General Education. And what about a googolplex, how many zeros does it have? The search website Google did get their name from this very large number. A googol, officially known as ten-duotrigintillion or ten thousand sexdecillion, is a 1 with one hundred zeros after it.
Written out, a googol looks like this: 10,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, The scientific notation for a googol is 1 x 10 Therefore, the only times a googol is a somewhat accurate estimate of anything is for hypotheticals.
After each chess player makes their first move, there are potential board setups. After each player has made two moves, there are , setups, after three moves there are over million, and the number continues to increase exponentially from there. One of them is a googolplex, which is a 1 followed by a googol of zeros. Counting to a googolplex would be even more impossible. As a comparison, counting to a trillion would take roughly 31, years, and a trillion is only a 1 followed by twelve zeros!
Kasner discussed googol and googolplex to show the difference between incredibly large numbers and infinity. Guess what? There are even larger numbers than a googolplex, although not many.
If you want to learn about all the large numbers and see a chart that makes it easy to compare them to each other, check out our guide to large numbers. Skewes was especially interested in prime numbers, and when his number was introduced in , it was described as the largest number in mathematics. What is a googol? A googol is a 1 followed by zeros. But if you zoom in on just Ann, Bryan and David, they are all joined by red edges.
This red triangle is an example of order hiding in the messy overall network. The more ordered a system is, the simpler its description. The most ordered friendship network is one that has all the edges the same colour: that is, everyone is friends or everyone is strangers.
Ramsey discovered that no matter how much order you were looking for — whether it was three people who were all friends and strangers or twenty people who were all friends and strangers — you were guaranteed to find it as long as the system you were looking in was large enough.
To guarantee yourself a group of three people who are all friends or all strangers you need a friendship network of six people: five people isn't enough as this counterexample shows. The number of people you need to guarantee that you'll find three friends or three strangers is called the Ramsey number R 3,3.
But we hit a wall very quickly. For example we don't know what R 5,5 is. We know it's somewhere between 43 and 49 but that's as close as we can get for now. Part of the problem is that numbers in Ramsey theory grow incredibly large very quickly. If we are looking at the relationships between three people, our network has just three edges and there are a reasonable 2 3 possible ways of colouring the network. Mathematicians are fairly certain that R 5,5 is equal to 43 but haven't found a way to prove it.
One option would be to check all the possible colourings for a network of 43 people. But each of these has edges, so you'd have to check through all of the 2 possible colourings — more colourings than there are atoms in the observable Universe! Big numbers have always been a part of Ramsey theory but in mathematician Ronald Graham came up with a number that dwarfed all before it.
He established an upper bound for a problem in the area that was, at the time, the biggest explicitly defined number ever published. Rather than drawing networks of the relationships between people on a flat piece of paper as we have done so far, Graham was interested in networks in which the people were sitting on the corners of a cube.
In this picture we can see that for a particular flat diagonal slice through the cube, one that contains four of the corners, all of the edges are red. But not all colourings of a three-dimensional cubes have such a single-coloured slice. Luckily, though, mathematicians also have a way of thinking of higher dimensional cubes.
The higher the dimension, the more corners there are: a three-dimensional cube has 8 corners, a four-dimensional cube has 16 corners, a five-dimensional cube has 32 corners and so on.
Graham wanted to know how big the dimension of the cube had to be to guarantee that a single-coloured slice exists. Graham managed to find a number that guaranteed such a slice existed for a cube of that dimension. But this number, as we mentioned earlier, was absolutely massive, so big it is too big to write within the observable Universe.
Graham was, however, able to explicitly define this number using an ingenious notation called up-arrow notation that extends our common arithmetic operations of addition, multiplication and exponentiation.
We can carry on building new operations by repeating previous ones. The next would be the triple-arrow. See here to read about the up-arrow notation in more detail.
The number that has come to be known as Graham's number not the exact number that appeared in his initial paper, it is a slightly larger and slightly easier to define number that he explained to Martin Gardner shortly afterwards is defined by using this up-arrow notation, in a cumulative process that creates power towers of threes that quickly spiral beyond any magnitudes we can imagine.
But the thing that we love about Graham's number is that this unimaginably large quantity isn't some theoretical concept: it's an exact number. We know it's a whole number, in fact it's easy to see this number is a multiple of three because of the way it is defined as a tower of powers of three.
And mathematicians have learnt a lot about the processes used to define Graham's number, including the fact that once a power tower is tall enough the right-most decimal digits will eventually remain the same, no matter how matter how many more levels you add to your tower of powers. Graham's number may be too big to write, but we know it ends in seven. Mathematics has the power not only to define the unimaginable but to investigate it too.
Rachel Thomas and Marianne Freiberger are the editors of Plus. This article is an edited extract from their new book Numericon: A journey through the hidden lives of numbers. OK now the question is how does the expansion for different numbers work. Googolplexitoll, Googolplexigong is larger than googolplex. Omega is even larger than infinity!
They all follow the same pattern. For up arrows, you just have one less up arrow than in the original problem. Put that back in your other problem. Unimaginably big. And that's only with only 4 up arrows. Thanks for posting the correct definition. People should NOT comment unless they know what they are talking about.
If you take a little time and watch Numberphile on YouTube, you can see Graham himself define the number. Your explanation was the correct one. Thank you for not saying something stupid.
Basically the number of arrows tell you the amount of times you repeat the operations that come before it. Long story short the arrow just tells you to take the process used for the previous number of arrows and use a repeated form of that. The difference between the towers is the height. The first tower is 3 high the same as the base number , and once that is multiplied out, that is the HEIGHT of the next one.
This continues forth. It just iterates upwards. Great explanation, but I'm somewhat puzzled by the fact that you never actually told us how Graham's number is defined. It's like there's a missing paragraph after you finish explaining the rapid recursive growth of up-arrows. That's "Step 1". Well beyond numbers anyone can really hope to imagine in any meaningfully representative way without deep mathematical understanding, this completes Step 2.
Anyway, that's what I think an explanation might look like; I'm surprised something like this wasn't included. Otherwise, great explanation,. Graham went one step further. However, that number is called "g1". That is "g2". Are you sitting down? Graham's Number is g64!!! They are not the same thing. Actual up arrows shouldn't be shown in it because like you said they don't mean the same thing. All 7. That would give you about. That's much better than before, but still not putting a dent on the.
Once again let's put the number of zeros the world would have wrote in terms of googol-grains. All those That's literally it. So let's move way way further from the realm of things that could actually happen. Imagine every person couldn't just write three zeros a second, but they could write a zero every Planck time! Calling this superhuman is an absurd understatement.
If you could write a zero every Planck time in only a second you'd write about 1. There's no way that could happen at all! Not only that, but everyone would write nonstop continuously from the Big Bang to the present!
With this ungodly ability we'll definitely be able to write a googolplex, right? Let's find out! With 1. That number has exactly 71 digits, but it's still a tiny fraction compared to the. We would have only gotten roughly a nonillionth a million trillion trillionth of the way to writing a googolplex! Writing a googolplex that way would take the whole world 2. That is unbelievably long and insane, especially considering that we're writing a digit each Planck time here! With the insane unbelievable size of a googolplex, surely there's no number that represents anything in the real world that comes anywhere near a googolplex, right?
Actually, it's possible to surpass a googolplex with several ways. A common example is considering the number of possible parallel universes. Sbiis Saibian has calculated an estimate of how many different parallel universes there can be, assuming sub-Planck units aren't meaningful. That is a number that, interestingly, has real-world meaning AND surpasses a googolplex we'll examine more numbers like this in a little bit. That shows that as vast as a googolplex is, it can have meaning in the real world!
Googolplex in culture. Because of its huge size and simple explanation, a lot of attention has been given to the googolplex, and it's become a classic benchmark for large numbers, and the largest number with a name for many people. I was no exception for quite a while until I learned of a googolduplex, 1 followed by a googolplex zeros which for a little while I knew as a googolplexplex. Right: A googolplex regularly appears in pop culture like this movie theater in The Simpsons.
Other examples include Googolplex Mall in the kids' TV show Phineas and Ferb, and astronomer Carl Sagan has referenced a googolplex when discussing the vast scales of the universe. Googolplex in googology. Probably the biggest innovation in googology to come out of the name "googolplex" is the -plex suffix.
Sbiis Saibian coined many other numbers with the -plex suffix like googolplexigong, and weirder numbers with -plex have been coined like piplex, ten to the pi-th power. However, Jonathan Bowers seems to interpret -plex a little differently. For example, he defines the following numbers and many others similarly to googol and googolplex:. This shows that Bowers seems to interpret "plexing" a number as replacing the number's largest argument with the original number. Golapulusplex, however, is an exception.
We'll examine that notation in section 3. Some people have extended the familiar -illion sequence to name very large numbers in terms of -illions. The two best known such systems are Conway and Guy's illions and Jonathan Bowers' -illions we'll discuss those later.
While Conway and Guy's is simpler and has shorter names, Jonathan Bowers' is more extensible with a colorful naming system. In terms of Conway and Guy's -illions more on those in the next article , a googolplex can be named:.
And in terms of Bowers' -illions, a googolplex can be named:. What about one followed by a googolplex zeros, or one followed by that many zeros? Continuing the Googolplex. One followed by a googolplex zeros has two widely recognized names: googolduplex and googolplexian. I prefer the name googolduplex because unlike "googolplexian" which is arbitrary, "duplex" in googolduplex literally means two plexes, and that's exactly what a googolduplex is.
There are other names for a googolduplex, such as gargoogolplex or googolplusplex. It is unknown who coined any of those names. A googolduplex is, well, unimaginably huge. There are also two well-known large numbers known as the Skewes' numbers used as upper-bounds in a mathematical proof, we'll go into them in detail a little later , one of which surpasses a googolduplex. F or more on a googolduplex see its entry on part 3 of my number list.
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