Chapter 1: What Constitutes the Largest Number?

In the realm of abstraction we inhabit, how many digits do we truly require? This exploration delves into the longest conceivable total in a non-technical manner.

TLDR: The largest number can be represented in 16 digits.

If you consult an astronomer, they might take you on an expansive journey through the universe. A microbiologist, however, will lead you into the enigmatic world of quarks and preons, the fundamental components of all matter. Then there's my five-year-old, who would simply say "643."

This query invites a wide range of guesses, yet when we scrutinize it closely, it becomes strikingly personal.

Infinity Excluded

I choose to exclude infinity from this discussion, treating it like imaginary numbers; it serves more as a concept than a practical numeral. It can never be represented on a measuring tape.

Parameters for Consideration

1. Each digit must have significance:

   While it's awe-inspiring to note that the universe spans about 93 billion light-years, this measurement could easily be 93.4 or 93.5 without losing its meaning. To achieve the precision required to validate such a figure necessitates a context where it matters. Traveling across the universe is intriguing, yet…

"You're really not going to like it."

— The Hitchhiker's Guide to the Galaxy, Douglas Adams

Even if there’s a pressing need for this number, we know it is subject to change. The figure of 93 billion light-years serves primarily as a monument to the concept of enormity, rather than a figure we need to know to an extraordinary degree of precision.

2. The number should have practical, real-world applications:

   Various groups pursue the discovery of long numbers simply for the sake of having them. As of this writing, Pi has been calculated to an astonishing 62.8 trillion digits.

For context, to measure the diameter of the observable universe with the accuracy of a single hydrogen atom, only 38 digits of Pi are necessary. Thus, this extensive research into trillions of digits is essentially a display of numerical prowess.

Binary Numbers: A Unique Perspective

Let's consider binary numbers. In a sequence, 1s and 0s hold as much validity as any other digits. When executing a program, the speed of counting through a series of numbers is crucial. Although the letter 'a' may be used in coding, it translates to 1100001 in ASCII (or 97 in binary). Regardless of the programming language, all paths eventually lead back to C, where 1s and 0s form the fundamental basis.

However, regarding the integrity of a program's string as a singular number, this is only partially true. Most users only skim the surface of a number sequence, shifting through them at incredible speeds.

Even the most complex programs, like search engines and AI, often contain hidden flaws from hastily adapted code. The world of coding is largely about cut and paste, and even after a lifetime of usage, one may not uncover these issues.

Thus, Google's vast 2 billion lines of code cannot simply be dismissed as a singular number. Within its intricate network lies a potential enigma, a Schrodinger's cat that could just as well be a 1… or a 0. This implies that it does not represent a definitive single number where every digit counts.

When dealing with sequential data, such as media files, each pixel in a frame corresponds to a number that must be accurate. With 2 million pixels per frame at 60 Hz, that’s 120 million pixels per second.

The largest well-documented single file I could locate is one that addresses the mathematical problem of the Boolean Pythagorean Triples, weighing in at an impressive 200 Terabytes. This file effectively contains vast amounts of numbers demonstrating every possible example of a particular mathematical theory. More information on that can be found here.

As a linear binary file, this represents a significant number: 200 quadrillion bytes, which translates to approximately 1,600,000,000,000,000,000 individual digits. However, it remains largely unseen, as this number does not serve any practical purpose, stemming from a $100 wager made by a professor back in the 1980s.

The Realm of Prime Numbers

Prime numbers form the backbone of Bitcoin theory and the internet security systems we rely on daily. RSA encryption employs two primes to securely transmit hidden messages through massive computations, effectively outpacing processing speeds due to their sheer size. Primes are both practical and valid at every digit.

The Great Internet Mersenne Prime Search team, often referred to as GIMPS, is dedicated to discovering the largest known primes. The current record stands at 2^(82,589,933) – 1, which comprises 24,862,048 digits.

However, advancements in technology may soon make these numbers less relevant, particularly with the advent of quantum computing. This new technology threatens to render current algorithms obsolete due to its potential for rapid processing.

Hypothetically, the largest numbers we compute may emerge from this technological leap. Yet, they will not conform to traditional understandings of numbers and will instead rely on digits existing in a superstate, rather than strictly as 1 or 0.

This hints at the core of our inquiry. What do we mean when we speak of colossal numbers? While some may be undeniably practical, their usefulness often lies within a more intricate context than a mere digit.

Take, for instance, media files. These undoubtedly represent astonishing numbers, but how valuable is that number when expressed as a written digit? The latest Aquaman film in 4K boasts 1,889,785,610,240 individual digits. Although the film may not be well-received, it presents those digits more engagingly than an incomprehensible matrix of digital gibberish.

The Human Memory Limitation

The longest number that any individual has managed to memorize is an impressive 111,700 digits, achieved by Akira Haraguchi. This serves as a notable example of human capability at its limits, as it touches the boundaries of our numerical capacity. This must serve as our ultimate benchmark.

We can only engage with larger numbers because we’ve devised methods to express them beyond mere numerical forms. Enormous numbers permeate our daily lives in ways that cannot be expressed in digits.

Consider the sky above us. Weather forecasting is one of the few areas that remain elusive. The latest endeavor involves a $1.7 billion supercomputer in the UK, aimed at understanding global weather patterns (a monumental number), and generating precise variables of interest to us—such as the temperature to the nearest degree or predicting rain at a specific time.

Contextualizing the Sandbox of Reality

At this juncture, we approach a clearer understanding of what constitutes the largest number. The relevance of a number is often determined by its relationship to our environment and our own experiences.

Our perception of weather—ranging from -20 to 80—is an excellent illustration of how we prefer our numbers simplified in everyday life, rarely exceeding three digits.

This provides context for some of the more peculiar numerical systems we still adhere to—such as imperial measurements:

• The Yard: The distance from the fingertips of King Henry I to his nose.
• An Inch: Three grains of barley laid end to end.
• The French Foot: You guessed it, Charlemagne's foot.

This haphazard system, prone to disputes and errors, surprisingly endured for centuries, leading to the odd architectural designs of Tudor homes.

Eventually, standardized measurement systems emerged, utilizing bits of wood to define inches, which were kept in London's guildhall.

However, even an inconsistent system like this sufficed for daily needs. Historically, there was seldom a requirement for calculations beyond six digits. Concepts like a chain (66 feet) or an acre (1/640th of a square mile) were reserved for select professions.

Only recently has the metric system gained traction, revealing the advantages of working in tens rather than threes and dozens:

• There are 100 million square cm in a hectare.
• There are 6,272,640 square inches in an acre.

(Interestingly, the best definition of an imperial unit often relies on its metric equivalent.)

Even a system that was largely disjointed for much of its history proved adequate for our needs. In modern times, the small numbers dominate our screens. What is the most significant number you’re likely to encounter in your tabs? News headlines are typically streamlined to three digits for a reason.

Even the most complex production projects, such as Lego, are calibrated to microns (one-hundredth of a millimeter). Within the scientific realm, the longest numbers in practical use hover around Intel’s new microchips, measuring a mere 2 nanometers, or 0.000000002 meters. While astonishing, it requires just 10 digits to represent.

With the burden of larger numbers comes greater responsibility, and even the most astute can suffer from number blindness. In 2022, a careless mistake by a trader triggered a 'flash crash' across European stock markets, resulting in losses of up to €300 billion. In this context, even a single digit in a series of twelve can be critical.

The sense of anxiety felt by a conveyancer (housing lawyer) when handling substantial sums at completion is palpable. Payday accounts in large corporations can reach into the billions, yet this still represents only a 12-digit number (including cents).

The Final Number

Consider the device you are currently using, be it a computer or phone.

A standard Apple calculator will only provide answers accurate to 16 digits, rounding the rest to zeros. Similarly, Excel documents round off at 15 digits. For calculations requiring greater precision, specialized tools like Wolfram Alpha are necessary.

Even at the pinnacle of accuracy, NASA’s Jet Propulsion Laboratory utilizes 3.141592653589793 as Pi for its vast calculations in interplanetary navigation. This level of precision ensures that over an orbit of 150 billion kilometers, the margin of error is no more than the width of a finger.

This illustrates the upper limits of what the most brilliant minds operate with—16 digits.

For many, the largest numbers discussed often start with childhood numerals and escalate to the daunting six-digit figures involved in house purchases (or eight with additional costs).

On average, 16 digits encompass any number likely to be encountered in everyday life. This exploration has been a rewarding journey to undertake and share. If you engage with even larger numbers—such as a 17-digit overdraft—feel free to share your experience in the comments.

See also:

The first video titled "What is the BIGGEST NUMBER?" provides an engaging exploration of numerical magnitude, examining various perspectives on what constitutes the largest number.

The second video, "What's the Biggest Number That You Could Count To?" delves into the concept of counting and numerical limits, offering insights into how we perceive and interact with large numbers.