I’m reading the biography of Laura Fermi1¹ Atoms in the Family: My Life With Enrico Fermi; Fermi; University of Chicago Press; 1995. and in it she mentions Ettore Majorana – an incredibly talented and intelligent physicist, who unfortunately seems to have run into troubles with his health, and eventually disappeared mysteriously.

One of the things Fermi noticed about Majorana is that when he was available, the other physicists didn’t reach for their slide rules – they just stated out loud the calculation they would like to do, and then Majorana did it mentally.

One example she gives is

How much is the square root of 243 by 578 cubed?

I interpret this as

\[\sqrt{243 \cdot 578^3}\]

I have a hypothesis regarding how Majorana might have done it. First, we convert it to geometric scale. The logarithms of 243 and 578 are approximately 2.38 and 2.77 respectively – since a little over a year back, we can do this conversion quickly in our heads. Then in geometric scale, the equivalent expression is

\[\frac{2.38 + 3 \cdot 2.77}{2}\]

I’m not very fast at mental multiplication, but when I try I eventually figure out this is 5.35. Converted back to regular arithmetic scale, this ought to be something near 225,000.

Thus, we suspect the answer is

\[\sqrt{243 \cdot 578^3} \approx 225\,000\]

It turns out the value of this expression is actually 216,618 meaning we were almost 4 % off. But it’s not bad for a relatively complicated expression involving some huge numbers (doing the calculations in full would involve an 11-digit number!)

Again, I have no idea if this was how Majorana did it, but given that he was apparently good at logarithms, I think it’s a fair guess.