From time to time, I participate in the weekly Indie Ink writing challenge. I had not yet published a response on this website, as my previous challenges did not lend themselves to the general theme that I have in my head for what belongs here. However, this week I found inspiration in our system of numbers. The result is a response to the prompt “This is what it’s like..,” which was given to me by Lazidaisical. I, in turn, asked Barb Black to write about the quadratic formula (I’ve been preparing lectures for summer algebra and, apparently, have numbers on the brain).
In many ways, this is also a response to the questions that I get from my algebra students. After spending a good deal of the course building toward it, we introduce the quadratic formula. One of the difficulties of the formula is that the solutions it produces often involve the square roots of negative numbers. This confuses many students, as they have been told for years that negative numbers don’t have square roots. This is made even more difficult by the unfortunate fact that these square roots are referred to as “imaginary” numbers.
These imaginary numbers are no more nor less abstract or imaginary than negative numbers or rational numbers. In Where Mathematics Comes From, the authors Lakoff and Nuñez argue that only some very basic counting and addition are innate to human cognition, and that everything beyond that is built through metaphor. From that perspective, there is nothing particularly special about imaginary numbers—they simply build upon existing structure to extend our understanding of how already abstract numbers interact.
Thus everything we know about numbers is the result of metaphor, analogy, and simile. It is difficult to explain what numbers are, but we can get a sense of what they are like.
Ode to the Infinite
The infant mind knows: one and two is three—
An intuition, inborn and innate.
Add one, and one, on to infinity;
By successors, the naturals relate.
Abstraction beyond the tangible world,
With which we add, subtract, and multiply;
But division the sublime scheme unfurled,
And debts the naturals can’t rectify.
Thus greater abstraction our minds invent:
Integers and rationals without end;
So-called reals of even greater extent;
The complex plane; quaternions! Attend,
For this is how it is: the infinite
Is ably seen by reason’s wondrous light.