Large Numbers, Infinities & God (1/3)

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This is the first of three posts, originally I wanted to cover it all in one post but humorously there was too much to write about. I will talk about the three subjects seperately but they are building towards the final post.

1. Large Numbers
I think my favourite large number has to be the Sagan unit. A sagan is a unit of measurement equal to at least four billion which pays homage to the glorious Carl Sagan’s exstensive use of the term “billions and billions”. As large numbers goes, four billion is pretty weak. Luckly Carl happens to have another number to his name, in astronomy the Sagan Number is the approximate amount of stars in the observable universe. It is roughly 70 sextillion, which looks like: 1000000000000000000000 (10^{21}). Is that big? The observable universe is said to be 10^{26}meters and there are in the region of 10^{118} particles in it…

One of the most well known large numbers is googolplex which is 10^{googol}. The number googol was popularized in the 1940s book Mathematics and the Imagination. It is defined as 10^{100} which is a one followed by 100 zeros. A googol is larger than the amount of atoms in the observable universe which is close to 10^{80}. Googolplex is simply 10^{10^{100}}, a one followed by googol zeros.

Notation is a helpful tool when dealing with extremely large numbers, the most standard is scientific notation. Here I will explain up arrow notation, a creation from the brilliant mind of Donald Knuth. Knuth basically condensed powers of powers into readable text:

\begin{matrix} a\uparrow b &=& \underbrace{a \times a \times a \times \dots a} &=& {a^b} \\ & & \text{b copies of a} \\ \end{matrix}

\begin{matrix} a\uparrow\uparrow b &=& \underbrace{a\uparrow (a\uparrow (\dots\uparrow a))} &=& {^{b}a} \\ & & \text{b copies of a} \\ \end{matrix}

\begin{matrix} a\uparrow\uparrow\uparrow b &=& \underbrace{a\uparrow\uparrow (a\uparrow\uparrow (\dots\uparrow\uparrow a))} \\ & & \text{b copies of a} \\ \end{matrix}

\begin{matrix} a \underbrace{\uparrow_{}\uparrow\!\!\dots\!\!\uparrow}_{n} b &=& \underbrace{a \underbrace{\uparrow_{}\!\!\dots\!\!\uparrow}_{n-1} ( a \underbrace{\uparrow_{}\!\!\dots\!\!\uparrow}_{n-1} ( \dots \underbrace{\uparrow_{}\!\!\dots\!\!\uparrow}_{n-1} a ) )} &=& a{\uparrow}^n b \\ & & \text{b copies of a} & \\ \end{matrix}

There are other notaions for large numbers such as tetration: {^{b}a} and also Conway’s chained arrow notation. Although we don’t see many large numbers in our day to day activities – mathematicians, especially pure mathematicians, have to use them when tackling all sorts of problems. For example when dealing with n-dimensional hypercubes we come across the final large number I will talk about: Graham’s Number.

Graham’s Number is another contender for the most well-known large number, claimed to be the largest number ever used in a serious mathematical proof. In the form of an upper bound to a problem in Ramsey Theory, this number can’t be expressed in digits even if each digit was planck’s length and it was written across the observable universe. But it can be defined through recursive up arrow notation.

\begin{matrix} G&=&3\underbrace{\uparrow\uparrow\cdots\cdots\cdots\cdots\cdots\uparrow}3 \\ & & 3\underbrace{\uparrow\uparrow\cdots\cdots\cdots\cdots\uparrow}3 \\ & & \underbrace{\qquad\;\; \vdots \qquad\;\;} \\ & & 3\underbrace{\uparrow\uparrow\cdots\cdots\uparrow}3 \\ & & 3\uparrow\uparrow\uparrow\uparrow3 \end{matrix} \text{64 Layers.}

In the above equation the lowest term 3↑↑↑↑3 with four arrows is known as g1 and it defines the amount of arrows in the next term g2, this carries on in the same fasion so we get \boldsymbol{G} = f^{64}(4),\text{ where }f(n) = 3 \uparrow^n 3, unusually with Graham’s number, we don’t know how it starts, as a matter of fact we only know how it ends (thanks to power towers of 3) the last ten digits of Graham’s number are …2464195387.

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9 thoughts on “Large Numbers, Infinities & God (1/3)”

  3. Great post man!
    I had known about the Sagan unit, googol and googoplex, but the up-arrow notation and Grahams’s number was a real delight to learn 🙂
    waiting for the other two parts

    Reply ↓

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  6. You’re giving a good explanation of tower powers, but in fact in the last fifty years there has even been a more impressive kind of number, which is way more hugely gigantically large than the largest tower power you’ve ever known. In fact, it’s a sequence of numbers that grows so large so quickly, that it’s logically impossible for a computer to determine what the sequence of numbers is, even if it had an infinite amount of memory and an infinite amount of time to complete the task. Note that these theoretical computers definitely *can* do that with the numbers you’ve described here. They’re called “Busy Beaver” numbers, and I’ll be writing a post about them soon. Keep an eye out for it.

    Great blog!

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