Life
Born 30 April 1777 in Brunswick, Gauss was a prodigy. At age 7 he summed 1 to 100 instantly by pairing (1+100)+(2+99)+…=5050. The Duke of Brunswick funded his studies at Collegium Carolinum and Göttingen.
Key milestones
- 1796 (age 19): constructs regular 17-gon with compass and straightedge; begins diary.
- 1799: doctoral thesis at Helmstedt — first rigorous proof of the Fundamental Theorem of Algebra.
- 1801: Disquisitiones Arithmeticae — founds modern number theory.
- 1801–1809: predicts orbit of Ceres; publishes Theoria Motus; develops least squares (used since 1795).
- 1807: director of Göttingen Observatory.
- 1820s: Hanover geodetic survey; invents heliotrope.
- 1827: Disquisitiones generales circa superficies curvas — Theorema Egregium.
- 1833: with Wilhelm Weber builds electromagnetic telegraph; defines magnetic units.
- 1855: dies in Göttingen, 23 February.
17-gon constructible
Disquisitiones published
of FTA across lifetime
Number Theory — Disquisitiones Arithmeticae (1801)
At 21, Gauss systematized number theory. He introduced the congruence symbol and theory, the classification of quadratic forms, composition of forms, and the first complete proofs of deep facts others had only conjectured.
Modular Arithmetic
Gauss wrote a ≡ b (mod m) to mean m | (a−b). This "clock arithmetic" turned messy divisibility arguments into algebra.
Example: 7 + 8 ≡ 3 (mod 12). Congruences behave like equations: you can add, subtract, multiply — and divide by numbers coprime to the modulus.
Quadratic Reciprocity — the Golden Theorem
For odd primes p≠q, the Legendre symbol (p|q)=1 if p is a square mod q, else −1. Gauss proved:
(p|q)·(q|p) = (−1)^{(p−1)(q−1)/4}
He called it theorema aureum and gave six proofs in his lifetime (ultimately eight by his notes). It governs when quadratic equations have modular solutions — the backbone of modern cryptography.
Gaussian Integers ℤ[i]
Numbers a+bi with a,b∈ℤ. Norm N(a+bi)=a²+b² is multiplicative. Gauss proved unique factorization in ℤ[i] and classified its primes:
- a+bi with a,b≠0 is prime iff a²+b² is an ordinary prime.
- rational prime p≡3 mod 4 stays prime in ℤ[i]; p≡1 mod 4 splits as (a+bi)(a−bi).
Least Squares & the Normal Distribution
To recover the asteroid Ceres (1801), Gauss minimized sum of squared errors — the method of least squares, used since 1795, published 1809. As the error law, he derived the Gaussian bell curve:
φ(x) = (1/σ√2π) exp(−(x−μ)²/(2σ²))
This justified the arithmetic mean as the most probable value under independent, small errors — founding modern statistics and geodesy.
Fundamental Theorem of Algebra
Every non-constant polynomial with complex coefficients has at least one complex root. Gauss's 1799 thesis gave the first essentially complete proof (geometric-topological). He returned to it three more times, each proof introducing new tools.
Differential Geometry — Theorema Egregium (1827)
In his survey work, Gauss studied curved surfaces. He defined Gaussian curvature K = k₁·k₂ (product of principal curvatures) and proved it is intrinsic: bending without stretching preserves K.
Consequence: a flat sheet (K=0) can become a cylinder or cone (K=0) but never a sphere (K>0) without distortion — why perfect maps are impossible.
Non-Euclidean Geometry
Gauss privately explored geometry where Euclid's parallel postulate fails — anticipating Bolyai and Lobachevsky by decades. He withheld publication, writing in 1832: "I fear the cry of the Boeotians." His notes show clear understanding of constant negative curvature surfaces.
Magnetism, Geodesy, and Physics
- With Weber (1833): electromagnetic telegraph across Göttingen; sensitive magnetometer.
- Gauss's law for magnetism: ∇·B = 0; magnetic flux through closed surface is zero.
- Absolute magnetic intensity measurement; "gauss" unit later named for him.
- Invented heliotrope for triangulation; error theory guided Hanover survey.
Interactive Demos
1) Modular Clock
2) Gaussian Primes in ℤ[i]
Dots are a+bi with |a|,|b|≤R that are Gaussian primes. Symmetry reflects units ±1, ±i.
3) Least Squares Fit
Click to add points. Gauss minimized Σ(y−mx−b)².
4) Curvature — Theorema Egregium
Surface z=k₁x²+k₂y². Gaussian curvature at origin K=4k₁k₂. Bending without stretch keeps K.
Octave / MATLAB Examples (Gauss-inspired)
Copy-paste into GNU Octave. No toolboxes required.
1) Modular arithmetic
% Gauss congruences a ≡ b (mod n)
n = 12;
a = 7; b = 8;
c = mod(a + b, n) % 7+8 ≡ 3 (mod 12)
% Solve 5x ≡ 3 (mod 12) using inverse
[~, u, ~] = gcd(5, n); % u is inverse of 5 mod 12
x = mod(u*3, n)2) Quadratic reciprocity check
% Legendre symbol via Euler's criterion
legendre = @(a,p) 0*(mod(a,p)==0) + (mod(a,p)~=0)*(2*(powmod(a,(p-1)/2,p)==1)-1);
% Octave lacks powmod; simple fast power:
function r = powmod(a,e,m), r=1; a=mod(a,m); while e>0, if bitand(e,1), r=mod(r*a,m); end; a=mod(a*a,m); e=bitshift(e,-1); end; end
p = 11; q = 19;
lhs = legendre(p,q) * legendre(q,p);
rhs = (-1)^((p-1)*(q-1)/4);
printf("LHS=%d RHS=%d (should match)\n", lhs, rhs);3) Gaussian primes plot
R = 30;
[x,y] = meshgrid(-R:R, -R:R);
n2 = x.^2 + y.^2;
isPrime = isprime(n2);
gaussPrime = (x~=0 & y~=0 & isPrime) | ...
(x==0 & isprime(abs(y)) & mod(abs(y),4)==3) | ...
(y==0 & isprime(abs(x)) & mod(abs(x),4)==3);
imagesc(-R:R, -R:R, gaussPrime); axis xy equal tight;
colormap([0 0 0; 0 1 0.7]); title('Gaussian primes');4) Least squares — Ceres style
% Fit y = b0 + b1*x by normal equations (Gauss 1809)
x = [1 2 3 4 5]'; y = [2.1 3.9 6.2 7.8 10.1]';
X = [ones(size(x)) x];
beta = (X'*X) \ (X'*y); % [intercept; slope]
xp = linspace(min(x),max(x),100);
yp = beta(1) + beta(2)*xp;
plot(x,y,'o', xp,yp,'-','LineWidth',2); grid on;
xlabel('x'); ylabel('y'); title('Least squares fit');5) Normal (Gaussian) distribution
mu = 0; sigma = 1;
x = linspace(-4,4,400);
gauss = 1/(sigma*sqrt(2*pi)) * exp(-(x-mu).^2/(2*sigma^2));
plot(x,gauss,'LineWidth',2); grid on;
title('Gaussian bell curve'); xlabel('x'); ylabel('\phi(x)');