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One of the first mathematical concepts that most of us grasp when we're children is that there's no such thing as the biggest number. No matter what number you pick, you can always add one to it. Now, you might think that such a simple idea wouldn't have any profound impact in mathematics, yet it does. In fact, mathematicians have come up with numbers so mind bogglingly large that it's 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. This episode is sponsored by Quints. A new year is upon us, and that means new resolutions, new goals, and maybe a new wardrobe. If you're craving a winter reset, start with pieces truly made to last season after season. Quince springs together premium materials, thoughtful design and enduring quality so you stay warm, look sharp and feel your best all season long. 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This episode will be about finite numbers. Absurdly large finite numbers, but finite numbers nonetheless. And 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 didn't develop words for numbers beyond 2 or 3 because their daily lives didn't require exact counting. Beyond that, instead of precise numbers, they often used qualitative terms such as 1, 2, and then many. And this wasn't a limitation of intelligence, but a reflection of practical needs. When activities like hunting, gathering, or sharing resources rarely depend on exact large quantities, there was little pressure to create or maintain a full counting system. But 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. 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. As the scale of both the atom and the universe came to be understood, scientific progress necessitated the use of increasingly large numbers. One of the first problems we had was how to express large numbers in writing. If you just write out a large number 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 actually counting the digits. 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. So 100 is 10 raised to the power of 2. As there are 2 zeros, 1 million is 10 to the power of 6. As there are 6 zeros, a billion is 10 to the 9, a trillion is 10 to the 12th, and so on. Each time the exponent increases by 1, it's considered an order of magnitude. And you've probably heard me use the expression order of magnitude on this podcast many times. A hundred 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 out in scientific notation, which is similar. 2,300,000 would be written as 2.3 times 10 to the power of 6. While exponential notation helps you compactly express large numbers in writing, we've also developed a naming system for large numbers, some of which you're probably already familiar with. The prefix naming system for very large numbers extends the familiar thousand, million and billion pattern used by Latin and Greek numeral roots combined with the suffix ilion in the modern short scale used in the United States and most English speaking countries. Each new ilion represents a power of 10 that is three orders of magnitude greater than the previous. 1 million is 10 to the 6th, billion is 10 to the 9th, trillion is 10 to the 12th, quadrillion is 10 to the 15th, quintillion is 10 to the 18th, and so on. The prefixes quad, quint, sex, sept, oct, non, and dec indicate 4 through 10. For even larger numbers, more systematic constructions are used combining root forms to form names such as undecilian, vi, gentilian, and centillion, which are defined by convention rather than by the prefect system. The system is regular in structure, but it becomes impractical at very high magnitudes, which are why scientific notation and exponent based systems are preferred beyond a certain point. Likewise, there is an international standards for prefixes. When used as an adjective, the SI prefix system is to represent powers of 10 in a standardized way. Each prefix corresponds to a specific power of 10 and is attached to a unit such as meter, gram or byte to indicate scale. The commonly encountered large prefixes are Kilo for 10 to the 3rd, Mega for 10 to the 6th, Giga for 10 to the 9th, Tetra for 10 to the 12th, PETA for 10 to the 15th, Exa for 10 to the 18th, Zeta for 10 to the 21st and Yoda for 10 to the 24th. To provide some examples which are a bit more tangible, here are some references in real world terms for extremely large numbers. The age of the universe is about 13.8 billion years, which corresponds to roughly 4 times 10 to the 17th seconds. Estimates for the number of sand grains on Earth usually fall around 10 to the 18th to 10 to the 20th. Astronomers estimate that there are roughly 10 to the 22 to 10 to the 24 stars in the observable universe. A typical human body contains on the order of 10 to the 27 atoms. The Earth contains roughly 10 to the 50 atoms and commonly cited estimate for the number of atoms in the Observable universe is 10 to the 80 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 to the power of 80 is about as big as we can go as far as counting actual things. However, mathematically, we're 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 10 to the power of 100, which is 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 Serrata, to invent a new name for the number 10 100. The child suggested Google, which may have come from the cartoon character Barney Google. The company Google was named after the number Google, although they are spelled differently. The company Google 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 10 to the power of 120. Of course, once a googol was defined, we went into the realm of stupidly huge numbers. And 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 binary form. A regular book can hold about 1 million zeros. If you write down a googolplex, you would need more books than there are particles in the universe. Now, you might think that a googolplex is about as big as it's worth bothering to define. But brother, we are just getting started. Exponents are just a short way of doing multiplication. So 10 to the third is just a short way of saying 10 times 10 times 10. However, you can raise exponents two exponents. A googolplex is 10 to the power of googol, which is just 10 to the power of 10 to the power of 100. And once you start putting exponents on exponents, you get ridiculous really quick. And once exponents themselves become too small to describe growth, mathematicians move to iterated exponentiation, which leads to Something called tetration. If exponentiation is just repeated multiplication, tetration is repeated exponentiation. To describe this growth systematically, computer scientist Donald Kunt introduced up arrow notation. The symbol is literally just an arrow pointing up. A single up arrow represents exponentiation. So a up arrow B means A to the power of b. Two up arrows represent tetration. So a double up B means A to a power tower of height B. So to put that in numbers, 2 up arrow 4 is just 2 to the power of 4, which is 16 easy. But 2 double up 4 is 2 raised to the power of 2 raised to the power of 2 raising to the power of 2, which equals 65,536 10. Double up 3 would be 10 to the power of 10 to the power of 10, which is 10 to the power of 10 billion, which is far larger than a GOOGOL. Three up arrows represent repeated tetration and so on. So each additional arrow moves to an entirely new insane level of growth. So, for example, 3, 3 up 3 is not just large, it's beyond a googolplex. And there are still larger numbers that can be expressed even beyond this, which is now in the realm of higher mathematics. Some famous insanely large numbers come from logic and combinatorics. One well known example is Gram's number, which arose as an upper bound in a problem in what's called Ramsey theory. Graham's number is defined using iterated arrow notation, where even the number of arrows is defined recursively. It's so large that its last few digits are the only part that can actually be meaningfully discussed in decimal form. Yet it's 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 all of 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 heck, I have a popular podcast, so I figured this was as good a time as any to describe it. While traveling around the world with my camera, I had an idea. My digital camera had a set number of pixels and each pixel can have a set color value. My 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 that 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 existed or ever could exist in every possible angle, with every combination of every possible person, and all of those images then mixed and combined together with all other photos. Surely there's 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 raised to the power of 50 million. Putting this into a normal exponent form raised to the powers of 10, it would be 10 to the power of 240,309,000. So even if every atom in the universe stored a unique image, and the universe were recreated again and again trillions of times, you still wouldn't come close to exhausting the number of possible images. To be fair, 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 possibly exist in reality. But mathematically, they're 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 Erds number to go along with my Kevin Bacon number. Insanely ridiculous. Absurdly large numbers can be difficult to wrap your head around, and that's okay, because our brains truly cannot grasp their size. The main thing that you should take away is that despite their massive sizes, they are not infinite, and that many things we may think of as being infinite are actually just really, really, really, really big. The executive producer of Everything Everywhere Daily is Charles Daniel. The associate producers are Austin Otkin and Cameron Kiefer. Today's review comes from Summer Homework Hater on Apple Podcasts in the United States. They write Help me with my AP Language homework. This podcast is amazing without a doubt. I wasn't going to write a review, but this podcast is pretty amazing and it deserves a shout out. It helped me with my AP Language Summer homework. I know EW Summer homework, but it was able to give me something interesting to write about and listen to. Keep up the amazing work. I love the new episodes. Well, thanks Homework Hater. You have discovered what more and more students are finding out. This podcast is a way to make school easy and make learning enjoyable. Remember, if you leave a review of the podcast on any of the major podcast apps. You too can have it read on.