In mathematics, a 'perfect number' is equal to the sum of the numbers it is divisible by, excluding itself, such as 28, which can be evenly divided by 1, 2, 4, 7, and 14, and 1 + 2 + 4 + 7 + 14 equals 28.

The most bizarre numbers in the universe

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...

Within mathematics, some infinities are bigger than others

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.

Infinity can come in different sizes and be ordered by these sizes

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.

Data suggests that claims about the golden ratio underlying the geometry seen in nature, from mollusk shells to human proportions, are largely exaggerated.

Calculators convert input values into a form of scientific notation—a fixed set of digits with a decimal multiplied by a base to some exponent—to store and process very large or very small quantities efficiently at the cost of rounding errors.

How the number 15 solved a longstanding math challenge

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...

In 2000, the Clay Mathematics Institute placed million-dollar bounties on seven unsolved problems in mathematics—the Millennium Prize Problems—in the hopes of spurring solutions.

The Ishango bone suggests mathematics was born about 20,000 years ago

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.

The history of the 10 axioms of Zermelo–Fraenkel set theory with choice, regarded as the most universal foundations of math

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.

The odds of a perfect March Madness bracket are worse than finding one specific grain of sand on Earth

With 63 games to predict, random guessing produces odds of 1 in 9.2 quintillion. Even an informed picker who gets 67% of games right faces odds of about 1 in 120 billion, meaning if every American filled out a unique bracket, we'd expect a perfect one in roughly 366 years.

A mathematical breakdown of how RSA encryption uses prime numbers to encrypt data

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.

Mathematical breakthroughs influenced Shakespeare’s stories

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.

One of the first mechanical calculators—the Pascaline—was built in the 1640s by French mathematician Blaise Pascal to help his father, who worked as a tax collector.

How quantum computers can empower Shor's algorithm to break digital encryption

Secure communications largely rely on numeric keys generated by multiplying two enormous prime numbers together, and it is computationally impractical to determine these primes by trying to divide by every possible one. While Shor's algorithm can provide better guesses for this brute-force approach, it relies on a number that is difficult to determine on classical computers but much easier to calculate using the simultaneous computations enabled by quantum superposition.

Most people will probably never hit a perfect 850 credit score for a variety of reasons, including the fact that younger borrowers are at a disadvantage as credit history length accounts for 15% of one's score.

Even in a perfect vacuum, empty space is full of quantum activity

Scientists have tried to create “nothing” by removing all matter, energy, and heat from a container, but the uncertainty principle and relativity prevent such conditions within any period of time. This means the universe is full of quantum foam, which can produce forces from nothingness.

A 14-year-old "human calculator" shatters six mathematical world records in one day

A teenage prodigy from Maharashtra, India named Aaryan Shukla set the record for the fastest time to mentally add 50 five-digit numbers. He did so in just 25.19 seconds–roughly equivalent to one addition every 0.5 seconds. The math whiz said he practices calculations for around five to six hours a day.

The first algorithmic random number generator was developed in 1946

Although previous methods of randomness involved physical tools with limited outcomes, such as dice, or relied on chaotic variables for number generation, such as electronic noise, John Von Neumann's middle-square method turned the problem into a mathematical computation. It has since been replaced by methods that generate numbers from unique initial values, such as the time on a clock.

Mathematicians often solve problems using wordless mental processes

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.

Abaci, the plural of abacus, allowed individuals to perform arithmetic and tally quantities by moving pebbles called calculi, from which the word 'calculator' is derived.

The Koch snowflake has an infinite perimeter enclosing a finite area

Counterintuitively, although it would be impossible to build a fence along the edges of the shape because it would require an infinite amount of materials, this perimeter can be contained in a larger shape with a finite perimeter and limited area. This is equivalent to enclosing an infinitely long spiral that never reaches the center within a circle.

Taxicab geometry defines displacement as the sum of vertical and horizontal distances

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.

As of 2025, the College Board has banned the use of calculators with computer algebra systems on the SAT and PSAT.

Although there are 10,000 possible combinations of four-digit personal identification numbers, data suggests that about 27% of all PINs are one of 20 combinations, including 1234, 1111, and 0000.

How a competition honoring a king's birthday uncovered history's first chaotic system

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.

There are more chess game variations than observable atoms in the universe

Mathematician Claude Shannon estimated there are 10 to the 120th power variations of chess games possible. Known as the "Shannon number," this is more than the estimated number of atoms in the cosmos.

Logic gates—digital versions of conditional statements, such as 'if 0 and 1, then 1'—are specialized circuits that combine multiple digital signals, like the binary code representing numbers, into a single output, thereby allowing machines to perform math.

Explore over 30,000 Olympiad-level math problems from around the world.

It took 70 years for computer scientists to determine how to cut a cake fairly

The cake-cutting problem explores how to divide something fairly, which is not solely about cutting evenly-sized pieces. A protocol that accounted for personal preferences (e.g., a corner piece) between just three people took 16 years to develop.

No map of the Earth is perfect because each introduces inaccuracies in the shape, size, or orientation of landmasses and oceans by trying to flatten the planet's curved surface.

By identifying patterns in the periodic movements of the sun, moon, and Earth, ancient civilizations realized that identical eclipses repeat about once every 6,585 days, known as the saros cycle.

Try counting like the Maya in this interactive math game

The Maya numerical system used dots for 1, bars for 5, and a shell symbol for zero. This interactive resource lets users practice counting with the Maya base-20 system using the same symbols once common in the Yucatán.

Compound interest has been called the eighth wonder of the world

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.

Once called 'aces,' quarks are fundamental pieces of matter that combine to create over 100 known composite particles, including the proton and neutron.

In a standard deck of 52 cards, shuffling produces one of approximately 8.07 × 10⁶⁷ potential card orders—a number about as large as the number of atoms in the Milky Way.

Four astronomical discoveries from ancient Greece

Modern astronomy relies on making discoveries without being present on the planets, stars, galaxies, and other objects studied. Imagine the difficulty the ancient Greeks had studying them without computers and telescopes. Yet somehow they calculated precise measurements of planetary distances, sizes, and more. Read this page to learn more about early triumphs in astronomical science.

The term 'algorithm' originated from the Latinized name of al-Khwārizmī, a 9th-century mathematician whose works introduced Arabic numerals and the decimal-based numeral system to Europe and whose treatise 'Al-Jabr' gave algebra its name.

Explore the fundamental constants of the metric system

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.

The first commercially successful calculating machine—the arithmometer—was patented in 1820 but not mass-produced for over three decades, likely because its inventor was focused on his insurance business.

Earth's total mass is equivalent to 13 quadrillion Egyptian pyramids of Khafre

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.

Despite the name, infinity mirrors—mirrors that reflect each other—do not produce reflections to infinity because mirrors absorb some of the incoming light with each reflection.

Antimatter's rarity across the cosmos is one of the biggest problems in modern physics because the high-energy environment of the early universe should have produced equal amounts of matter and antimatter.

Astronomers inferred Neptune's existence through mathematics while attempting to explain irregularities in Uranus' orbit, but Neptune had been directly observed 234 years earlier by Galileo.

Antimatter was predicted to exist four years before it was observed, though the mathematics responsible for the prediction described it as matter with negative energy.

Figuring out how to get a sofa around a corner has been a math problem since 1966

A sofa with the largest possible area that can make a 90-degree turn is known as Gerver's sofa, named after the mathematician who determined the shape in 1992. In 2025, a 119-page paper provided the proof confirming this shape.

Einstein created a mathematical term for anti-gravity to keep the universe static

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.

One model of the universe expresses all constants using only time

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.

A temperature of absolute zero is impossible to reach

Zero on the Kelvin scale—equivalent to -273.15 degrees Celsius—describes a system where all atomic motion completely stops, which is forbidden according to thermodynamics. As of 2025, the closest scientists have gotten to absolute zero is 0.00000000004 K.

Why are there as many males as females?

Pretty much every species breaks down with an equal sex-ratio, with roughly the same number of males as females. But on the surface, it appears that most species wouldn't need many males to keep the population growing, with their energy-efficient, abundant sex cell production. So why is it equal? Scientists argue an equal ratio is the most evolutionarily stable. Learn about this quirk of biology here.

Modern physics suggests that many universes with different physical laws could exist

Physicists suggest that quantum fluctuations in the early universe could have given rise to these universes. Ancient Greek philosophers like Chrysippus proposed a version of the multiverse over 2,000 years ago—millennia before its adoption into various theoretical frameworks today.