Problems

We’ll look at ways of making big numbers today. Hopefully you know about powers of numbers. Most of the biggest numbers you’ve seen probably involve powers. Powers are typically thought of as ’repeated multiplication’. You could think of this as being similar to how multiplication is ’repeated multiplication’.

We might use powers when decimal representation is far too long to be useful for understanding the size of an object.

But what if powers aren’t helpful enough? Mathematician and computer scientist Donald Knuth introduced Knuth’s up-arrow notation. A single up-arrow means ‘raise to the power of’. So \(2\uparrow5=2^5=2\times2\times2\times2\times2=32\). Recall that the \(5\) means we have five \(2\)s. Two arrows means ‘tetration’. For example \(2\uparrow\uparrow4=2\uparrow(2\uparrow(2\uparrow2))=2^{(2^{(2^2)})}=2^{(2^4)}=2^{16}=65536\). The \(4\) means that we have four \(2\)s. Three arrows means pentation. So \(2\uparrow\uparrow\uparrow3=2\uparrow\uparrow(2\uparrow\uparrow2)\). Here the \(3\) means that there are three \(2\)s. Remember that \(2\uparrow\uparrow2=2\uparrow2=2^2=4\). So \(2\uparrow\uparrow\uparrow3=2\uparrow\uparrow4=65536\).

A million is \(10^6\) and a billion is \(10^9\). A million seconds is about eleven and a half days. A billion seconds is about 31 years. Other famous big numbers are googol, Skewes’ number, Graham’s number, busy beaver and TREE(3). Another classic big number is the number of atoms in the observable universe - about \(10^{80}\). This is less than \(4\uparrow\uparrow3\), \(3\uparrow\uparrow\uparrow3\), or \(2\uparrow\uparrow\uparrow\uparrow3\).

A couple of these problems are Fermi problems, named after physicist Enrico Fermi. This is where you try to estimate a quantity in the real world. We’re not expecting an exact answer (indeed, we don’t know the exact answer), but using some intelligent estimation, you can get a good idea of the answer.

What’s \(3\uparrow\uparrow3\)?

Is \(4^{15}\) more or less than a billion?

Approximately how many footsteps do I take in a year? (estimate to the nearest power of \(10\))

What’s \(2\uparrow\uparrow\uparrow\uparrow\uparrow\uparrow\uparrow\uparrow\uparrow\uparrow2\)?

What’s bigger out of \(5^6\) and \(2^{14}\)?

What’s \(3\uparrow\uparrow3\) as a power?

Is \(45^{45}\) bigger or smaller than \(10^{80}\)?

How many times have the people in this room blinked in their lives in total? Find an answer to the nearest power of 10.

What’s bigger out of \(99!\) and \(50^{99}\)?