## Apathetic Numbers

My son is 11, and he likes to think about numbers. In this time of working and learning from home, I’ve had him reading Isaac Asimov’s *Realm of Numbers*.

The other day my son stumbled upon this numerical curiosity:

How fascinating that the numbers don’t care whether they are being added or multiplied. Both operations yield the same result.

My son eagerly reported his discovery to me. I wondered out loud with him if there are other numbers that don’t care about whether they are being added or multiplied, and we worked out this relationship:

It looks like we can choose any number $a \neq 1$ and calculate its mate as $\frac{a}{a – 1}$. Choosing $a = 3$ gives us the pair $(3, \frac{3}{2})$. We also have $(4, \frac{4}{3})$, $(5, \frac{5}{4})$, and so on.

As Asimov writes in *Realm of Numbers*, mathematicians like to give names to numbers or sets of numbers that have certain properties. We are calling numbers that don’t care whether they are being added or multiplied *apathetic numbers*.

My son went away giddy.