Carl Friedrich Gauss’ Contributions in Mathematics

Carl Friedrich Gauss

Johann Friedrich Carl Gauss aka Carl Friedrich Gauss was a Greek mathematician and Physicist. He was sometimes referred to as the “Princeps mathematicorum“, the Foremost of mathematics or the Prince of mathematics. He was born on 30th April 1777, in Brunswick, Duchy of Brunswick (now a part of Germany). A German geologist Wolfgang Sartorius wrote,

 When Gauss was barely three years old he corrected a math error his father made; and that when he was seven, solved an arithmetic series problem faster than anyone else in his class of 100 pupils.”

This arithmetic problem was to sum the integers from 1 to 100. In 1788, Guass attended Gymnasium (a senior secondary school in Germany) and then in 1792 entered Brunswick Collegium Carolinum. In 1798, at the age of 21, he completed his most famous work “Disquisitiones Arithmeticae” which was published in 1801. Section VII of this work was mainly devoted to number theory. Gauss also predicted the position of a then-new small planet ‘Ceres’ by using the least square approximation method. In 1802, Gauss visited Olbers, who discovered Pallas, and then Gauss discovered its orbit. After the death of the Duke of Brunswick, in 1807, Gauss left Brunswick to take up the position of director of the Göttingen observatory. After he arrived in Göttingen, within two years both his father and wife died and he was completely shattered. But his problems never affected his work. In 1809, he published his second book, “Theoria Motus corporum coelestium in sectionibus conicis Solem ambientium”, a two-volume series on the motion of celestial bodies. The first volume covered the topics, differential equations, conic sections, and elliptical orbits. In the second volume, he showed how to calculate and refine the estimate of a planet’s orbit.  His publications include “Disquisitiones generales circa seriem infinitam”, a rigorous treatment of series, and an introduction of the hypergeometric function, “Methodus nova integration valores per approximationem inveniendi”, a practical essay on approximate integration, “Bestimmung der Genauigkeit der Beobachtungen”, a discussion of statistical estimators, and “Theoria attractionis corporum sphaeroidicorum ellipticorum homogeneorum methodus nova tractate.”In 1822, Gauss won the university prize for Theory of attraction and also for the idea of mapping one surface onto another so that the two are similar in their smallest parts. The paper “Theoria combinationis observationum erroribus minimis obnoxiae (1823), with its supplement (1828) was about mathematical statistics, particularly the least square method. Gauss also gave proof of some axioms which suggested that he believed in the existence of non-Euclidean geometry. He never published any paper on non-Euclidean geometry, as he feared that doing so will affect his reputation. After a decade, when Gauss came to know that Lobachevsky had published a paper on non-Euclidean geometry, he wrote a letter to Schumacher in 1846 saying,

had the same convictions for 54 years.”

which shows that he knew the existence of non-euclidean geometry when he was just 15 years old. Gauss also published several papers on differential geometry. One of the papers also included the most famous Gauss theorem. He was equally interested in the study of Physics and he also published several papers. On 23 February 1855, Gauss died of a heart attack. Rudolf Wagner preserved the brain of Gauss and examined it. He found that the mass of his brain was slightly above average that explained him being a genius.

1. Construction of Heptadecagon

Gauss constructed a regular 17-sided polygon, heptadecagon with the help of straightedge and compass only, when he was just 19. His proof says that the number of sides of the regular polygon is distinct Fermat primes, which are of the form {F}_{n} = {2}^{[katex]{2}^{n}}[/katex] for some nonnegative integer n. Gauss discovered a general formula for construction polygon beyond heptadecagon. He further founded, 51, 85, 255, 257,….., and 4, 294, 967, 295 – sided figures. Gauss was so happy with his result that he asked the stones man to inscribe a heptadecagon over his tomb. However, the Stoneman declined by saying that the difficult construction of a heptadecagon will look like a circle only.

File:01-FünfzehneckSeite-Animation.gif - Wikipedia

2. Integers as Triangular Numbers

In 1796, Gauss discovered that every positive integer can be expressed as the sum of at most three triangular numbers. He wrote this result in his diary as, “ΕΥΡΗΚΑ num = Δ + Δ + Δ“. This diary was lost after 40 years of his death. Gauss was supposed to be the last person who mastered every aspect of mathematics.

3. Complex Numbers

However, Complex numbers are known from the 16th century but Gauss was the first mathematician to give a clear picture of complex numbers and functions of complex variables. Although Euler worked on imaginary and complex numbers, there was still no proper explanation of how real and imaginary numbers are related. Gauss took into practice the complex numbers and gave the standard notation a+ib for complex numbers. Then onwards, more concepts of complex numbers were unleashed.

complex numbers

4. Fundamental theorem of Algebra

Gauss gave the proof of a fundamental theorem of algebra when he was just 22. He stated that,

 Every non-constant single-variable polynomial over the complex numbers has at least one root.”

He also proved that the field of complex numbers is algebraically closed (a polynomial with complex coefficients will yield a complex solution only) unlike the field of real numbers, where a polynomial with real coefficients can yield a complex solution.

5. Number Theory

Gauss made many significant contributions to Number theory. He used to say that “Mathematics is the queen of sciences and number theory is the queen of mathematics.” In 1801, at the age of 24, Gauss published his most famous book “Disquisitiones Arithmeticae” which is considered the most influential book in the field of number theory. This book includes,

  • Triple bar symbol (≡) for congruence
  • Presentation of Gauss’ method of modular arithmetic
  • The first two proofs of the law of quadratic reciprocity
  • Theories of binary and ternary quadratic forms
  • Gauss class number problem
  • Proved Fermat’s polygonal number theorem for n=5
  • Descartes’ rule of signs
  • Kepler’s conjecture for regular arrangements
  • Calculating date of Easter
  • Discovered Cooley-Tukey FFT algorithm for calculating Discrete Fourier transforms

6. Properties of Pentagramma Mirificum

Some properties of Pentagramma mirificum were discussed by Gauss. Pentragramma mirificum is a star-shaped polygon on a sphere whose sides are great circle arcs and all the interior angles are right angles.

Pentagramma mirificum - Wikipedia

Gauss introduced the following expression

(α, β, γ, δ, ε) = (tan² TP+ tan²PQ+ tan²QR+ tan²RS+ tan²ST), where the following identities hold and three quantities can be determined by the remaining two.

1 + α = γδ, 1+ β = δε, 1 + γ = αε, 1 + δ = αβ, 1 + ε = βγ

Gauss proved the following identity and also gave a solution as (α, β, γ, δ, ε) = (9, 2⁄3, 2, 5, 1⁄3)

αβγδε = 3 + α + β + γ + δ +ε = √(1+α)(1+β)(1+γ)(1+δ)(1+ε), and area of polygon PQRST is given by,

{A}_{PQRST} = 2π – (|PQ| + |QR| + |RS| + |ST| + |TP|)

7. Gaussian Function and The Gaussian Error Curve

Gauss introduced Gaussian distribution and demonstrated how probability can be represented graphically by a bell-shaped or a normal curve. This curve has the highest peak around mean and expected value and gradually falls off near plus and minus infinity. The graph of the normal curve is given below.

normal curve

Also, Gaussian distribution is a type of continuous probability distribution for a real-valued random variable and the general form of the density function is given by,

Probability density function

8. Theorema Egregium (“Remarkable Theorem”)

Gauss’ Theorema Egregium is a result of differential geometry that talks about the curvature of surfaces. The theorem states that,

 The Gaussian curvature of a surface does not change if one bends the surface without stretching it.”

Thus the theorem means that the Gaussian curvature can be determined entirely by measuring angles, distances and their rates on a surface, without reference to the particular manner in which the surface is placed, in the ambient 3-dimensional Euclidean space. The theorem is said to be remarkable because the result does not depend on its embedding even after undergoing all bending and twisting deformations.

9. Gauss’ Theorem

Gauss theorem is also known as the Divergence theorem or Ostrogradsky’s theorem. In vector calculus, this theorem states that,

The surface integral of a vector field over a closed surface, which is called the flux through the surface, is equal to the volume integral of the divergence over the region inside the surface.”

This theorem establishes a relationship between the flux of a vector field over a closed surface and the volume integral of the divergence of the field. Graphical representation of the Gauss theorem is given by,

Gauss divergence theorem

10. Gauss’ Lemma

Gauss’s lemma states that the product of two primitive polynomials is primitive (a polynomial with integer coefficients is primitive if it has 1 as the greatest common divisor of its coefficients). This lemma over the integers is stated as,

If  F(X)= {a}_{0} + {a}_{1}X +…..+ {a}_{n} {X}^{n} is a polynomial with integer coefficients, then F is called primitive if the greatest common divisor of all the coefficients {a}_{0}, {a}_{1},…., {a}_{n} is 1; in other words, no prime number divides all the coefficients.

Further, he gave Gauss primitive lemma and irreducibility lemma. Gauss primitive lemma states that,

If P(X) and Q(X) are primitive polynomials over the integers, then product P(X)Q(X) is also primitive.”

Gauss irreducibility lemma states that,

A non-constant polynomial in Z[X] is irreducible in Z[X] if and only if it is both irreducible in Q[X] and primitive in Z[X].”

11. Gauss Approximation Method

Gauss had given Gauss-Siedel iterative method in numerical algebra used to solve a system of linear equations. This method is named after German mathematicians Gauss and Phillip Ludwig von Siedel. This method is used to solve a system of n linear equations with unknown x.

Ax = b and the iteration is given by,

L{x}^{k+1} = b – U{x}^{k} , where {x}^{k} is the kth approximation of x and {x}^{k+1} is the next iteration of x. A is decomposed into a lower triangular matrix L, and a strictly upper triangular matrix U, i.e., A = L + U.

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