Lessons from teaching gifted elementary school students (Part 5c)

Every so often, I’ll informally teach a class of gifted elementary-school students. I greatly enjoy interacting with them, and I especially enjoy the questions they pose. Often these children pose questions that no one else will think about, and answering these questions requires a surprisingly depth of mathematical knowledge.

A bright young student of mine noticed that multiplication is repeated addition:

$x \cdot y = x + x + x \dots + x$,

$x^y = x \cdot x \cdot x \dots \cdot x$,

So, my student asked, why can’t we define an operation that’s repeated exponentiation? Like all good explorers, my student claimed naming rights for this new operation and called it $x \hbox{~playa~} y$:

$x \hbox{~playa~} y = x^{x^{x^{\dots}}}$

For example,

$4 \hbox{~playa~} 3 = 4^{4^4} = 4^{256} \approx 1.34 \times 10^{154}$

Even with small numbers for $x$ and $y$, $x \hbox{~playa~} y$ gets very large.

Unfortunately for my student, someone came up with this notion already, and it’s called Knuth’s up-arrow notation. I’ll give some description here and refer the interested reader to Wikipedia and Mathworld (and references therein) for more information. Surprisingly, this notion has only become commonplace since 1976 — within my own lifetime.

Let’s define $x \uparrow y$ to be ordinary exponentiation:

$x \uparrow y = x \cdot x \cdot x \dots \cdot x$.

Let’s now define $x \upuparrows y$ to be the up-arrow operation repeated $y$ times:

$x \upuparrows y = x \uparrow x \uparrow x \dots \uparrow x$.

In this expression, the order of operations is taken to be right to left.

Numbers constructed by $\upuparrows$ get very, very big and very, very quickly. For example:

$2 \upuparrows 2 = 2 \uparrow 2 = 2^2 = 4$.

Next,

$2 \upuparrows 3 = 2 \uparrow (2 \uparrow 2)$

$= 2 \uparrow (2^2)$

$= 2 \uparrow 4$

$= 2^4$

$= 16$

Next,

$2 \upuparrows 4 = 2 \uparrow (2 \uparrow (2 \uparrow 2))$

$= 2 \uparrow 16$

$= 2^{16}$

$= 65,536$

Next,

$2 \upuparrows 5 = 2 \uparrow (2 \uparrow (2 \uparrow (2 \uparrow 2)))$

$= 2 \uparrow 65,536$

$= 2^{65,536}$

$\approx 2.0035 \times 10^{19,728}$

Next,

$2 \upuparrows 6 = 2 \uparrow (2 \uparrow (2 \uparrow (2 \uparrow (2 \uparrow 2))))$

$= 2 \uparrow 2^{65,536}$

$= 2^{2^{65,536}}$

$\approx 10^{6.031 \times 10^{19,727}}$

We see that $2 \upuparrows 6$ is already far larger than a googolplex (or $10^{10^{100}}$), which is often (and erroneously) held as the gold standard for very large numbers.

I’ll refer the interested reader to a previous post in this series for a description of how logarithms can be used to write something like $2^{65,536}$ in ordinary scientific notation.

Knuth’s up-arrow notation can be further generalized:

$x \uparrow^3 y = x \upuparrows x \upuparrows x \dots \upuparrows x$,

repeated $y$ times. The numbers $x \uparrow^4 y$, $x \uparrow^5 y$, etc., are defined similarly.

These numbers truly become large quickly. For example,

$2 \uparrow^3 2 = 2 \upuparrows 2 = 4$, from above.

Next,

$2 \uparrow^3 3 = 2 \upuparrows (2 \upuparrows 2)$

$= 2 \upuparrows 4$

$= 65,536$, from above

Next,

$2 \uparrow^3 4 = 2 \upuparrows (2 \upuparrows (2 \upuparrows 2))$

$= 2 \upuparrows 65,536$

$= 2 \uparrow 2 \uparrow 2 \uparrow \dots \uparrow 2$,

where there are 65,536 repeated 2’s on this last line. It’d be nearly impossible to write this number in scientific notation, and we’ve only reached $2 \uparrow^3 4$.