One centillion
1,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000
,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000
,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000
,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000
,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000
,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000
We finally reach a centillion, the 100th -illion, equal to 1 followed by 303 zeros. It's the largest -illion with an official name in English. Since it's about 10118 times the number of Planck times in the observable universe, surely a centillion is too big to represent anything in the real world, right?
Actually, there are several ways to get numbers as big as a centillion. One thing you could consider is that since the portion of the universe that is visible is always expanding, we can consider the future volume of the universe, which is also something Sbiis Saibian talks about in an article of his[1C]. In this case, the observable universe would have a volume of about a centillion Planck volumes roughly 1050 years in the future.
What will the universe be like in 1050 years? There are several theories. According to Wikipedia[8], there is a theory that after an EXTREMELY long time (believed to be between 1035 and 1043 years) protons will decay into a pion (which quickly decays into pure energy) and a positron, causing all the matter in the universe to eventually cease to exist. Even if protons do not decay, the universe will be very different from how it is now in 1050 years. Maybe all stars will be somehow thrown into black holes, and maybe even rigid matter will start rearranging atoms as if it was liquid on a very long scale. It's all pretty mind-boggling, and quite scary as well. Therefore this isn't the best way to get a feel of how much a centillion is.
Another not-as-scary way to consider numbers as big as a centillion is with something called quartic hypervolume. What exactly is that? It's what you get when considering time as a fourth dimension, along with the three spatial dimensions.
A simple example of quartic hypervolume would be as follows: Imagine a cube a light-year wide, and considering it only for a year. Then the quartic hypervolume of that cube for a year is one quartic light-year. Now, imagine a cube 2 light years wide, considering it for a year. Then the quartic hypervolume of that cube for a year would be 8 quartic light-years. But if you considered that same 2-light-year-cube considering it for 2 years, then the quartic hypervolume would be 16 quartic light-years.
With that in mind, we can now consider quartic hypervolumes, but on a Planck scale. We'll use the Planck length, but what time unit will we use? The unit we can use is the Planck time. What is a Planck time? The Planck time is the time analog of the Planck length, defined as the amount of time it takes for light to travel a Planck length, about 5.39*10-44 seconds. Planck times are very analogous to Planck lengths. When dealing with sub-Planck times it is theorized that the concepts of past and future get scrambled up. Therefore two events that occur less than a Planck time apart can be considered simultaneous.
How would you work with that for the observable universe? You need to do a bit of math. The real formula for quartic hypervolume of the entire observable universe is pretty complicated, but a decent aproximation formula given by Robert Munafo[9] is:
1/4 * a * 4/3 * π * r^3
where a is the age of the universe in Planck times and r is its radius in Planck lengths.
The current radius of the observable universe in Planck lengths is 2.75*1061 and the age in Planck times is 8.03*1060. The values are quite close, but relation between the values is complicated.
But for simplicity's sake, we can assume that at any time the Planck radius will generally be 3.42 times the Planck age. Though this is not strictly true, it's a decent estimation which allows us to simplify the formula to:
1/4 * a * 4/3 * π * (3.42*a)^3
Simplifying further gives us the formula for quartic hypervolume based on age:
41.89*a4
Now, things just got a lot simpler. Now all we need to is solve the equation:
10^303 = 41.89*a^4
a = 2.2104e75
Keep in mind that a is in Planck times, so what we need to do ... all we need to is convert to years. That gives us 3.778 septillion years. Then we can say:
The quartic hypervolume of the observable universe from the Big Bang to 3.778 septillion years in the future is about a centillion quartic Planck units.
However, another way we can consider a centillion's size is by probability, but we'll get to that in a later article. For now we'll push the idea of measuring things to its very limits by going hypothetical!
Beyond?
How would we get numbers beyond a centillion in the real world? By now it's difficult, but the best option we have goes as follows:
First off, it is theorized that the universe encompasses an area far greater than the observable universe. It is not known how big that area is. Some believe that the size of the whole universe is infinite, but that would be boring to us since that would not provide a real example of a large finite number. Something more interesting, however, is that the size of the entire universe, based on extrapolations from the theory of inflation in the Big Bang, may be something on the order of 1010^12 meters wide. That's quite big alright, since this estimate is one followed by a trillion zeros. This number is horrifying to even complicate. To get the size of this entire universe you need to dwarf the diameter of the observable universe by a factor of its diameter in meters, 37,000,000,000 times! If you did such a gigantic dwarfing factor once every second, then it would take 1190 years to get the size of what could be the entire universe!! This diameter is so big that it doesn't really matter what units we use (it could be Planck lengths or yottameters); the estimate is still around 1010^12.
However, chaotic inflation is a theory devised by Andrei Linde that allows for even bigger estimates of the size of the universe. With chaotic inflation, there can be something known as a Grand Universe, which can be imagined as not just one single universe, but a sort of "grand universe" with local Big Bangs constantly going off. Not a lot is known about how that theory would work in the real world, but it can lead to some even bigger numbers. Sbiis Saibian lower-bounded the chaotic inflationary size of the universe by assuming that it expanded by a factor of 1010^12 every Planck time and extending it to 13.7 billion years, to be on the order of 1010^64 [10]. This is an interesting figure especially since it's a lower bound. With such a Grand Universe it could well be that our Big Bang was not even the first ... and yet, with that theory the universe still must be finite. It's impossible to really know how big the Grand Universe might be, or if it even exists ...
Is it possible to go further than this Grand Universe? Maybe theoretically, but anything past measuring the entire observable universe is quite theoretical, especially with the idea of chaotic inflation. For now let's review what we've seen.
Conclusion
As we saw, millions, billions, and trillions are way bigger numbers than you probably thought, and the even bigger numbers like a vigintillion are absolutely incredible! Our world is full of large numbers, but it's easy to make numbers so big that our world can't even keep up with their size! But hold on, those are for later. For now let's continue with looking at the legendary numbers, the googol and googolplex, with their history, size, and cultural impact, and after that we'll return to the idea of large numbers in the real world with probability.
1,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000
,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000
,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000
,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000
,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000
,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000
We finally reach a centillion, the 100th -illion, equal to 1 followed by 303 zeros. It's the largest -illion with an official name in English. Since it's about 10118 times the number of Planck times in the observable universe, surely a centillion is too big to represent anything in the real world, right?
Actually, there are several ways to get numbers as big as a centillion. One thing you could consider is that since the portion of the universe that is visible is always expanding, we can consider the future volume of the universe, which is also something Sbiis Saibian talks about in an article of his[1C]. In this case, the observable universe would have a volume of about a centillion Planck volumes roughly 1050 years in the future.
What will the universe be like in 1050 years? There are several theories. According to Wikipedia[8], there is a theory that after an EXTREMELY long time (believed to be between 1035 and 1043 years) protons will decay into a pion (which quickly decays into pure energy) and a positron, causing all the matter in the universe to eventually cease to exist. Even if protons do not decay, the universe will be very different from how it is now in 1050 years. Maybe all stars will be somehow thrown into black holes, and maybe even rigid matter will start rearranging atoms as if it was liquid on a very long scale. It's all pretty mind-boggling, and quite scary as well. Therefore this isn't the best way to get a feel of how much a centillion is.
Another not-as-scary way to consider numbers as big as a centillion is with something called quartic hypervolume. What exactly is that? It's what you get when considering time as a fourth dimension, along with the three spatial dimensions.
A simple example of quartic hypervolume would be as follows: Imagine a cube a light-year wide, and considering it only for a year. Then the quartic hypervolume of that cube for a year is one quartic light-year. Now, imagine a cube 2 light years wide, considering it for a year. Then the quartic hypervolume of that cube for a year would be 8 quartic light-years. But if you considered that same 2-light-year-cube considering it for 2 years, then the quartic hypervolume would be 16 quartic light-years.
With that in mind, we can now consider quartic hypervolumes, but on a Planck scale. We'll use the Planck length, but what time unit will we use? The unit we can use is the Planck time. What is a Planck time? The Planck time is the time analog of the Planck length, defined as the amount of time it takes for light to travel a Planck length, about 5.39*10-44 seconds. Planck times are very analogous to Planck lengths. When dealing with sub-Planck times it is theorized that the concepts of past and future get scrambled up. Therefore two events that occur less than a Planck time apart can be considered simultaneous.
How would you work with that for the observable universe? You need to do a bit of math. The real formula for quartic hypervolume of the entire observable universe is pretty complicated, but a decent aproximation formula given by Robert Munafo[9] is:
1/4 * a * 4/3 * π * r^3
where a is the age of the universe in Planck times and r is its radius in Planck lengths.
The current radius of the observable universe in Planck lengths is 2.75*1061 and the age in Planck times is 8.03*1060. The values are quite close, but relation between the values is complicated.
But for simplicity's sake, we can assume that at any time the Planck radius will generally be 3.42 times the Planck age. Though this is not strictly true, it's a decent estimation which allows us to simplify the formula to:
1/4 * a * 4/3 * π * (3.42*a)^3
Simplifying further gives us the formula for quartic hypervolume based on age:
41.89*a4
Now, things just got a lot simpler. Now all we need to is solve the equation:
10^303 = 41.89*a^4
a = 2.2104e75
Keep in mind that a is in Planck times, so what we need to do ... all we need to is convert to years. That gives us 3.778 septillion years. Then we can say:
The quartic hypervolume of the observable universe from the Big Bang to 3.778 septillion years in the future is about a centillion quartic Planck units.
However, another way we can consider a centillion's size is by probability, but we'll get to that in a later article. For now we'll push the idea of measuring things to its very limits by going hypothetical!
Beyond?
How would we get numbers beyond a centillion in the real world? By now it's difficult, but the best option we have goes as follows:
First off, it is theorized that the universe encompasses an area far greater than the observable universe. It is not known how big that area is. Some believe that the size of the whole universe is infinite, but that would be boring to us since that would not provide a real example of a large finite number. Something more interesting, however, is that the size of the entire universe, based on extrapolations from the theory of inflation in the Big Bang, may be something on the order of 1010^12 meters wide. That's quite big alright, since this estimate is one followed by a trillion zeros. This number is horrifying to even complicate. To get the size of this entire universe you need to dwarf the diameter of the observable universe by a factor of its diameter in meters, 37,000,000,000 times! If you did such a gigantic dwarfing factor once every second, then it would take 1190 years to get the size of what could be the entire universe!! This diameter is so big that it doesn't really matter what units we use (it could be Planck lengths or yottameters); the estimate is still around 1010^12.
However, chaotic inflation is a theory devised by Andrei Linde that allows for even bigger estimates of the size of the universe. With chaotic inflation, there can be something known as a Grand Universe, which can be imagined as not just one single universe, but a sort of "grand universe" with local Big Bangs constantly going off. Not a lot is known about how that theory would work in the real world, but it can lead to some even bigger numbers. Sbiis Saibian lower-bounded the chaotic inflationary size of the universe by assuming that it expanded by a factor of 1010^12 every Planck time and extending it to 13.7 billion years, to be on the order of 1010^64 [10]. This is an interesting figure especially since it's a lower bound. With such a Grand Universe it could well be that our Big Bang was not even the first ... and yet, with that theory the universe still must be finite. It's impossible to really know how big the Grand Universe might be, or if it even exists ...
Is it possible to go further than this Grand Universe? Maybe theoretically, but anything past measuring the entire observable universe is quite theoretical, especially with the idea of chaotic inflation. For now let's review what we've seen.
Conclusion
As we saw, millions, billions, and trillions are way bigger numbers than you probably thought, and the even bigger numbers like a vigintillion are absolutely incredible! Our world is full of large numbers, but it's easy to make numbers so big that our world can't even keep up with their size! But hold on, those are for later. For now let's continue with looking at the legendary numbers, the googol and googolplex, with their history, size, and cultural impact, and after that we'll return to the idea of large numbers in the real world with probability.
