Insanely Ridiculously Absurdly Large Numbers

Podcast Transcript

One of the first mathematical concepts that most of us grasp when we are children is that there is no such thing as the biggest number. No matter what number you pick, you can always add one to it.

You might think that such a simple idea wouldn’t have any profound impact in mathematics, but it does.

In fact, mathematicians have come up with numbers so mind-bogglingly large that it is difficult to even grasp their size, and new forms of notation had to be developed to even write them down.

Learn more about Insanely Ridiculously Absurdly Large Numbers on this episode of Everything Everywhere Daily.

Let me start by saying that this episode is not about infinity. I’ve previously done an episode on infinity and how it is handled by mathematics.

This episode will be about finite numbers. Absurdly large finite numbers, but finite numbers nonetheless.

I’ll start by noting that society’s need for large numbers has changed as civilization has evolved.

Some small-scale or historically isolated societies did not develop words for numbers beyond two or three because their daily lives did not require exact counting beyond that. Instead of precise numerals, they often used qualitative terms such as “one,” “two,” and then “many.”

This was not a limitation of intelligence, but a reflection of practical needs. When activities like hunting, gathering, or sharing resources rarely depended on exact large quantities, there was little pressure to create or maintain a full counting system.

As we progressed, we needed to worry about things like money. Even in the ancient world, the idea of a million people or a million pieces of silver was not unheard of.

Many of you might remember a time when a billion dollars was an astronomical amount of money. Today, there are companies with valuations over a trillion dollars, and the US national debt is approaching 40 trillion dollars.

As the scale of both the atom and the universe came to be understood, scientific progress necessitated the use of increasingly larger numbers.

One of the first problems was how to express large numbers in writing.

If you just write out large numbers as they appear, it becomes a massive string of digits that is difficult to read. It might be easy to tell the difference between a million and a hundred at first glance, but it would be hard to tell the difference between a septillion and an octillion without counting the digitis.

To solve this problem, mathematicians, scientists, and engineers use exponential notation.

This is usually expressed as 10 to the power of something because we use a base-10 numbering system. The exponent reflects the number of zeros in the number.

100 is 10² as there are two zeros. One million is 10⁶, as there are six zeros.

A billion is 10⁹, a trillion is 10^12, and so on.

Each time the exponent increases by 1, it is considered an order of magnitude. You’ve probably heard me use the expression on this podcast about an order of magnitude. 100 is an order of magnitude greater than 10.

A million is four orders of magnitude greater than 100.

If you need greater precision, you can write it in scientific notation, which is similar. 2,300,000 would be written 2.3 x 10⁶.

While exponential notation helps you compactly express large numbers in writing, we have also developed a naming system for large numbers, some of which you are probably familiar with.

The prefix naming system for very large numbers extends the familiar thousand, million, and billion pattern by using Latin and Greek numerical roots combined with the suffix -illion.

In the modern short scale used in the United States and most English-speaking countries, each new “-illion” name represents a power of ten that is three orders of magnitude larger than the previous one.

Million is 10?, billion is 10?, trillion is 10¹², quadrillion is 10¹?, quintillion is 10¹?, and so on. The prefixes quad-, quint-, sext-, sept-, oct-, non-, and dec- indicate four through ten.

For even larger numbers, more systematic constructions are used, combining roots to form names such as undecillion, vigintillion, and centillion, which is defined by convention rather than by the prefix system.

The system is regular in structure, but it becomes impractical at very high magnitudes, which is why scientific notation and exponent-based systems are preferred beyond a certain point.

Likewise, there are international standards for prefixes when used as an adjective. The SI prefix system is to represent powers of ten in a standardized way. Each prefix corresponds to a specific power of ten and is attached to a unit such as meter, gram, or byte to indicate scale.

The commonly encountered large prefixes are kilo for 10³, mega for 10?, giga for 10?, tera for 10¹², peta for 10¹?, exa for 10¹?, zetta for 10²¹, and yotta for 10²?.

To provide some examples that are a bit more tangible, here are some references in real-world terms for these extremely large numbers.

The age of the universe is about 13.8 billion years, which corresponds to roughly 4 × 10¹? seconds.

Estimates for the number of sand grains on Earth usually fall around 10¹? to 10²?

Astronomers estimate that there are roughly 10²² to 10²? stars in the observable universe.

A typical human body contains on the order of 10²? atoms.

The Earth contains roughly 10?? atoms, and a commonly cited estimate for the number of atoms in the observable universe is about 10??. This figure is sometimes called the Eddington number, named after Arthur Eddington, a British astrophysicist who first estimated its value in 1940.

The Eddington number of 10?? is about as big as we can go as far as counting actual things.

However, mathematically, we are just getting started.

If you can express large numbers as exponents, we can create even bigger numbers.

Such a bigger number that most of you are familiar with, even if you don’t know it, is 10¹⁰⁰, also known as a googol.

The word googol was coined in 1920 by American mathematician Edward Kasner. While discussing the idea of extremely large numbers, Kasner asked his nine-year-old nephew, Milton Sirotta, to invent a name for the number 10¹??. The child suggested “googol,” which might have come from the cartoon character Barney Google.

The company is Google was named after the number googol, although they are spelled differently. The company is G-o-o-g-l-e, whereas the number is g-o-o-g-o-l.

While a googol is far larger than the number of particles in our universe, no matter how you define it, there are still things that are larger than a googol.

The number of distinct possible chess games, known as the Shannon number, is often estimated to be around 10¹²?.

Of course, once a googol was defined, we went into the realm of stupidly huge numbers. Another one that some of you might be familiar with is a googolplex.

A googolplex is 10 raised to the power of a googol. A googolplex is so large that even if every particle in the universe were one bit in a massive computer, you couldn’t even express the number in a binary form.

A regular book can hold about one million zeros. If you write down a googolplex, you’d need more books than there are particles in the universe.

You might think that a googolplex is about as big as it is worth bothering to define.

Brother, we are just getting started.

Exponents are just a short form of doing multiplication. 10³ is just a short way of saying 10x10x10.

However, you can raise exponents to exponents. A googolplex is 10^googol, which is just 10^{10^100}.

If you start putting exponents on exponents, you get ridiculous really quick.

Once exponents themselves become too small to describe growth, mathematicians move to iterated exponentiation, which leads to tetration.

If exponentiation is repeated multiplication. Tetration is repeated exponentiation.

To describe this growth systematically, computer scientist Donald Knuth introduced up arrow notation. A single up arrow represents exponentiation, so a ? b means a^b. Two up arrows represent tetration, so a ?? b means a power tower of height b.

So 2 ? 4 is just 2^4, which is 16. Easy.

2??4 is 2 raised to the power 2, raised to the power of 2, raised to the power of 2, which equals 65,536.

10??3 would be 10 to the power of 10 to the power of 10, which is 10 to the power of ten billion, which is far larger than a googol.

Three up arrows represent repeated tetration, and so on. Each additional arrow moves to an entirely new level of growth. For example, 3 ??? 3 is not just large; it is far beyond numbers like a googolplex.

There are still larger numbers that can be expressed even beyond this, which is in the realm of higher mathematics.

Some famous “insanely large” numbers come from logic and combinatorics. One well-known example is Graham’s number, which arose as an upper bound in a problem in Ramsey theory.

Graham’s number is defined using iterated up arrow notation, where even the number of arrows is defined recursively.

It is so large that its last few digits are the only part that can be meaningfully discussed in decimal form, yet it is still finite and precisely defined.

Another example is the TREE(3) number from graph theory, which is vastly larger than Graham’s number. Its definition is short and rigorous, but the resulting value grows faster than almost any function commonly encountered in mathematics.

I want to end this episode with my own little contribution to the world of insanely large numbers. I’ve never published this, but I do have a popular podcast, so I figured this was as good a time as any to describe it.

While traveling the world with my camera, I had an idea. My digital camera has a set number of pixels, and each pixel can have a set color value.

The question was, how many different photos could I take with my camera?

Most people, when asked such an absurdly open-ended question like this, would probably say there are an infinite number of photos you could theoretically take.

Think of every possible scene, in every possible angle, in every possible lighting condition. Every person who ever was or ever could be, in every possible angle, with every combination of every possible person, and all of those images are then mixed and combined together with all the other photos.

Surely, there is no end to the number of photos that could be taken. However, there is. Because the number of pixels and the number of colors are finite, the total possible number of photos that could be taken has to be finite as well.

In the case of a 50 megapixel camera with 64,000 color possibilities per pixel, that would be 64,000^50,000,000.

Putting this into a normal exponent form raised to the powers of 10, it would be 10^240,309,000.

Even if every atom in the universe stored a unique image, and the universe were recreated again and again trillions of times, you would still not come close to exhausting the number of possible images.

Most of these images would appear to be pure noise. Only an infinitesimal fraction would resemble anything meaningful, like landscapes, faces, or photographs that could exist in reality. But mathematically, they are all valid images a camera could theoretically produce.

The important takeaway is that the total number of possible photos isn’t infinite; it’s finite. Stupidly overwhelmingly large, but finite nonetheless.

If any mathematicians out there want to use this in a paper, please give me co-author credit so I can earn a Paul Erd?s number to go with my Kevin Bacon Number.

Insanely ridiculously absurdly large numbers can be difficult to wrap your head around, and that’s OK because we can’t truly grasp their size. The main thing you should take away is that, despite their massive sizes, they are not infinite, and that many things we think of as being infinite are actually just really, really, really big.

The Executive Producer of Everything Everywhere Daily is Charles Daniel. The Associate Producers are Austin Oetken and Cameron Kieffer.

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