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.

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...

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

On a basic calculator, the keys that start with M, such as M+ and M-, are memory keys that allow users to store and modify a number in the device's memory.

Slide rules—ruler-sized mechanical calculators with a component that slides out to show relationships between numbers—were used by engineers at NASA during the Apollo missions.

The Year 2038 problem is the next potential computer time rollover bug

Many older systems store time as a signed 32-bit integer counting seconds since Jan. 1, 1970. That counter maxes out on Jan. 19, 2038—overflowing into negative time and sending clocks back to 1901, potentially crashing any older software that depends on accurate dates.

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.

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.

Since computers rely on the distribution of electrons in microchips to represent binary code, cosmic rays and ionizing radiation from radioactive elements can introduce software errors by detaching electrons from atoms.

Watch how the S&P 500 is actually calculated

After accounting for each company's float-adjusted market capitalization (a metric that combines the price of a company's stock, the total number of shares, and the stock's float, which means how many shares are actually available), the S&P 500 formula uses a divisor to create the index number. However, what the divisor is and what it represents are secrets.

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.

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.

How 'vibes' factor into scientists' decisions about using certain units

Although there are absolute quantities from which units can be defined, such as the speed of light and the corresponding light year for distance, the usefulness of units comes from their ability to easily communicate values, rather than how fundamental their origins are. For example, while Kelvin measures absolute temperature, Celsius is scaled to the temperatures of boiling and freezing water, making them more practical for everyday usage.

A battery management system determines the displayed battery percentage

The software in modern devices provides values by combining estimates based on voltage—akin to knowing how much air is left in a tire based on the pressure of the escaping air—and coulomb counting, where the rate of electrical current is calibrated to the ratio of remaining charge to full capacity.

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.

With many smaller computing cores, graphics processing units process many identical tasks in parallel, whereas central processing units have fewer, larger cores built to handle a wide variety of more complex tasks sequentially.

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.

Despite the abundance of free graphing software, Texas Instruments' de facto monopoly and the continued demand for its use in standardized testing have kept the cost and design of graphing calculators largely unchanged for decades.

See how computer memory really works—from binary bits to modern RAM

The video explains how computers store information as 1s and 0s, how RAM functions as fast working memory, and why systems utilize different types of storage. The video demonstrates how this process enables computers to operate.

An explanation of Turing machines—the foundations of digital algorithms and computers

The device reads data, overwrites or ignores it, and then moves on to the next piece of data based on a set of programmed rules. By imagining this data as zeros and ones—binary that makes up computer code—Turing machines became the theoretical models for how any algorithm could be run on a computer if the procedure could be coded into a set of digital rules.

'Floating rate' loans periodically adjust their interest rates in accordance with economic conditions

When the Federal Reserve began raising interest rates in 2022 to combat inflation, interest payments borrowers owed to private credit lenders increased accordingly. Private credit funds benefited as loan returns increased, making the asset class attractive to investors at the time.

Misunderstandings about how quantum computing works may have overhyped their ability

Quantum computers are often viewed as machines that can solve all problems instantly by trying every solution simultaneously. Instead, they amplify correct answers and cancel out wrong ones, which only works for specific kinds of problems with special structures.

All computers are defined by their ability to input, store, process, and output data

From wooden machines to wearable tech, computers have changed a lot in appearance, but they all still perform the same essential tasks. Early computers were made of wood and metal—some were so big that they filled entire rooms just to solve simple math problems.

Qubits let quantum computers process massive datasets all at once

The quantum phenomena of superposition and entanglement enable qubits to act in 0 and 1 binary states simultaneously and influence each other instantly. These features give quantum computers incredible power, though keeping them stable is a major challenge.

Within quantum mechanics, the Heisenberg uncertainty principle limits how precisely one can simultaneously know the values of pairs of complementary properties for an object, such as position and momentum.

Designed in 1937 by Harvard graduate student Howard H. Aiken, the 'Harvard Calculator' was used by the US Navy to solve advanced physics problems.

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.

Explore chaos in this simulation of two double pendulums, where small differences in the release angles produce wildly different outcomes.

Since 1967, a second has been defined as how long it takes an electron to transition between two energy levels of a cesium-133 atom 9,192,631,770 times.

Explore interactive simulations of strange attractors, the graphs that help visualize chaos

Although simple, linear equations reach an explicit value for a given input, such as *f(x) \= x* + 1 equaling 4 when *x* equals 3, some chaotic, nonlinear systems tend to evolve towards multiple values simultaneously. These simulations allow users to observe the unusual mathematical artifacts created by systems as their outputs evolve toward these distinct values.

Unsatisfied with the designs provided by an Apple engineer, Steve Jobs designed the original Mac calculator in about 10 minutes.

Compounding, explained

During conversations about personal finance, the term "compounding" gets thrown around—but not everyone understands how compound interest and compound returns work. Broadly speaking, compounding is what happens when you not only earn returns or interest on your original investment, but also on the accumulated returns or interest you receive over time. This article explains how both of these processes function, and what compounding could mean for your wallet.

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.

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

See how a good VantageScore and a good FICO score can differ

While VantageScores and FICO scores both offer credit scores between 300 and 850 and consider the same core factors, they weigh those factors differently, and specific numbers can have different meanings depending on the type of score.

When leap seconds are needed to resynchronize clocks with Earth's rotation, instead of shifting time all at once, technology companies implement a leap smear, which evenly distributes the change over a 24-hour period.

More than 2,000 years ago, a hand-powered model of the solar system called the Antikythera mechanism—the oldest example of an analog computer—could predict solar eclipses.

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.

Accurate climate models incorporate calculus to account for complex feedback loops

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.

Uncertainties in measurements make finding gravity's Newtonian limit unclear

Although Newton's "universal" law is unreliable in strong gravitational fields when compared with the field equations of general relativity, it is still taught for its accuracy on everyday scales. However, gravity's effect on small scales is too weak to be measured reliably, its accuracy at atomic scales remains unknown.

VisiCalc was the first electronic spreadsheet, released with Apple II's personal computer in 1979

Before 1979, managers were essentially information processors working off paper ledgers. Scenario modeling was slow, manual, and expensive. Companies were understood as organizations that built things with financial analysis as secondary. VisiCalc made calculation so cheap that you could work iteratively, setting out forecasts and tweaking assumptions until you got the answer you wanted. Oks argues It created a new way of seeing the world.

Lasers, semiconductors, medical imaging, and atomic clocks rely on quantum mechanics

Semiconductors, crucial in electronics, operate based on quantum superposition. Lasers use stimulated emission, a quantum process, and MRI machines leverage the quantum spin of hydrogen atoms for imaging. Atomic clocks measure time with unprecedented precision using quantum transitions.

Use this interactive calculator to see if you could rebalance the federal budget by adjusting taxes and spending.

Take a virtual tour inside the NVIDIA GA102 graphics processing unit to explore the architecture that enables it to perform over 35 trillion calculations per second.

If the temperature of water continuously decreases but it is not given room to expand, it will eventually contract and form exotic forms of ice that sink in liquid water.

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.

Virtual particles are bookkeeping tools used to describe force interactions

Physicists mathematically describe the electromagnetic, weak nuclear, and strong nuclear forces as the result of these particles being exchanged between particles of matter. Despite not being real, observable particles, this method has provided experimental predictions with an accuracy of over 12 decimal places.

Why atoms are used to keep track of time and define the meaning of a second

Small variations in the production of human-made devices, such as mechanical watches and pendulums, make them unreliable for the ultraprecise timekeeping required by modern technology, including financial transaction timestamping and GPS. Because atoms obey fundamental laws of nature and never wear out, their light emissions when energized provide a consistent timekeeping signal.

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.

It's tough to measure just how much the global commodities market is worth

That's partly because of all the different ways and places commodities are traded—such as futures exchanges, spot markets, and over-the-counter deals. Additionally, the same commodity can be traded multiple times in different markets, making it easy to risk duplicating trade values in calculations.

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.