Described as the most beautiful equation in mathematics, Euler's identity combines five unrelated but fundamental constants into one relationship.

Often depicted as one formula, Einstein's equation represents a suite of 16 nonlinear partial differential equations, which can be simplified to six unique ones, that relate the geometry of spacetime to the presence of mass and energy.

Even for math-y types, some numbers are incomprehensible. This article dives into several kinds of numbers more complicated than even irrational numbers with endless decimals like "pi" or Euler's number: Noncomputable numbers. Read the bizarre...

Although working independently of one another, the two exchanged correspondence in the lead up to November 1915, when Hilbert submitted their gravitational field equations five days before Einstein presented their work. An error in Hilbert's formulation and its nonpublication until March 1916 would leave credit for general relativity to Einstein.

General relativity predicted that the gravity of all the matter in the universe would cause it to collapse, contradicting the belief at the time that the universe was constant and unchanging. Albert Einstein added the cosmological constant to his equations, which acted like anti-gravity, to "fix" his math.

Newton's law of universal gravitation was used to calculate the value after Henry Cavendish torsion balance experiments determined the missing value of the universal gravitational constant from Newton's equation.

In 2019, the International System of Units was revised, such that all seven fundamental units are defined in terms of combinations of seven fundamental constants of the universe. These include the speed of light in a vacuum, Planck's constant, Avogadro's number, and the elementary charge of protons and electrons.

Within the pages are calculations, sketches and his first attempts at deriving what would be the field equations for gravity. It also documents some dead ends and the difficulties Einstein encountered as he describes part of the mathematics of what would become general relativity, describing it as "too involved."

That's because it has the near-magical quality of helping financial investments grow exponentially. Rather than accruing a standard lump of simple interest annually, compound interest takes the previous year’s yield and reinvests it as part of the principal.

In 1935, Einstein suggested that quantum mechanics was incomplete due to its strange implications about entangled particles. Their challenge led to decades of experiments that eventually proved distant particles could affect each other instantly.

Like postulates in geometry—accepted truths that form the logical building blocks for formal proofs of more complex theorems—axioms cannot be proven, but set the underlying logic and rules for systems. The ZFC axioms resulted from painstaking work to identify the underlying rules of mathematics and are built on the premise of sets, or well-defined collections of elements.

About seven base units, such as meters for length, have historically been used to describe everything in our physical reality. However, researchers have shown that, through relativistic space-time, clocks can serve as replacements for rulers, scales, and other instruments, allowing everything to be measured in terms of time.

The “packing coloring” problem asks how many numbers are needed to fill an infinite grid so that identical numbers never get too close to one another. A new computer-assisted proof finds a surprisingly straightforward answer—only 15. While...

It's a common technique among expert mathematicians, where thinkers rely on intuition, blurry perceptions and subconscious work—no writing, no formulas. The state is distinct from daydreaming, and often relies on sudden illuminations stemming from extended periods of subconscious reflection.

New ideas in math were transforming Europe at the time Shakespeare penned his plays, including newly popular conceptions of zero via philosophers Descartes and Leibniz.

The Wheeler-DeWitt equation, a modified version of the Schödinger equation—the formula that describes the evolution of the quantum configuration of a system—describes the quantum state of space-time across the universe. The transformation causes the variable of time to disappear, suggesting it is not a fundamental, physical parameter of the cosmos.

In this mathematical system, pi is valued at four, and circles aren't round—they appear as diamonds with corners instead. The system simplifies pixel distances in digital imaging and helps engineers with robot navigation and chip design.

Recognizing the mathematical complexities of generalizing special relativity with a new picture of gravity and acceleration, Albert Einstein had Grossman introduce him to tensor calculus. Years before developing the precursor to GR together, Einstein would use Grossman's lecture notes to prepare for his exams.

A committee established a contest for King Oscar II's 60th birthday, featuring four unsolved problems in mathematics, including one on proving the stability of a system of more than two objects held together by gravity. Focusing on three celestial objects, Henri Poincaré won the prize for his proof, which he later learned was incorrect and, upon correcting it, instead proved these systems to be unstable.

The discovery of chaos theory came from Edward Lorenz's computer model of the atmosphere, which used three equations to quantify atmospheric circulation. However, the chaotic evolution of this system creates wandering figure-8-like shapes—some resembling butterfly wings—as it repeatedly attempts to settle on one of two values along unique trajectories.

The simplest models assume uniformity across the Earth, with incoming energy from the sun being offset by the planet's albedo, or reflectiveness, which prevents the radiation from being fully absorbed. Incorporating elements such as atmospheric layers, latitude bands, geographic variability, and emissivity yields coupled differential equations quantifying Earth's interconnected systems.

Russian mathematician Georg Cantor demonstrated that certain sets of numbers (e.g., the natural numbers) form countable infinities, while others cannot be counted (e.g., the real numbers) and are therefore uncountable, representing larger versions of infinity. Through the axiom of choice, German mathematician Ernst Zermelo proved this concept of ordering infinities.

The thought experiment explains why every electron has the same charge and mass as every other electron, and describes every positron—the antimatter version of the electron—as that same electron traveling backwards in time. Among other foundational challenges, the model suggests that the universe contains equal numbers of electrons and positrons, yet normal matter is much more common than antimatter.

The total amounts of properties like energy, charge, and angular momentum within a system have been experimentally observed to remain constant, even as changes occur. Particle interactions also conserve specific quantities, such as baryon and lepton numbers. Inconsistencies in these values within isolated systems have historically hinted at the existence of new physics.

The shape—called an analemma—can be obtained by photographing the sun from the same location and time daily for a year. The Earth's eccentricity created the analemma's width, and its tilt shifts the sun's height in the sky over the seasons.

Ancient civilizations developed mathematical systems for various purposes, including the Egyptians for the construction of the pyramids and the Ancient Greeks for logical reasoning in theorems. However, these systems were predated by the Ishango bone—found in the Democratic Republic of the Congo—whose notches suggest a base ten system, prime numbers, and simple mathematical operations.

While the concept of infinity refers to a countless set of items, these infinite sets can be compared to one another by matching an item from one set to an item in another, one by one. This analysis shows that certain sets must be larger than others because some infinite sets have leftover, unmatched items.

Stokes waves—a solution to the mathematics of water waves—were once assumed to be stable. Italian mathematicians have proven disturbances cause these waves to collapse and such collapses follow an infinite repeating pattern. To complete the proof, the team offered a $100 donation to solicit online help.

This form of asymmetric cryptography uses mathematical functions that rely on the product of two large prime numbers. Because traditional computers struggle to factor such products, obtaining the private keys needed to decrypt messages becomes effectively impossible. This security can be reinforced with larger prime numbers, though this makes encryption more computationally taxing.