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:

$$3 \times 1.5 = 3 + 1.5$$

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:

$$\begin{eqnarray}a \times b &=& a + b \\a \times b – b &=& a \\b \times (a – 1) &=& a \\b &=& \frac{a}{a – 1} \\\end{eqnarray}$$

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.

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