Charles Hermite, French mathematician and theorist (b. 1822)

Charles Hermite, born on December 24, 1822, and passing away on January 14, 1901, was an exceptionally influential French mathematician whose profound contributions reshaped several core areas of mathematics. Recognized for his intellectual prowess and elected as a Fellow of the Royal Society (FRS), a Fellow of the Royal Society of Edinburgh (FRSE), and a Member of the Imperial Academy of Sciences (MIAS), among other distinguished academies, Hermite's extensive research spanned number theory, quadratic forms, invariant theory, orthogonal polynomials, elliptic functions, and algebra, leaving an indelible mark on the landscape of 19th-century mathematics. His name, pronounced as [ʃaʁl ɛʁˈmit] in French, continues to resonate through fundamental mathematical concepts and tools.

Pioneering Research and Enduring Contributions

Hermite's analytical brilliance led to groundbreaking discoveries and the development of essential mathematical frameworks. His work often interconnected diverse fields, revealing deeper structures within mathematics.

Number Theory and the Transcendence of e

One of Charles Hermite's most celebrated achievements was his groundbreaking proof in 1873 that the mathematical constant e, the base of the natural logarithm (approximately 2.71828), is a transcendental number. A transcendental number is a real or complex number that is not a root of any non-zero polynomial equation with integer coefficients. In simpler terms, it cannot be expressed as the solution to an algebraic equation involving only integers and basic operations like addition, subtraction, multiplication, and powers. This was a monumental step in mathematics, as it provided the first example of a transcendental number that was not constructed specifically for that purpose (like Liouville numbers). Hermite's innovative methods and logical framework were subsequently adapted and utilized by German mathematician Ferdinand von Lindemann in 1882 to prove the transcendence of π (pi), a discovery that famously resolved the ancient Greek problem of "squaring the circle" – demonstrating its impossibility using only a compass and straightedge.

Exploring Quadratic Forms and Invariant Theory

Hermite made significant advancements in the study of quadratic forms, which are homogeneous polynomials of degree two, foundational to number theory and geometry, particularly in the classification of surfaces and in optimization problems. His work in invariant theory focused on properties of algebraic expressions that remain unchanged under certain transformations, a field crucial for understanding symmetries in mathematics and physics.

Orthogonal Polynomials and Elliptic Functions

His research on orthogonal polynomials led to the development of the widely used Hermite polynomials, which are essential in areas such as probability theory (describing Hermite distribution), numerical analysis, and quantum mechanics (specifically in the solution of the quantum harmonic oscillator). Hermite's contributions to the theory of elliptic functions, complex functions that generalize trigonometric functions, were also highly influential, providing new insights into their properties and applications in number theory and algebraic geometry.

A Lasting Legacy: Concepts Bearing His Name

The profound and pervasive impact of Charles Hermite's work is evident in numerous mathematical concepts, operators, and methods that bear his name, serving as a testament to his enduring influence across diverse mathematical disciplines and even in physics and computer science.

Hermite Polynomials

These are a classical sequence of orthogonal polynomials, crucial for solving differential equations, particularly the quantum harmonic oscillator in quantum mechanics, and in the study of probability distributions.

Hermite Interpolation

A method in numerical analysis that finds a polynomial that not only matches given function values at specific points but also matches the values of its derivatives at those points, leading to a smoother interpolation than standard Lagrange interpolation.

Hermite Normal Form

A canonical form for matrices with integer entries, used in computational number theory and lattice basis reduction algorithms, providing a unique representation for integer matrices.

Hermitian Operators

Fundamental in functional analysis and quantum mechanics, these are linear operators on a complex vector space that are equal to their own adjoints, representing observable quantities (like position, momentum, energy) in quantum systems because their eigenvalues are always real.

Cubic Hermite Splines

Widely used in computer graphics and geometric modeling, these are piecewise cubic polynomials used to interpolate a set of data points, ensuring both position and tangent continuity, resulting in smooth curves ideal for rendering and animation.

Beyond his direct mathematical output, Hermite's influence extended through his students. Notably, one of his most brilliant students was Henri Poincaré, who would himself become one of the most significant mathematicians and theoretical physicists of all time, further propagating Hermite's legacy and methods.

A Remark on the Frontiers of Mathematics

Charles Hermite, a product of his era's mathematical rigor, sometimes expressed a cautious perspective on revolutionary new concepts. In a revealing letter to his fellow mathematician Thomas Joannes Stieltjes, Hermite famously remarked about Weierstrass functions, "I turn with terror and horror from this lamentable scourge of continuous functions with no derivatives." This quote reflects the initial discomfort many mathematicians felt towards functions that, while continuous (meaning their graphs can be drawn without lifting the pen), did not have a derivative at any point – a concept then considered "pathological" and counter-intuitive to the classical smooth functions they were accustomed to. The Weierstrass function, first published in 1872, challenged traditional notions of smoothness and differentiability, foreshadowing the development of fractal geometry and deeper insights into real analysis.

Recognition Beyond Earth

Charles Hermite's immense contributions are also commemorated beyond our planet. The Hermite crater, a large lunar impact crater located near the Moon's north pole, is officially named in his honor. This extraterrestrial tribute underscores his global and lasting impact on the scientific world.

Frequently Asked Questions About Charles Hermite

Who was Charles Hermite?

Charles Hermite was a distinguished French mathematician (1822-1901) known for his extensive research in number theory, quadratic forms, invariant theory, orthogonal polynomials, elliptic functions, and algebra, and for proving that e is a transcendental number.

What was Charles Hermite's most famous mathematical proof?

His most famous proof, accomplished in 1873, was demonstrating that the mathematical constant e (the base of the natural logarithm) is a transcendental number. This was a pioneering achievement in number theory.

What is a transcendental number, and why was Hermite's proof significant?

A transcendental number is a number that is not algebraic, meaning it cannot be the root of any non-zero polynomial equation with integer coefficients. Hermite's proof for e was significant because it provided the first well-known example of such a number, paving the way for Lindemann's proof of the transcendence of π and resolving long-standing mathematical problems like squaring the circle.

Which famous mathematician was a student of Charles Hermite?

One of Charles Hermite's most renowned students was Henri Poincaré, an incredibly influential French mathematician, theoretical physicist, and philosopher of science, often described as the last universalist in mathematics.

Are there real-world applications of Hermite's work?

Absolutely. Concepts named after him, like Hermite polynomials, are fundamental in quantum mechanics and probability theory. Hermitian operators are essential in quantum physics, and cubic Hermite splines are widely used in computer graphics for creating smooth curves and surfaces.