How Many Zeroes Are There In A Googolplexian

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How Many Zeroes Are There In A Googolplexian

Fracture: The ground is rough. Awakened: The universe is made up of high school math equations. Photo: Triff/Shutterstock

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. By the way. What is the biggest number you can think of? 700? A million? What’s in Elon Musk’s bank account today? It doesn’t matter – add one. Congratulations – you’ve made an even bigger number! You played yourself.

A consequence of this unfortunate fact is that, frankly, there are a very large number. Numbers like googol, which looks like this:

It’s a sequence of 100 zeros, or 10100, and it’s actually bigger, much, much bigger than the number of gold in the entire universe. Not a molecule, you will see; not even an atom; but

: electrons, leptons, quarks, etc. which have taken the place of atoms in physics as the basic elements of the universe.

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However, for googologists (nerds who spend their time reading these huge numbers) one googol is not much. As soon as the term was coined, it became one for an even larger and related number: the googolplex, defined as 10.

Unlike googol, we will never try to type a googolplex, because – well, a quick test puts the number of zeros that can be typed by simply holding down the 0 key for ten seconds to 320. fast, and indeed, if you could connect all the people who have ever existed to hold the zero key for about 72.5 billion years, you could probably finish writing it. Of course, this is between the Big Bang and the point in the distant future where the entire universe becomes a quantum tunnel in a strange vortex of molten liquid, which will probably have some difficulty maintaining its momentum. .

I hope we’ve made our point: It’s really, really hard to get your head around how big these kinds of numbers are. Here’s the thing:

Theoretically, there are more than a googolplex with the same numbers in the sequence. However, in practice, understanding these large numbers is not so easy, and not many of us even have the words.

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Take Skewes’ number, for example. It has a name and a number theory definition, so you’d think we’d know something about its size, and we do. you know

If you think that’s a huge difference, you’re right: the first two guesses can be completely written in a few minutes if that’s your type, but the last one is the equivalent of a googolplex raised to the power of 500. To put it in perspective, raising just two (2) to the power of 500 produces a number up to the top googol… and then half the same length again. The reason for this is a little thing called the Riemann hypothesis: the first guess assumes that the hypothesis is true, while the other guesses are false.

Then there is Graham’s number. It comes from combinatorics—specifically, an area now known as Ramsey theory—and it’s so big it doesn’t really fit in the universe. Even if we could write it down with a number for a Planck volume, which is the smallest measurable size an object can have, we would still run out of space long before we reached the end of the number.

Physicist and Numberphile Tony Padilla said in one of five videos on his YouTube channel dedicated to the number: “If you really try to imagine Graham’s number in your head, then your head will go into a black hole. “

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“It’s not just kind of crazy,” he explained. “You can’t store that much information in your head…the maximum amount of entropy you can store in your head depends on the size of the black hole, and the entropy of the black hole depends on the size of your head .Brings the least amount of information you need to write a Graham number.

In fact, not only is Graham’s number so large that his numbers cannot be written in the size of the universe, but

Number, and thus in all ways, for levels higher than the total number of Planck volumes in the universe to observe.

All this means that the Graham number cannot be written as a set of powers, as we did earlier with the Skewes number. Mathematicians had to create a new type of notation to deal with such large numbers

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Like Graham number, the following series of integers are expressed as elements of series, but with a big conceptual difference.

While googol and googolplex were invented simply to exist and be large, and Skewes and Graham numbers appeared as answers to difficult questions in unrelated areas of mathematics, numbers like TREE (3) – another number on our list – or SSCG. (3), an even greater number were found almost by chance.

“DAR (3) […] definitely puts Graham’s number to shame,” Padilla said. “I mean, really, Graham’s number is effectively zero according to DARA (3).”

It’s a wonderful name, isn’t it? In this case, the number is not named after the boffin who discovered it, but is completely descriptive: it is the third element of the TREE array.

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To understand what this means, we need to look into an area of ​​mathematics known as graph theory. If you’re not a mathematician, you might think you know what a graph is: it’s one of those graphs where you compare two objects on a pair of axes. Like this:

You may not like it, but this is high-end graphics performance. Image: Public domain via Wikipedia

Graphs, like graph theorists, are made up of edges and vertices. This is the most basic definition, although there are special names for graphs where every vertex is connected to every other vertex (a complete graph), or graphs where pairs of vertices are connected only in one direction and not in the other (direct one). ). and such things.

Trees are one such case: a “tree” is defined as a graph in which any two vertices are connected by exactly one edge (because mathematicians are so funny, a set of connected trees is called a forest).

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Now you know what a tree is, you can also find a WOOD (3) – just play.

“There are three types of seeds,” Padilla explains in the video below. “Mathematicians don’t call them seeds, they will call them ninety, but we will call them seeds.”

From those three initial nodes, according to some rules the goal is to grow a forest.

“The first tree cannot have more than one seed,” says Padilla. “The second tree can’t have more than two seeds…yeah, okay? That’s rule number one.”

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“The other rule is that if you plant a tree [and] find that there is already a tree left in that tree, the whole forest dies,” he concludes.

“The first tree should have at most one seed, right?” Padilla explains. “Can I write anything else? I can’t… I have to stop now.”

– additional work, we can see that DAR(2) is three. Nor are the numbers particularly large. So what should we expect from DAR (3)?

“THE TREE (3) [is] … so big that, you know, even just to show that it’s finished, there’s not enough time in the universe to do that,” Padilla said.

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Really great at everything; We don’t even know how many numbers it has. Is that the biggest number we know?

For a real-life look at the kinds of surprises mathematicians can come up with when they want to, look no further than the Lightning Number, a number created solely to win a “Who Can Find the Biggest Number” contest between MIT professor Agustín Lightning to create. , named after the Mexican Multiplier and Princeton Professor Adam Elga, or Dr. bad None of this is a joke.

“The Mexican Multiplier raised his hands in victory, smiling, as Dr. Evil whispered, ‘I’ve been crushed.’ The war is finally over,” the January 31, 2007 issue of The Tech told its readers.

Towards the end of a grueling battle, tense and tight marking, Rayo seems to have given up – he left with his head hanging in shame. So, a stroke of inspiration. He “angrily” wrote his now-famous definition of number on the blackboard: “The smallest number greater than any number that can be expressed in the language of first-order set theory with as few googol symbols.” each, with 2,500 digits per page (except the first, which will have 2,501).

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