The 11 Most Beautiful Mathematical Equations (2023)

By Clara Moskowitz


Live Science asked physicists, astronomers and mathematicians for their favorite equations. Here's what we found.

The 11 Most Beautiful Mathematical Equations (1)

Mathematical equations aren't just useful — many are quite beautiful. And many scientists admit they are often fond of particular formulas not just for their function, but for their form, and the simple, poetic truths they contain.

While certain famous equations, such as Albert Einstein's E = mc^2, hog most of the public glory, many less familiar formulas have their champions among scientists. LiveScience asked physicists, astronomers and mathematicians for their favorite equations; here's what we found.

Related: What's the Universe Made Of? Math, Says Scientist

General relativity

The 11 Most Beautiful Mathematical Equations (2)

The equation above was formulated by Einstein as part of his groundbreaking general theory of relativity in 1915. The theory revolutionized how scientists understood gravity by describing the force as a warping of the fabric of space and time.

(Video) The Most Beautiful Equation in Math

"It is still amazing to me that one such mathematical equation can describe what space-time is all about," said Space Telescope Science Institute astrophysicist Mario Livio, who nominated the equation as his favorite. "All of Einstein's true genius is embodied in this equation." [Einstein Quiz: Test Your Knowledge of the Genius]

"The right-hand side of this equation describes the energy contents of our universe (including the 'dark energy' that propels the current cosmic acceleration)," Livio explained. "The left-hand side describes the geometry of space-time. The equality reflects the fact that in Einstein's general relativity, mass and energy determine the geometry, and concomitantly the curvature, which is a manifestation of what we call gravity." [6 Weird Facts About Gravity]

"It's a very elegant equation," said Kyle Cranmer, a physicist at New York University, adding that the equation reveals the relationship between space-time and matter and energy. "This equation tells you how they are related — how the presence of the sun warps space-time so that the Earth moves around it in orbit, etc.It also tells you how the universe evolved since the Big Bang and predicts that there should be black holes."

The Standard Model

The 11 Most Beautiful Mathematical Equations (3)

Another of physics' reigning theories, the standard model describes the collection of fundamental particles currently thought to make up our universe.

The theory can be encapsulated in a main equation called the standard model Lagrangian (named afterthe 18th-century French mathematician and astronomerJoseph Louis Lagrange), which was chosen by theoretical physicist Lance Dixon of the SLAC National Accelerator Laboratory in California as his favorite formula.

"It has successfully described all elementary particles and forces that we've observed in the laboratory to date — except gravity," Dixon told LiveScience. "That includes, of course, the recently discovered Higgs(like) boson, phi in the formula. It is fully self-consistent with quantum mechanics and special relativity."

The standard model theory has not yet, however, been united with general relativity, which is why it cannot describe gravity. [Infographic: The Standard Model Explained]


The 11 Most Beautiful Mathematical Equations (4)

While the first two equations describe particular aspects of our universe, another favorite equation can be applied to all manner of situations. The fundamental theorem of calculus forms the backbone of the mathematical method known as calculus, and links its two main ideas, the concept of the integral and the concept of the derivative.

"In simple words, [it] says that the net change of a smooth and continuousquantity, such as a distance travelled,over a giventime interval (i.e. the difference in the values of the quantityat the end points of the time interval) is equal to the integral of the rate of change of that quantity, i.e. the integral of the velocity," said Melkana Brakalova-Trevithick, chair of the math department at Fordham University, who chose this equation as her favorite. "The fundamental theorem of calculus (FTC) allows us to determine the net change over an interval based on the rate of change over the entire interval."

The seeds of calculus began in ancient times, but much of it was put together in the 17th century by Isaac Newton, who used calculus to describe the motions of the planets around the sun.

Pythagorean theorem

The 11 Most Beautiful Mathematical Equations (5)

(Video) The 11 Most Beautiful Mathematical Equations

An "oldie but goodie" equation is the famous Pythagorean theorem, which every beginning geometry student learns.

This formula describes how, for any right-angled triangle, the square of the length of the hypotenuse, c, (the longest side of a right triangle) equals the sum of the squares of the lengths of the other two sides (a and b). Thus, a^2 + b^2 = c^2

"The very first mathematical fact that amazed me was Pythagorean theorem," said mathematician Daina Taimina of Cornell University. "I was a child then and it seemed to me so amazing that it works in geometry and it works with numbers!" [5 Seriously Mind-Boggling Math Facts]

1 = 0.999999999….

The 11 Most Beautiful Mathematical Equations (6)

This simple equation, which states that the quantity 0.999, followed by an infinite string of nines, is equivalent to one, is the favorite of mathematician Steven Strogatz of Cornell University.

"I love how simple it is —everyone understands what it says — yet how provocative it is," Strogatz said. "Many people don't believe it could be true. It's also beautifully balanced. The left side represents the beginning of mathematics; the right side represents the mysteries of infinity."

Special relativity

The 11 Most Beautiful Mathematical Equations (7)

Einstein makes the list again with his formulas for special relativity, which describes how time and space aren't absolute concepts, but rather are relative depending on the speed of the observer. The equation above shows how time dilates, or slows down, the faster a person is moving in any direction.

"The point is it's really very simple," said Bill Murray, a particle physicist at the CERN laboratory in Geneva. "There is nothing there an A-level student cannot do, nocomplexderivatives and trace algebras. But what it embodies is a whole new way of looking at the world, a whole attitude to reality and our relationship to it. Suddenly, the rigid unchanging cosmos is swept away and replaced witha personal world, related to what you observe. You move from being outside the universe, lookingdown, to one of the components inside it. But the concepts and the maths can be grasped byanyone that wants to."

Murray said he preferred the special relativity equations to the more complicated formulas in Einstein's later theory. "I could never follow the maths of general relativity," he said.

Euler's equation

The 11 Most Beautiful Mathematical Equations (8)

This simple formula encapsulates something pure about the nature of spheres:

(Video) Your Daily Equation #11: Euler's Identity or The Most Beautiful of all Equations

"It says that if you cut the surface of a sphere up into faces, edges and vertices, and let F be the number of faces, E the number of edges and V the number of vertices, you will always get V – E + F = 2," said Colin Adams, a mathematician at Williams College in Massachusetts.

"So, for example, take a tetrahedron, consisting of four triangles, six edges and four vertices," Adams explained. "If you blew hard into a tetrahedron with flexible faces, you could round it off into a sphere, so in that sense, a sphere can be cut into four faces, six edges and four vertices. And we see that V – E + F = 2. Same holds for a pyramid with five faces — four triangular, and one square — eight edges and five vertices," and any other combination of faces, edges and vertices.

"A very cool fact! The combinatorics of the vertices, edges and faces is capturing something very fundamental about the shape of a sphere," Adams said.

Euler-Lagrange equations and Noether's theorem

The 11 Most Beautiful Mathematical Equations (9)

"These are pretty abstract, but amazingly powerful," NYU's Cranmer said. "The cool thing is that this way of thinking about physics has survived some major revolutions in physics, like quantum mechanics, relativity, etc."

Here, L stands for the Lagrangian,which is a measure of energy in a physical system, such as springs, or levers or fundamental particles. "Solving this equation tells you how the system will evolve with time," Cranmer said.

A spinoff of the Lagrangian equation is called Noether's theorem, after the 20th century GermanmathematicianEmmy Noether. "This theorem is really fundamental to physics and the role of symmetry," Cranmer said."Informally, the theorem is that if your system has a symmetry, then there is a corresponding conservation law.For example, the idea that the fundamental laws of physics are the same today as tomorrow (time symmetry) implies that energy is conserved. The idea that the laws of physics are the same here as they are in outer space implies that momentum is conserved.Symmetry is perhaps the driving concept in fundamental physics, primarily due to [Noether's]contribution."

The Callan-Symanzik equation

The 11 Most Beautiful Mathematical Equations (10)

"The Callan-Symanzik equation is a vital first-principles equation from 1970, essential for describing how naive expectations will fail in a quantum world," said theoretical physicist Matt Strassler of Rutgers University.

The equation has numerous applications, including allowing physicists to estimate the mass and size of the proton and neutron, which make up the nuclei of atoms.

Basic physics tells us that the gravitational force, and the electrical force, between two objects is proportional to the inverse of the distance between them squared. On a simple level, the same is true for the strong nuclear force that binds protons and neutrons together to form the nuclei of atoms, and that binds quarks together to form protons and neutrons. However, tiny quantum fluctuations can slightly alter a force's dependence on distance, which has dramatic consequences for the strong nuclear force.

"It prevents this force from decreasing at long distances, and causes it to trap quarks and to combine them to form the protons and neutrons of our world," Strassler said. "What the Callan-Symanzik equation does is relate this dramatic and difficult-to-calculate effect, important when [the distance] is roughly the size of a proton, to more subtle but easier-to-calculate effects that can be measured when [the distance] is much smaller than a proton."

The minimal surface equation

The 11 Most Beautiful Mathematical Equations (11)

(Video) Proving the Most Beautiful Equation Bob Ross Style

"The minimal surface equation somehow encodes the beautiful soap films that form on wire boundaries when you dip them in soapy water," said mathematician Frank Morgan of Williams College. "The fact that the equation is 'nonlinear,' involving powers and products of derivatives, is the coded mathematical hint for the surprising behavior of soap films. This is in contrast with more familiar linear partial differential equations, such as the heat equation, the wave equation, and the Schrödinger equation of quantum physics."

The Euler line

The 11 Most Beautiful Mathematical Equations (12)

Glen Whitney, founder of the Museum of Math in New York, chose another geometrical theorem, this one having to do with the Euler line, named after 18th-century Swiss mathematician and physicist Leonhard Euler.

"Start with any triangle," Whitney explained. "Draw the smallest circle that contains the triangle and find its center.Find the center of mass of the triangle — the point where the triangle, if cut out of a piece of paper, would balance on a pin. Draw the three altitudes of the triangle (the lines from each corner perpendicular to the opposite side), and find the point where they all meet. The theorem is that all three of the points you just found always lie on a single straight line, called the 'Euler line' of the triangle."

Whitney said the theorem encapsulates the beauty and power of mathematics, which often reveals surprising patterns in simple, familiar shapes.

Follow Clara Moskowitz on Twitter @ClaraMoskowitzor LiveScience @livescience. We're also on Facebook & Google+.

Clara Moskowitz

Clara has a bachelor's degree in astronomy and physics from Wesleyan University, and a graduate certificate in science writing from the University of California, Santa Cruz. She has written for both and Live Science.


What is the most beautiful equation in math? ›

Euler's Identity is written simply as: e^(iπ) + 1 = 0, it comprises the five most important mathematical constants, and it is an equation that has been compared to a Shakespearean sonnet. The physicist Richard Feynman called it “the most remarkable formula in mathematics”.

What's the answer to x3 y3 z3 K? ›

The equation x3+y3+z3=k is known as the sum of cubes problem. For decades, a math puzzle has stumped the smartest mathematicians in the world. x3+y3+z3=k, with k being all the numbers from one to 100, is a Diophantine equation that's sometimes known as "summing of three cubes." ∴ The required result will be 3xyz.

What are the top 5 formulas in math? ›

  • Completing the square: x2+bx+c=(x+b2)2−b24+c.
  • Quadratic formula: the roots of ax2+bx+c are −b±√b2−4ac2a.
  • Circle: circumference=2πr, area=πr2.
  • Sphere: vol=4πr3/3, surface area=4πr2.
  • Cylinder: vol=πr2h, lateral area=2πrh, total surface area=2πrh+2πr2.

What is the most beautiful equation in science? ›

The formula most commonly rated as beautiful in the study, in both the initial survey and the brain scan, was Euler's equation, e+ 1 = 0.

What are the 7 hardest math problems? ›

Clay “to increase and disseminate mathematical knowledge.” The seven problems, which were announced in 2000, are the Riemann hypothesis, P versus NP problem, Birch and Swinnerton-Dyer conjecture, Hodge conjecture, Navier-Stokes equation, Yang-Mills theory, and Poincaré conjecture.

Why 0.99999 is 1? ›

The meaning of the notation 0.999... is the least point on the number line lying to the right of all of the numbers 0.9, 0.99, 0.999, etc. Because there is ultimately no room between 1 and these numbers, the point 1 must be this least point, and so 0.999... = 1.

What is the longest formula ever? ›

What is the world's longest equation? Answer – The Boolean Pythagorean Triples issue was initially introduced in the 1980s by California-based mathematician Ronald Graham is the longest arithmetic equation, according to Sciencealert, and includes roughly 200 gigabytes of text.

What's the hardest math problem? ›

Today's mathematicians would probably agree that the Riemann Hypothesis is the most significant open problem in all of math. It's one of the seven Millennium Prize Problems, with $1 million reward for its solution.

What is the hardest math problem solved? ›

“There are no whole number solutions to the equation xn + yn = zn when n is greater than 2.” Otherwise known as “Fermat's Last Theorem,” this equation was first posed by French mathematician Pierre de Fermat in 1637, and had stumped the world's brightest minds for more than 300 years.

What are the 6 hardest math problems? ›

Millennium Prize Problems
  • Birch and Swinnerton-Dyer conjecture.
  • Hodge conjecture.
  • Navier–Stokes existence and smoothness.
  • P versus NP.
  • Riemann hypothesis.
  • Yang–Mills existence and mass gap.

What are the 9 formulas of algebra? ›

Here is a list of Algebraic formulas –
  • a2 – b2 = (a – b)(a + b)
  • (a + b)2 = a2 + 2ab + b2
  • a2 + b2 = (a + b)2 – 2ab.
  • (a – b)2 = a2 – 2ab + b2
  • (a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca.
  • (a – b – c)2 = a2 + b2 + c2 – 2ab + 2bc – 2ca.
  • (a + b)3 = a3 + 3a2b + 3ab2 + b3 ; (a + b)3 = a3 + b3 + 3ab(a + b)

What does <_ mean in math? ›

< Less Than and > Greater Than

This symbol < means less than, for example 2 < 4 means that 2 is less than 4. This symbol > means greater than, for example 4 > 2. ≤ ≥ These symbols mean 'less than or equal to' and 'greater than or equal to' and are commonly used in algebra.

What is the maths equation for I love you? ›

143. This one is the most common mathematical way of saying I love you and you probably know of it already! The numbers 1,4,3 represent the number of alphabets that are present in each of the words of the phrase 'I love you'. That is: I = 1, love = 4 and you = 3.

What is a unique equation? ›

A unique solution means only one solution. If a linear equation has a unique solution means only one solution set exists for the equation. A system of linear equations a 1 x + b 1 y = 0 a 2 x + b 2 y = 0 has a unique solution, if a 1 a 2 ≠ b 1 b 2 .

What is the equation of life? ›

E=mc^2 : Equation of Life and Death - Gonit Sora.

Has 3X 1 been solved? ›

After that, the 3X + 1 problem has appeared in various forms. It is one of the most infamous unsolved puzzles in the word. Prizes have been offered for its solution for more than forty years, but no one has completely and successfully solved it [5].

What is the 1 million dollar math problem? ›

The Riemann hypothesis – an unsolved problem in pure mathematics, the solution of which would have major implications in number theory and encryption – is one of the seven $1 million Millennium Prize Problems. First proposed by Bernhard Riemann in 1859, the hypothesis relates to the distribution of prime numbers.

What is the oldest unsolved math problem? ›

But he doesn't feel bad: The problem that captivated him, called the odd perfect number conjecture, has been around for more than 2,000 years, making it one of the oldest unsolved problems in mathematics.

Why is Euler's number so special? ›

Key Takeaways. Euler's number is an important constant that is found in many contexts and is the base for natural logarithms. An irrational number represented by the letter e, Euler's number is 2.71828..., where the digits go on forever in a series that never ends or repeats (similar to pi).

Why 1 0 does not exist? ›

As much as we would like to have an answer for "what's 1 divided by 0?" it's sadly impossible to have an answer. The reason, in short, is that whatever we may answer, we will then have to agree that that answer times 0 equals to 1, and that cannot be ​true, because anything times 0 is 0. Created by Sal Khan.

Why does Euler's number exist? ›

It's used to calculate compounding interest, the rate of radioactive decay, and the amount of time it takes to discharge a capacitor. As Stefanie Reichert puts it in Nature Physics, “we cannot escape Euler's number.”

What is the longest chemical symbol? ›

The IUPAC name for Titin. This is the largest known protein and so has the longest chemical name. Written in full, it contains 189,819 letters.

What's the maximum size for a formula? ›

8,192 characters

What is the longest math proof? ›

The Stampede supercalculator used for solving the "Boolean Pythagorean triples problem." Researchers use computers to create the world's longest proof, and solve a mathematical problem that had remained open for 35 years. It would take 10 billion years for a human being to read it.

What is the smartest math problem? ›

For decades, a math puzzle has stumped the smartest mathematicians in the world. x3+y3+z3=k, with k being all the numbers from one to 100, is a Diophantine equation that's sometimes known as "summing of three cubes."

What is the answer for 3x 1? ›

In the 3x+1 problem, no matter what number you start with, you will always eventually reach 1. problem has been shown to be a computationally unsolvable problem.

Are there 7 unsolved math problems? ›

The Clay Mathematics Institute officially designated the title Millennium Problem for the seven unsolved mathematical problems, the Birch and Swinnerton-Dyer conjecture, Hodge conjecture, Navier–Stokes existence and smoothness, P versus NP problem, Riemann hypothesis, Yang–Mills existence and mass gap, and Poincaré ...

What is the largest number in the world? ›

A "googol" is the number 1 followed by 100 zeroes. The biggest number with a name is a "googolplex," which is the number 1 followed by a googol zeroes.

What is the hardest math in high school? ›

What is the Hardest Math Class in High School? In most cases, you'll find that AP Calculus BC or IB Math HL is the most difficult math course your school offers. Note that AP Calculus BC covers the material in AP Calculus AB but also continues the curriculum, addressing more challenging and advanced concepts.

Is there any odd perfect number? ›

While even perfect numbers are completely characterized, the existence or otherwise of odd perfect numbers is an open problem. We address that problem and prove that if a natural number is odd, then it's not perfect.

What are some perfect numbers? ›

perfect number, a positive integer that is equal to the sum of its proper divisors. The smallest perfect number is 6, which is the sum of 1, 2, and 3. Other perfect numbers are 28, 496, and 8,128.

What is 1 1 in math? ›

Answer: 1 : 1 ratio means when two quantities are measured or expressed in the same proportion. The ratio a : b helps us to know how much one part of a is equivalent to one part of b. Explanation: When two quantities are taken in the same proportion, they are said to be in the ratio of 1:1.

What is famous math formula? ›

E=mc^2. For our first, we'll take perhaps the most famous equation of all. Albert Einstein's 1905 equation relating mass and energy is both elegant and superficially counterintuitive. It says that energy is equal to the mass of an object in its rest frame multiplied by the speed of light squared.

What does □ mean in math? ›

□ (the Halmos tombstone) means “QED”, which is an abbreviation for the Latin phrase quod erat demonstrandum (“which was to be proven”). “QED” has been the most common way to symbolize the end of a logical argument for many centuries, but the modern convention in mathematics is to use the “tombstone” in place of “QED”.

What does € mean in texting? ›

In Statistics Explained articles the symbol '€' should be used for euro in the text if it is followed by a number.

What are these symbols called in English * {} []? ›

Curly brackets, also known as braces, are rarely used punctuation marks that are used to group a set.

What is the answer to 128 √ e980? ›

128√e980 = I love You.

What is the Infinity equation? ›

x × ∞ = ∞

What is the 42 equation? ›

It took over a million hours of computing time, but the two mathematicians found their solution. So, the full equation is (-80538738812075974)3 + 804357581458175153 + 126021232973356313 = 42.

What is mathematical number for beauty? ›

Can you give a number to beauty? The answer is yes – it is roughly 1.62 and it is called the Golden Ratio of beauty! The Golden Ratio (also known as Phi, or the Fibonacci number) is the mathematical symmetry algorithm that underlies our perception of attractiveness.

What is the world's hardest math equation? ›

For decades, a math puzzle has stumped the smartest mathematicians in the world. x3+y3+z3=k, with k being all the numbers from one to 100, is a Diophantine equation that's sometimes known as "summing of three cubes."

Does Euler's identity prove God? ›

While Leonhard Euler was himself a Christian, it is completely inconceivable that he himself actually thought this was proof of God. It's a non-sequitur, intended to be humorously surprising. Euler knew damn well that it was not really proof of anything at all, much less God.

What is the most famous saying in math? ›

Famous math quotes
  • Pure mathematics is, in its way, the poetry of logical ideas. ...
  • Without mathematics, there's nothing you can do. ...
  • Mathematics is the most beautiful and most powerful creation of the human spirit. ...
  • We will always have STEM with us.
Jul 17, 2019


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