Large numbers

Practical numbers

OK, 1 is not exactly a large number. However, with numbers, it depends a lot on what you mean by them. If there is a large unit, even 1 can be a lot. 1 Tesla (unit of magnetism), for example, means a very strong magnet.

1 also has an important property that any two numbers higher than 1 multiplied together lead to a larger number. Multiplying two numbers further down cannot get you backwards.

1 circle:

Ten is not usually thought to be a large number. But ten people together are a group, a ten meters tall house is not exactly small and it's not easy to lift ten kilograms of something. Ten is not exactly a large number, but it's large enough to be noticed. Ten dollars buys you a lunch (depending on where you live).

10 circles:

oooooooooo

You have to stop for a moment to count them.

100

A human can only have about a hundred relationships at the same time ("relationship" is any friend you see often). It's a bit strange to see a hundred identical things at one place. The only reason this ever happens is that humans like neat numbers like 100 and because 10x10 is easy to pack (as opposed to 97, for example, which cannot be divided into rows). While this may seem trivial, it is also true about 81, for example (9x9), and the only reason to favor 100 over 81 is its "neatness". A one in a hundred chance is not worth very much. 100 kilogams is obesity (or just a bit overweight if you are very tall). Anyway, a hundred things is hard to imagine (without imagining a 10x10 square) and sometimes, you find out that it's more than you expect.

100 circles:

ooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo

ooooooooooooo

Still quite okay.

1 000

A thousand is unwieldy. It's hard to imagine anything under it, unless you cheat by imagining a cube 10x10x10, or any similar 3D structure. Imagining a thousand things in a line is hard. Whenever a thousand appears somewhere, people try to remove it. The SI prefix kilo- is exactly for that. A thousand grams is hard to imagine, a kilogram isn't. A thousand meters is hard, a kilometer is easy. A thousand things need to be compared to something, it cannot be imagined directly.

1 000 circles:

ooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo

ooooooooooooooooooooooooooooooooooooooooooo

See? It's in several lines, which is not perfectly representative. In one line it would be about a meter long.

1 000 000

A million is about the largest number humans can imagine wwithout resorting to abstract ideas like money. I wanted to put a million circles to a different site (with a link here), but it was "too much data". So here are 100 000. Imagine that times ten. Hard. A million objects, while possible to imagine, is an unlikely occurence. So let's get abstract anyway: A million years is about the number of years humans have been around (depends on what you count as humans), a million dollars is enough for a fancy house and a million kilometers is found often in astronomy (within the solar system). In a full vacuum cleaner's hold there are about a million dust specks.

A billion is often thought to be similar to a million (both are "very large"). This is only a case of people not understanding large numbers. Look up, to the thousand circles. If each of them represent a million, together they are a billion. Now look all the way to the single circle. It is quite a difference, right?

A million years ago, there were humans on Earth. A billion years ago, multicelluar life was quite young. A million seconds is about 11.5 days while a billion seconds is about 31 years.

1 000 000 000 000

A trillion (or a billion in the long scale). It can be used in astronomy (about the sizte of the solar system in meters) and in economics (GDP of countries, huge projects). The circles would take up a space of roughly 10 by 10 kilometers or all the way to the Moon and back, 5 times. Using the billion, trillion, quadrillion etc. scale gets weird with higher numbers. Who uses the word "septillion"?

Impractical numbers

1 000 000 000 000 000 000 000 000

10²⁴

Septillion (short scale)

I use the word septillion. A septillion is the largest prefix in the SI system. It's the prefix "yotta-" with a yottameter, for example, being noted Ym. It's useful with really big or strong things. 880 yottameters is about the diameter of the observable universe, the Earth weighs about 5 000 Yottagrams and the Sun has the power output of 500 Yottawatts (a light bulb can have about 50 watts, so it's about 5 to 10 septillion light bulbs). Larger numbers can only be used in odd ways like "number of sand grains that would fill the observable universe".

10¹⁰⁰

Also called a googol.

The name Google comes from a misspeled version of googol, by which the company wanted to show how much information they can provide. They aren't there yet, a googol bits would take up a big chunk of the observable universe. It's unlikely that advances in technology will bring us there, but who knows?

A googol is the volume of the observable universe in cubic micrometers, which makes it more than the number of dust specks you could fit into it.

10¹⁸⁵

This is the largest number with any practical (or impractical) use as the amount of things. It's the number of the smallest things possible (cubic Planck lengths) you could fit into the largest thing possible (observable universe again).

Larger numbers can be used in the real world, but it's quite hard. For example, in quantum physics. If you leave a glass of beer on a table, it can (this is very simplified) spontanneously overcome an energetical barrier (complicated way to say "turn a bit for no reason") and fall over. It would, however, take about a googolplex years, so the beer would vaporise or freeze because of heat death of the universe much earlier that this would happen. In fact, it's enough time for the glass to vaporize.

Googolplex

Googol-1-plex

10^googol

You cannot even write this number out in the observable universe without using zeroes smaller than human cells. At this level it doesn't matter if we talk about seconds, days, or centuries. A googolplex seconds is about ^10googol-9 centuries, so the difference between a googolpex seconds and googolplex centuries is similar to the difference between a googol and a googol minus ten. Unnoticable.

Googolplexplex

Googol-2-plex

10^googolplex

While this number is sometimes called diferently, it's probably obvious how it is created. It's one with a googolplex zeroes. If you add a few more plexes (four should be enough, so a googolplexplexplexplexplexplex, or googol-6-plex), you get the largest number used in physics, the Poincaré recurrence time. After this many seconds (or years, or whatever), everything that can happeen would happen. In the post-heat death universe, a completely new one could reappear because of quantum mechanics. This would happen so many times that every possible state of the universe would happen.

Warning:

I am NOT an expert on quantum physics. Take my claims about that field with a grain of salt. If there's something wrong, correct me.

End of warning.

Even though a googol-6-plex satisfies any practical need, there are still larger numbers used in certain mathematical proofs - such as Graham's number.

Graham's number

There are many large numbers used in mathematical proofs. Many of them are larger than Graham's number (only G from now on), so it might be good to explain why G.

Any number worth mentioning here has to satisfy three basic conditions.

1) It has to be used in a proof of something not large-numbers-related. It's easy to make up huge numbers, but it's not so easy to use them.

2) It must be clear why the number is so large. It should be defined as a result of some calculations instead of being defined by a property.

3) It should be created using ordinary mathematical procedures (with extensions) instead of appearing out of thin air.

There are numbers to which dwarf G in a similar way as how G dwarfs the number 3. One such number is TREE(3). Let's compare them:

1) It's fine. TREE(3) is related to Kruskal's tree theorem, which is a problem in graph theory.

2) Nope. There are lower bounds on the number that are very large (they dwarf G easily), but it's not easy to explain how they were calculated.

3) Because the number is defined by a property, this is impossible.

G, on the other hand, satisfies all the conditions.

Reason for G

The problem G was created for is long, boring and confusing but it's luckily not necessary to understand the number. If you feel that you don't understand this passage, it's okay to skip it.

Let's assume we have a square, in which we connect all vertices with a line, which we color either blue or red. It's obviously possible to do this without there being four vertices on a plane (i.e. all four vertices) all connected with same-colored connections. Let's take a cube. Again, connect all vertices with either blue or red lines. This can still be done without there being four coplanar (i.e.on the same plane) vertices connected only with blue or only with red lines. We can try four dimensions.

Wait a moment. The fourth dimension in mathematics is NOT time. You can imagine it like this: A cube is like two squares, one above the other in the third dimension. A 4D cube (or tesseract) is like two cubes, one above the other in the fourth dimension. There's no point in imagining this, so we use an analogy, where we make the second cube smaller and put it inside the first cube. Here you are:

We try to do the same thing here. There might be a problem with determining whether four vertices are coplanar, but it's still doable. We can still succeed at coloring the lines.

It's clear now that we have to continue to more dimensions. It's very hard, but it has been proven that you can do this for a 10D cube. There is, however, a proof that for some number of dimensions, it won't be possible. That number is between 11 and G. This is what G means and why it's important. Once G was the largest number ever used in a serious mathematical proof (it was even in the Guinness book of world records), but there have been larger ones since then.

The path to G

The four basic operations +-*/ can be derived from just +* if you take substraction as adding the opposite (5-3=5+(-3)=2) and division as multiplying by the reciprocal (8/2=8*(1/2)=4). For most practical purposes multiplication is just repeated addition (5*3=5+5+5=15). This means that all of the ordinary arithmetics is just addition and its derivatives. G is understandable because it's also created using derivatives of addition, they're just taken further than usual.

Substraction and division are a step backwards, so they're not gonna be used here. Addition itself is not very powerful either, so we need to make it stronger. We do that by adding a lot of large numbers together. Repeated addition is just multiplication. 100*100 is a much smarter way to write 100+100+100+100... 100 times. Multiplication results in larger numbers, but gets impractical somewhere around a googol. That's why we use powers. The change from multiplication to powers is the same as from addition to multiplication. This means 5⁷ (that's how 5 to the power of 7 is written, it can also be written 5^7) means 5*5*5*5*5*5*5=78125. Powers give very large results, especially if the number at the top is large itself. Raising a number to the power of itself gives us the function f(n)=nⁿ. (If you don't know what a function is, it's similar to a formula. You put one number in and get a different one out. f(3) is 3³=27, for example, because f(n)=nⁿ and n is 3 in this case. The letter before the parentheses is the function's name and the number inside is the input. Functions are usually written f(x) or g(x) or similar.) This is the largest function you can get to using methods most people know about.

It's important to note that 4³ is not equal to 3⁴. For sufficiently large numbers (larger than 5), the output will be larger if the larger number is on the top. So 7⁸ is more than 8⁷.

If we want large results, we can use f(n) twice. This means raising a number to the power of itself and then doing the same to the result. This is not the best way to proceed, however, because of this:

Let's say we start with 5. 5⁵=3125 and 3125³¹²⁵ is a number with roughly 10 000 digits. That is large, indeed, but if instead of (5^5)^(5^5) we do 5^(5^(5^5)), we get

5^(5^(5^5))=5^(5^3125)=5^A NUMBER WITH ABOUT 2000 DIGITS=5^{A NUMBER WITH ABOUT 2000 DIGITS}, which is almost identical to 10^{A NUMBER WITH ABOUT 2000 DIGITS}, a number so huge that its number of digits has 2000 digits itself.

This means that it's more efficient to use power towers. In a power tower, we start from the top and work our way down. For example, consider a power tower of size 4 made out of 5's. We start by calculating 5⁵. We get 3125. Then, we calculate 5³¹²⁵. We get a huge number with about 2000 digits. Let's call it v. Then we calculate 5^v. The result is the result of this power tower. If the tower was of height 5, we'd have to calculate 5 to the power of this result.

Because the normal way to write a power tower is not very concise, we'll use the notation made by Donald Knuth:

3↑4=3⁴

One arrow is just a different way to write exponentiation (or powers). Nothing more about it. 4 to the power of 7 is 4↑7.

5↑↑4 is our power tower. It's made out of 5's and is 4 high. Notice that there are two arrows. a↑↑b is a power tower made out of a's, b high.

In the arrow notation, a↑↑b equals a↑(a↑(a↑(a...))) with b a's.

This is also called tetration. It can be written as ⁴5, with this meaning 5↑↑4, but that's impractical because it cannot be generalised.

Warning, super large numbers on the way.

Results of tetration can still be kind of comprehensible. That's why we have the three arrows operator, pentation.

4↑↑↑3 = 4↑↑(4↑↑4))

This might not be very easy to understand. Start by calcuzlating the number in the parentheses. It is the power tower of 4↑(4↑(4↑4)) = 4↑(4↑256) > 4↑(10↑150) > googolplex

This means that the number in the parentheses is larger than a googolplex, a number itself very hard to just write out, let alone use.

4↑↑googolplex is a power tower made out of fours (but if they were fives, it wouldn't matter much) way higher than a googolplex (this is the important part). As shown before, even a power tower with five numbers produces huge results. Here, we have a googolplex numbers. The result of this is completely incomprehensible.

It's a number for which only the number of digits of number of digits of number of digits of number of digits... (googolplex times "number of digits") makes any kind of sense. There's no point at trying to compare this number with anything humans can imagine. There are people who will say about G (still a long way there), that "a computer large enough to store G in its memory would immediately collapse into a black hole". It's correct, that would happen, but that can be said about a googolplex or googolplexplex, which are both ridiculously small compared to 4↑↑↑3. All the point of the list of numbers to googolplexplexplexplex was to show the limit of our ability to compare numbers with the real world. From this point on we are far above that limit and the only way to understand how large those numbers are is to understand the arrow system.

Three arrows can be written as ↑³ to save space.

a↑³b = a↑↑(a↑↑(a↑↑(...)) with b a's.

It's probably clear how to continue. Four arrows (or hexation):

3↑⁴3 = 3↑↑↑↑3 = 3↑³(3↑³3) = 3↑³(3↑↑(3↑↑3)) = 3↑↑↑(3↑↑(3↑(3↑3))) = 3↑↑↑(3↑↑(3²⁷)) = 3↑↑↑(3↑↑7 625 597 484 987)) = 3↑↑↑(a power tower with 7 625 597 484 987 threes) = 3↑↑↑TOWER

This got confusing. This will be 3↑↑(3↑↑(3↑↑(3↑↑3...))), TOWER times. It's hard to comprehend, so here's how to calculate that.

Step 1:

Calculate a power tower out of 3's, 3 high. That's 7 625 597 484 987.

Step 2:

Calculate a power tower from 3's of height 7 625 597 484 987. You get TOWER.

Step 3:

Calculate a power tower from 3's of height TOWER. You get a large number.

Step 4:

Calculate a power tower from 3's of height from last step. You get a much larger number.

And so on for TOWER steps. After you get the last, TOWERth, step done, you get the Graham's number.

That's not true. You get a large number. But G is far, far larger. This number, however, is an important step in calculating it.

It should be clear how arrows work by now, but here's the general rule, just in case:

a↑ⁿb = a↑^n-1(a↑^n-1(a↑^n-1(a↑^n-1...))) b a's.

So a↑^xb means b repetitions of ↑^x-1 between a's (We count right to left).

A few examples for illustration:

5↑↑3=5↑(5↑5)=5↑3125=5³¹²⁵=about 2000-digit number

2↑↑↑3=2↑↑(2↑↑2)=2↑↑(2↑2)=2↑↑(2²)=2↑↑4=2↑(2↑(2↑2))=2↑(2↑4)=2↑16=65 536

3↑↑↑2=3↑↑3=3↑(3↑3)=3↑27=7 625 597 484 987

4↑↑↑↑2=4↑↑↑4=4↑↑(4↑↑(4↑↑4))=4↑↑(4↑↑(4↑(4↑(4↑4))))=4↑↑(4↑↑(4↑(4↑(256))))>4↑↑(4↑↑(4↑10000000000...(over 150 zeroes)))>4↑↑(4↑↑100000...(a googol zeroes))=4↑↑(a power tower from 4's larger than a googolplex)=4↑↑A LARGER NUMBER THAN ANYBODY CAN IMAGINE=4↑↑ZILLION=a power tower from fours ZILION high=SQUILLION

Summary:

-1 arrow = addition:

Technically, the arrow system starts here, with multiplication being repeated addition and exponentiation being repeated multiplication. It's not practical to use -1 arrows.

You probably know how addition works. Numbers under 10 000 are at this level, after that, using addition to get high numbers is impractical.

0 arrows = multiplication:

It can create a large number from two relatively small numbers. Around a googol it stops being useful.

1 arrow = exponentiation:

If the number at the top is large enough (10 or more), you get a result that's useless but easy to write. Googolplex is somewhere around here. As a side note, for whatever reason, when people want to create a large number they often use a factorial. That doesn't make sense, because for example 6!=720 is less than 6⁶=46656. Both is 6 numbers multiplied together, but with a factorial they decrease (6!=6*5*4*3*2*1) and with a power they stay the same (6⁶=6*6*6*6*6*6). Anyway, the factorial is somewhere here.

2 arrows = tetration = power tower:

The results are useless anywhere in the real world even with a very wide understanding of "real world"(Planck lengths are not ordinary), but for very small numbers you can use the number in some obscure way. You can easily use tetration to create a number that cannot be (reasonably) written in the decimal notation.

3 arrows = pentation:

Hard to imagine and gives unwritable results even for very small numbers. Cannot be written as a power tower (reasonably). It's the first operator for which you really need the arrows, since power towers can still be written as a lot of numbers on top of each other. It's kind of doable even with pentation ("A power tower of height of power tower of height of power tower... etc. b times, all the towers are from a"), but it's very confusing and unnecessary.

4 arrows = hexation:

It's easy to get lost in it and the result cannot be compared with anything from the lower levels. Pentation still usually gives a result in the form of a large power tower, so you can understand the number using the number of times you have to say"number of digits of the number of number of digits...". Hexation, however, is too far from ordinary mathematics to be compared to it.

5 arrows+:

THE NUMBERS GET LARGE.

No further explanation needed, it wouldn't help anyway.

It took forever to get here, so it's tim to ask...

How much is G?

We start by creating a sequence. g₁is its first member and equals the above described 3↑↑↑↑3. g₁ is, therefore, huge by itself.

g₂ could be defined as g₁↑↑↑↑g₁. If we write it as g₁↑⁴ g₁, the mistake in that becomes visible. The numbers before and after the arrows are not that important, the important part is actually the number of arrows. This g₂ would be therefore smaller than 3↑↑↑↑↑↑3, that is six arrows (five would probably suffice). The correct way to continue is to make g₂ use g₁ as the number of arrows.

g₂ = 3↑^g13 = 3↑↑↑↑↑↑↑↑↑↑↑...(g₁ arrows)...↑↑↑↑↑↑↑↑↑↑↑↑↑3

If you properly understood the arrow system, this should feel crazy. With every next arrow, you get an operation that is far, far, far stronger than all the ones before it. Five arrows is far enough to make your head explode from the complexity of writing the result out in a way that fits on paper and is understandable by laymen. If you can't even comprehend how many arrows here are, the result is gonna be superhuge.

The best way to describe g₂ is to give up. We can stare at it, we can discuss is, we can try to figure out some of its properties, but that's about it. Let's just stare at it for a few moments and move on. We're not at G yet!

g₃ = 3↑^g23 = 3↑↑↑↑↑↑↑↑↑↑↑↑...(g₂ arrows)...↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑3

Of course. This, while it may seem similar to g₂, is a lot larger. Remember, with every arrow added we move further into the realm of craziness. Here, it's almost like adding g₂ of them (g₁ gets lost in g₂, more about that later). Howeer, g₃ has very similar properties to g₂. The difference is large, but the human brain cannot comprehend it and just labels both g₂ and g₃ as of similar size.

Moving on...

g₄ = 3↑^g33 = 3↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑...(g₃ arrows)...↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑3

We're almost there. Here is the general rule, just in case:

g_n = 3↑^g(n-1)3 = 3↑↑↑↑↑↑↑↑↑↑↑...(g_n-1 arrows)...↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑3

G is g₆₄. We repeat this lengthy process 64 times. Then, we get G.

Properties of G

For starters, g₁ is defined in a way that complicates expressing G concisely. It turns out that G could be expressed exactly by other notations (more on that in the second part). It has also been proven that G is not necessary as the upper bound for the proof mentioned earlier and that it can be replaced by a number as tiny (compared to G) as 2↑↑↑6. This decrease of g₁ could therefore easily be done. There's probably no reason for defining G in this impractical way. Graham, who coined the number, has not given any and all sources describe G as described here, so it's hardly a mistake.

The only reason for that might be that 27 arrows is very hard to explain, while four arrows are still kind of understandable (you at least kind of understand how it works).

The fact that G is so large, but can be described in a text of size of about a page is caused by the fact that it contains very little information. Imagine that it the definition of

g₁, instead of a three, there would be ten (10↑↑↑10). Everything else would be done normally. The result would be one and a lot of zeroes.The number of zeroes would be one and lots of zeroes agan. The number of zeroes in that? Same thing. The number of times we would have to do this to get to one? One and a lot of zeroes. The fact that any changes lead to one and a lot of zeroes is the cost for creating a number far larger than the text in which it is described. G is the same thing, just in base 3.

G is a large power tower from 3's (far over g₆₃ high). If you create a sequence 3↑↑1, 3↑↑2, 3↑↑3... you find out that the last digits of the numbers start repeating. All such towers higher than some number will end the same. A sufficiently large tower (far smaller than G) will end in ...2464195387

This can be done further, so we know the last 10 000 digits of G. There doesn't seem to be any structure in them and they're de facto random.

Even though G is finite, it has many properties by which it is more similar to infinity. Addition, multiplication, exponentiation, any reasonable number of arrows etc. all give almost exactly the larger number as a result. (g₅*g₇=g₇ almost exactly). Different numbers higher than g₂ are hard to distinguish. It's hard to say whether g₄₅ and g₄₆ are very different, because substraction doesn't make much sense here either. What number is the average of g₂ and g₃? Ordinary aritmethic average gives you almost exactly a half of g₃, which is not distinguishable from g₃. A useful average of g₂ and g₃ cannot be g₃! There are other ways to find an average (such as geometric average), but they all give the same result. Because there is no number "in between" g₂ and g₃ without being equivalent to one of them, there's not much difference between working with any number higher than g₂.

G is still just 0% of infinity! Living for G years is a weird concept. There's a limited number of states a human brain can be in (less than googolplex). After some time, it has to start repeating. In G years it would be in every possible state over g63 times. g63 times would a human live through every possible situation without having any idea that it had happened so many times before. Would that be good or bad? No idea. Decide that for yourself.

Conclusion

This text was written to explain Graham's number. The part before it was there just to show the limits of real life. There are, however, numbers far bigger than Graham's number. Since some readers may be interested, here's a second part about them. And if you're just wondering if there's any point in this, here you are.