# Mathematician:Carl Friedrich Gauss

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## Contents

## Mathematician

One of the most influential mathematicians of all time, contributing to many fields, including number theory, statistics, analysis and differential geometry.

According to legend, he was correcting his father's arithmetic at the age of $3$.

## Nationality

German

## History

- Born: 30 April 1777 in Braunschweig, in the Electorate of Brunswick-Lüneburg (now part of Lower Saxony, Germany)
- 1792 -- 1795: Attended the Collegium Carolinum (now Technische Universität Braunschweig)
- 1795 -- 1798: University of Göttingen
- 1807: Appointed Professor of Astronomy and Director of the astronomical observatory in Göttingen
- 1820: Embarked on an exercise to supervise a geodetic survey of the Kingdom of Hanover
- 1833: Constructed the first electromagnetic telegraph with Wilhelm Eduard Weber
- Died: 23 February 1855 in Göttingen, Hannover (now part of Lower Saxony, Germany)

## Theorems and Definitions

- Gauss's Constant
- Gauss Error Function
- Gauss Map
- Gaussian Binomial Coefficient
- Gaussian Curvature
- Gaussian Distribution
- Gaussian Domain, another term for Unique Factorization Domain
- Gaussian Elimination
- Gaussian Field
- Gaussian Hypergeometric Function
- Gaussian Integer
- Gaussian Rational

- Gauss-Bolyai-Lobachevsky Space (with János Bolyai and Nikolai Ivanovich Lobachevsky)
- Gauss-Jordan Elimination (with Wilhelm Jordan)
- Gauss-Laplace Pyramid (with Pierre-Simon de Laplace)
- Gauss-Kuzmin-Wirsing Constant (with Rodion Osievich Kuzmin and Eduard Wirsing)
- Gauss-Manin Connection (with Yuri Ivanovitch Manin)
- Gauss-Markov Process (with Andrey Andreyevich Markov)

- Gauss-Bonnet Theorem and Generalized Gauss-Bonnet Theorem (with Pierre Ossian Bonnet)
- Gauss-Codazzi Equations (with Delfino Codazzi)
- Gauss-Kronrod Quadrature Formula (with Alexandr Semenovich Kronrod)
- Gauss-Krüger Coordinate System (with Johann Heinrich Louis Krüger)
- Gauss-Kuzmin-Wirsing Operator (with Rodion Osievich Kuzmin and Eduard Wirsing)
- Gauss-Laguerre Quadrature (with Edmond Nicolas Laguerre)
- Gauss-Legendre Algorithm (with Adrien-Marie Legendre)
- Gauss-Lucas Theorem (with Édouard Lucas)
- Gauss-Markov Theorem (with Andrey Andreyevich Markov)
- Gauss-Newton Algorithm (with Isaac Newton)
- Gauss-Seidel Method (with Philipp Ludwig von Seidel)
- Ostrogradsky-Gauss Theorem (with Mikhail Vasilyevich Ostrogradsky)

- Gauss Composition
- Gauss's Continued Fraction
- Gauss's Digamma Theorem
- Gauss's Eureka Theorem
- Gauss's Formulas (also known as Delambre's Analogies, for Jean Baptiste Joseph Delambre)
- Gauss's Generalization of Wilson's Theorem
- Gauss's Hypergeometric Theorem
- Gauss's Interpolation Formula (also known as Gregory-Newton Interpolation, for James Gregory and Isaac Newton)
- Gauss's Lemma (Polynomials)
- Gauss's Lemma (Number Theory)
- Gauss Linking Integral
- Gauss Multiplication Formula
- Gauss's Principle of Least Constraint
- Gauss Sum
- Gauss's Theorem (otherwise known as the Divergence Theorem)
- Gaussian Binomial Theorem
- Gaussian Integral
- Gaussian Integration Rule
- Gaussian Isoperimetric Inequality

Also:

- Invented the Method of Least Squares
- Proved the Law of Quadratic Reciprocity
- Invented the field of modulo arithmetic
- Conjectured the Prime Number Theorem
- Demonstrated Construction of Regular Heptadecagon
- Proved Construction of Regular Prime $p$-Gon Exists iff $p$ is Fermat Prime
- 1799: Proved the Fundamental Theorem of Algebra

Results named for **Carl Friedrich Gauss** can be found here.

Definitions of concepts named for **Carl Friedrich Gauss** can be found here.

## Publications

- 1798:
*Disquisitiones Arithmeticae*(not published until 1801) - 1799:
*Demonstratio nova theorematis omnem functionem algebraicam rationalem integram unius variabilis in factores reales primi vel secundi gradus resolvi posse*(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) (doctorate thesis) - 1809:
*Theoria Motus Corporum Coelestium* - 1813:
*Disquisitiones generales circa seriam infinitam $1 + \frac {\alpha \beta} {1 \cdot \gamma} \, x + \frac {\alpha \paren {\alpha + 1} \beta \paren {\beta + 1} } {1 \cdot 2 \cdot \gamma \paren {\gamma + 1} } \, x \, x + \cdots $*(*Commentationes societatis regiae scientarum Gottingensis recentiores***Vol. 2** - 1827:
*Disquisitiones Generales circa Superficies Curvas* - 1828:
*Theoria residuorum biquadraticorum, Commentatio prima*( - 1832:
*Theoria residuorum biquadraticorum, Commentatio secunda*(

## Notable Quotes

*Mathematics is the Queen of the Sciences, and Arithmetic the Queen of Mathematics.*- -- Quoted in 1937: Eric Temple Bell:
*Men of Mathematics*:*They Say: What Say They? : Let Them Say*

- -- Quoted in 1937: Eric Temple Bell:

*The operation of distinguishing prime numbers from composites, and of resolving composite numbers into their prime factors, is one of the most important and useful in all of arithmetic. It has engaged the industry and wisdom of ancient and modern geometers to such an extent ... The dignity of science seems to demand that every aid to the solution of such an elegant and celebrated problem be zealously cultivated.*--*Disquisitiones Arithmeticae*, article $329$.- -- Quoted at the end of of 1969: Donald E. Knuth:
*The Art of Computer Programming: Volume 2: Seminumerical Algorithms*: Section $4.5$ - -- Quoted by David Wells in Section $257$ of his
*Curious and Interesting Numbers*of $1986$, requoting John Brillhart

- -- Quoted at the end of of 1969: Donald E. Knuth:

*I mean the word proof not in the sense of the lawyers, who set two half proofs equal to a whole one, but in the sense of the mathematician, where $\frac 1 2$ proof $= 0$ and it is demanded for proof that every doubt becomes impossible.*

*You know that I write slowly. This is chiefly because I am never satisfied until I have said as much as possible in a few words, and writing briefly takes far more time than writing at length.*

*It is not knowledge but the act of learning, not possession but the act of getting there, which grants the greatest enjoyment. When I have clarified and exhausted a subject, then I turn away from it in order to go into darkness again.*- -- Letter to Wolfgang Bolyai

*The higher arithmetic presents us with an inexhaustible store of interesting truths -- of truths too, which are not isolated, but stand in a close internal connection, and between which, as our knowledge increases, we are continually discovering new and sometimes wholly unexpected ties. A great part of its theories derives an additional charm from the peculiarity that important propositions, with the impress of simplicity upon them, are often easily discoverable by induction, and yet are of so profound a character that we cannot find their demonstration till after many vain attempts; and even then, when we do succeed, it is often by some tedious and artificial process, while the simpler methods may long remain concealed.*- -- Translated by H.J.S. Smith from Gauss's introduction to the collected papers of Ferdinand Eisenstein
- -- Quoted in 1937: Eric Temple Bell:
*Men of Mathematics*: Chapter $\text{IV}$: The Prince of Amateurs

*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 darkness that they defeat the sharpest enquiries.*- -- Quoted in 1986: David Wells:
*Curious and Interesting Numbers*: Introduction

- -- Quoted in 1986: David Wells:

## Critical View

*He is like the fox, who effaces his tracks in the sand with his tail.*

*The name of Gauss is linked to almost everything that the mathematics of our century*[ the nineteenth ]*has brought forth in the way of original scientific ideas.*

## Also known as

Full name: **Johann Carl Friedrich Gauss**.

Some sources (perhaps in error) report his first name as **Karl**.

## Sources

- John J. O'Connor and Edmund F. Robertson: "Carl Friedrich Gauss": MacTutor History of Mathematics archive

- 1937: Eric Temple Bell:
*Men of Mathematics*... (previous) ... (next):*They Say: What Say They? : Let Them Say* - 1937: Eric Temple Bell:
*Men of Mathematics*... (previous) ... (next): Chapter $\text{IV}$: The Prince of Amateurs - 1937: Eric Temple Bell:
*Men of Mathematics*: Chapter $\text{XIV}$ - 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): Introduction - 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): A List of Mathematicians in Chronological Sequence - 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $257$ - 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*... (previous) ... (next): Entry:**Gauss, Carl Friedrich**(1777-1855) - 1991: David Wells:
*Curious and Interesting Geometry*... (previous) ... (next): A Chronological List Of Mathematicians - 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {A}.25$: Gauss ($1777$ – $1855$) - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): Introduction - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): A List of Mathematicians in Chronological Sequence - 2008: David Joyner:
*Adventures in Group Theory*(2nd ed.) ... (previous) ... (next): Chapter $2$: 'And you do addition?': $\S 2.3$: Relations: Example $2.3.4$ - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next): Entry:**Gauss, Carl Friedrich**(1777-1855) - 2008: Ian Stewart:
*Taming the Infinite*... (previous) ... (next): Chapter $7$: Patterns in Numbers: Gauss - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**Gauss, Carl Friedrich**(1777-1855)