Gauss years before 1829 reveal him obscurely discussing the problem of parallel lines. Waldo Dunnington, a biographer of Gauss, argues in Gauss, Titan of Science that Gauss was in fact in full possession of non-Euclidean geometry long before it was published by Bolyai, but that he refused to publish any of it because of his fear of controversy.
Journey of Non-euclidean or meta geometry began when Euclidean fifth postulate was denied and its validity was questioned. Fifth postulate seemed to be somewhat vague and it could not be proved using all other postulates of Euclid. Many mathematicians showed that parallel postulate can be denied and yet consistent and logical geometries could be found. Such geometries are elliptic and hyperbolic geometries. On a surface of sphere given a point and a straight line an infinite number of parallel lines can be drawn.
In the field of non-euclidean geometry Gauss contributed more than anyone else. He studied curved surfaces what is called developable. That is to say, all the surfaces that can be developed on to each other without tearing and glueing with others are called developable surfaces. These kind of sufaces can be represented by a common mathematical expression.
ds^2 = g(uv)dx(u)dx(v) ---- where u , v = 1,2
Writing G(uv) as functions E, F and G the expression becomes
ds^2 = E(dx)(d) + 2 F (dx)(dy) + G(dy)(dy)
E, F and G are functions of the coordinates. Gauss thoery of surfaces led the development of differential geometry. The most notable application of such geometry can be seen in theory of general relativity. Eulcid's geometry is constant geometry where clocks and rulers does not change. In fact you can construct a geometry where clock and rulers can be made flexible. The functions E, F and G bear this fact. Such a geometry equiped with metric elements is called differential geometry.

Suppose R is a set of all sets that are not members of themselves. Now is this set R is a member of itself or not? If it is a member of itself then it is one of the sets which are not members of themseves. So it is not member of itself. And if this set R is not member of itself then by definition it is a member of itself. So in either way we come to a contradiction!

Gauss's ingenius sum

Gauss summed all numbers from 1 to 100 quickly when his teacher asked everyvody in the class to calculate it. This is how he did it
Gauss contribution to mathematics
Besides this, he invented a method to solve simulaneous sytem of equations. Here is an example

Gauss contribution to mathematics

Divergence and stoke's theorem

Gauss developed some theorems which are used in physics very often :

Gauss contribution to mathematics

Humor cartoon

A cartoon is looking at his back while thinking about the equation of it.

Gauss contribution to mathematics
Gaussian function is very important tool in analysing probabilistic events. Gaussian functions can be extended into two dimensions as follows:

Gauss contribution to mathematics
The integral of gaussian function is just π

Gauss contribution to mathematics
And the Gauss's law states that electric flux is equal to the charge divided electrical permitivity :
Gauss contribution to mathematics

Gauss contribution to mathematics

Prime number formula

It is possible to approximate the number of prime numbers which falls in the range of o and a large number say x. Gauss worked on such distribution of prime numbers at his early age and he found the function π(x) to calculate the number of prime numbers less than x.
prime number theorem
He came up with this formula by looking at a simple table of data as follows:
prime number distribution

Quadratic reciprocity

Gauss also invented many number theory formulas. One of those is the proof of Leonard Euler's qudratic reciprocity. Quadratic reciprocity states the conditions under which quadratic congruence equatiobs are solvable.
More specifically If p and q are prime numbers, then the two congruence equations x(square) = p mod q and x(square) = q mod p are either both solvable or both not solvable unless p and q each have remainder 3 on division by 4, in which case one equation is solvable, while the other is not.
Examples are : The equation x (square) = 5 mod 7 is not solvable; that is, there are no perfect squares in the sequence 5, 12, 19, 26, 33, 40, . . ., since the "reciprocal" equation x (square) = 7 mod 5 = 2 mod 5 has no solution. There is no square whose final digit is 2 or 7. The equation x(square) = 5 mod 11 has a solution, since it is clear that perfect squares appear in the sequence of numbers 5, 16, 27, 38, 49, 60, . . . . But then the reciprocal equation 11 mod 5 = 1 mod 5 must have at least one solution (there exist squares with 1 or 6 as final digit). The equation x(square) = 3 mod 11 is solvable, since in the sequence 3, 14, 25, 36, 47, 58, . . . there are some perfect squares. But then the reciprocal equation x(square) 11 mod 3 = 2 mod 3 has no solution, that is, in the sequence 2, 5, 8, 11, 14, 17, 20, 23, . . . are to be found no perfect squares.

Cyclotomic equation

An equation of the form x ^ n = 1 is called a cyclotomic equation (from the Greek kyklos = circle, temnein = to cut), since one can write down the n solutions in the form

Cyclotomic equation
If one then draws these points, as was GAUSS's practice after 1820, in the complex plane (also known as the GAUSSian plane), in which the number 1 is drawn to the right of the origin and the number i above it, then these points form the vertices of a regular n-gon on the unit circle. The different kind of solution will be graphically
Cyclotomic equation solutions
It is also known as the roots of unity. Roots of unity follows some basic identities of summation and multiplication.
roots of unity

Pappus's centroid law

It is not a theorem that Gauss developed but it is related to geometry and calculus. There are two theorems of Pappus's. First theorem states
The surface are of a surface-of-revolution generated by revolving a plance curve C about an axis external to C and on the same plance is equal to the arc length S of C multiplied by the distance travelled by the centroid of C.
that is A = sd
The second theorem states that the volume of a solid-of-rotation generated by a plane figure F about an external axis is equal to the surface area A of the figure F and the distance travelled by the centroid of F. That is
V = Ad.
An animated figure can be shown as an easy explantion of Pappus's theorem.
pappus's theorem

Computer science and Engineering

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RSA cryptography

RSA cryptography is done using modular arithmetic and number e.
RSA cryptography

Gaussian function

In an n-dimensional space a Gaussian function can be defined as
gaussian function

Reference materials:

Law of thermodynamics
A briefer history of time by S. Hawking
A brief history of time by S. Hawking
Quantum mechanics
Grand Design by Stephen Hawking
Higher Engineering Mathematics ( PDFDrive.com ).pdf
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