## Apathetic Numbers

*April 19, 2020 by Chris Johnson. Filed under math, public.*

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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