Buzz
A common question children ask is “What is the largest number?” This marks an important transition into understanding abstract concepts. The answer, of course, is that numbers are infinite, but at a certain point, numbers become so large that they lose practical significance. While I could simply list an enormous number followed by +1, +2, +3, I’ve chosen to focus on 10 numbers that do have some relevance to our world, even if that relevance is minuscule in comparison to the size of the numbers themselves. I’ll present these in ascending order with a brief explanation of their significance.
10. 10^80
Ten to the power of eighty – a 1 followed by 80 zeros – is an immense number, yet somewhat tangible from a relatively grounded perspective. This is the estimated count of fundamental particles in the known universe. And we’re not talking about just tiny particles, but far smaller ones like Quarks and Leptons – subatomic particles. In the U.S. and modern British English, this number is called “One Hundred Quinquavigintillion.” I wouldn’t even attempt to write the pronunciation, as it’s quite a mouthful. While the sheer scale of this number and the number of particles in the universe might seem overwhelming, it’s still the most comprehensible of all the numbers on this list.
9. A Googol
The term 'googol,' with a slightly altered spelling, has gained prominence in modern times as a verb, thanks to a popular search engine. The number itself has an intriguing history, which you can easily discover by simply googling it. Coined by Milton Sirotta in 1938 when he was only 9 years old, the term is somewhat abstract, existing mainly because it technically does. However, it occasionally appears in other contexts.
Mental calculator Alexis Lemaire set a world record by calculating the 13th root of a 100-digit number. For instance, the 13th root of 8,192 is 2, meaning 2 multiplied by itself 13 times. A 100-digit number is essentially a googol. One of the numbers Lemaire calculated would have been written as (3 googol, 893 Duotrigintillion, etc.). Another use of googol is in predicting that around 1 to 1.5 googol years after the Big Bang, the most massive black holes will have exploded. These will mark the final recognizable structure of the universe to disintegrate. Afterward, the universe will enter its 5th and final phase, known as the Dark Era, according to some scientific models.
8. 8.5 x 10^185
The Plank length is incredibly minuscule, approximately 1.616199 x 10^-35 meters, or 0.00000000000000000000000000000616199 meters in full notation. You could fit around a googol of these lengths into a 1-inch cube. Plank length and Plank volumes play a crucial role in quantum physics, especially in string theory. In some theories, such tiny scales help detect extra dimensions. So, how does this tie into the third smallest number on our list? There are roughly 8.5 x 10^185 plank volumes in the universe. While this number is enormous and its practical application is almost nonexistent, it still remains simpler compared to the other massive numbers listed here.
7. 2^43,112,609 – 1
The third largest number on this list, representing the number of plank volumes in the universe, contains 185 digits. In contrast, this number has nearly 13 million digits. Its significance lies in it being the largest known prime number, which was discovered in August 2008 by the Great Internet Mersenne Prime Search (GIMPS). From here, the numbers become far more challenging to comprehend.
6. Googolplex
Many have heard the term 'googolplex,' and if you’re a fan of the Back to the Future movies, you might recall Dr. Emmett L. Brown saying, 'she’s one in a million, one in a billion, one in a googolplex.' But what exactly is a googolplex? Remember how large a googol is? It’s a 1 followed by one hundred zeros. A googolplex, however, is a 1 followed by a googol of zeros. How massive is this number? If the entire universe were filled with paper, and each piece of paper contained zeros written in size 10 font, it would only account for half the zeros needed to write out this number in its full form. Even writing it in scientific notation isn't really practical. A number this enormous requires a special notation known as a 'power tower.' For instance, 10^80 is the base of our first power tower. As the tower grows, the next number is placed as a superscript above and to the right of the 80. This method isn’t always possible to write in standard digital text, so we use the shorthand notation ' ^.' For example, 10^80 (ten to the eighty) is the first in our power tower, and the googolplex is written as 10^10^100, or ten to the tenth to the one hundredth. We’ll be using these power towers for the upcoming numbers, so I hope you’re prepared to visualize them.
5. Skewe’s numbers
Skewe’s Number represents the upper bound for the mathematical problem that states: π(x) > Li(x). While this equation may appear straightforward, the function Li is much more complex. In essence, Skewe’s Number demonstrates that a number 'x' exists which breaks this rule. If Riemann's hypothesis holds true, this number 'x' is less than 10^10^10^36, a vastly larger number than a googolplex, largely due to the additional power tower. There is also an even larger Skewe number: without assuming Riemann’s hypothesis, 'x' is less than 10^10^10^963.
4. Poincare Recurrence Time
This concept is quite complex, but the core idea is relatively simple: 'given enough time, anything is possible.' Poincaré recurrence time refers to the period it would take for the universe to return to a state similar to its current configuration, due to random quantum fluctuations. In simpler terms, it’s the idea that 'History will repeat itself.' The maximum estimate for this duration is 10^10^10^10^10^1.1 years.
2. ∞ – Infinity
Most people are familiar with this concept, often used as hyperbole – like saying 'one zillion.' However, it's much more complex than most realize, and if you thought the previous numbers were unusual, this one is even more bizarre, and also quite controversial. In the realm of infinity, there are an infinite number of odd and even numbers, even though there can only be half as many odd numbers as there are total numbers. Interestingly, infinity plus one equals infinity, infinity minus one equals infinity, infinity plus infinity equals infinity, and dividing infinity by two still results in infinity. But infinity minus infinity is not fully understood, and infinity divided by infinity would likely equal 1.
Scientists estimate that there are 10^80 subatomic particles in the observable universe. However, many scientists believe that the universe is infinite, or at least accept the possibility. If the universe is indeed infinite, then mathematically speaking, there must be another Earth where every atom is positioned exactly as it is on our Earth. The likelihood of two identical Earths is extremely low, but in an infinite universe, it not only could happen, but it must happen. Furthermore, if the universe is infinite, there must be an infinite number of such Earths.
Not everyone accepts infinity as a reality. Israeli mathematics professor Doron Zeilberger argues that numbers do not go on forever, and that there exists a number so large that adding 1 to it would bring you back to zero. This number, he suggests, is far beyond human comprehension, and it may never be discovered or proven. This idea forms the foundation of a mathematical philosophy known as Ultrafinitism.
1. Graham’s Number
This number is colossal. In the 1980s, it earned a spot in the Guinness Book of World Records as the largest finite number ever used in a serious mathematical proof. Ron Graham created it as the upper bound for a problem in Ramsey Theory dealing with multicolored hypercubes. The number is so immense that even a power tower would be too cumbersome to represent it. The only efficient way to depict this number is by using Knuth’s Up-Arrow Notation and its own specific equation. Let’s break this down step by step.
Knuth’s Up-Arrow Notation is a system used to express extraordinarily large numbers. Explaining exactly how the arrows work here would be far too complex, but you can think of it like this: 3↑3 equals 33 or 27. 3↑↑3 means 3 raised to the power of 3 raised to the power of 3, or 7,625,597,484,987. Now, if you add another arrow, 3↑↑↑3, the result would be a power tower with more than 7.5 trillion levels. This alone dwarfs the Poincare Recurrence time, and the number can grow exponentially with an infinite number of arrows.
Graham’s number is represented as: G=f64(4), where f(n)=3↑^n3. To understand this, imagine it in layers. The first layer is 3↑↑↑↑3, which is already too large to represent in most other forms. The second layer has that many arrows between 3s, and then the resulting number is used to create another layer with that many arrows between 3s. This process continues for 64 layers. If you’re curious, the last ten digits of Graham’s number are 2464195387, but not even Graham himself knows the first digit.
3. ∞+1 – Infinity + 1
Apologies, I just had to go there.
