## Sunday, December 16, 2012

### Imagine new numbers

As a child I wrote poems but abandoned the craft until many years later when I was a math professor; at that later time some of my poems related to ideas pertinent to my classroom.  For Number Theory classes "A Mathematician's Nightmare" gave a story to the unsolved Collatz conjecture; in Abstract Algebra "My Dance Is Mathematics" gave the mathematical history a human component.
My editor-colleague (Strange Attractors), Sarah Glaz, also has used poems for teaching --  for example, "The enigmatic number e."  And Marion Cohen brings many poems of her own and others into her college seminar course, "Truth & Beauty: Mathematics in Literature."  Add another to these east-coast poet-teachers -- this time a California-based contributor: teacher, poet, and blogger (Math Mama Writes) Sue VanHattum.  VanHattum (or "Math Mama") is a community college math teacher interested in all levels of math learning.  Some of her own poems and selections from other mathy poets are available at the Wikispace, MathPoetry, that she started and maintains. Here is the poet's recent revision of a poem from that site, a poem about the invention (or discovery?) of imaginary numbers.

Imaginary Numbers Do the Trick      by Sue VanHattum

In an email group of 4,000 homeschoolers,
a member wrote:

My son asks, “The square root of 1 is 1,
So what's the square root of -1 ?”

What we call the real numbers
is everything on a number line,
positive, negative, zero.

If you're thinking about those real numbers,
on that number line,
none of the negative numbers can have a square root,
because anything times itself will come up positive
(or zero).

But, once upon a time (for real),
mathematicians dueled
by giving each other lists of thirty hard problems.
The winner got recognition
and perhaps a job.

All this dueling led to
a solution for cubic equations:
these mathematicians
created a formula
that would find the numbers
that would solve a thing
like 2x³-3x²+4x-5 = 0.

But that formula was a problem!
It came up with square roots of negative numbers,
which drove the mathematicians wild.
No, no, no. There is no such thing!
Well, maybe there could be…
and if there is,
what would it look like?

With a wave of the magic wand of imagination,
These mathematicians
which later got the name i.
(Imagine it written in a fancy script.)
i is the square root of -1,
so i squared must equal -1.
is the first step in creating …
the imaginary numbers.

Imagine, if you will, a new number line
of imaginaries
crossing the line of real numbers at 0,
with the real number line horizontal,
and this new one vertical.
(It looks just like x and y axes,
but now it's all one number system,
a bit more complex.)

i sits one step above zero.
Another step up this imaginary number line,
we see 2i,
2i is the square root of -4.
(It is?!
Why yes, 2i times 2i equals 4 times i squared,
and i squared equals -1,
so we get -4.
Cool, huh?)
And on it goes.

Now all of this wouldn't really solve much
if there were no square root of i,
and that seems too weird to think about.
But, once you study trigonometry
(how'd that get in here?!),
the solution to that little problem
is actually quite elegant.

VanHattum, based in Richmond, CA, is one of the organizers for a poetry reading sponsored by the Journal of Humanistic Mathematics and scheduled at to be held at the Joint Mathematics Meetings in San Diego on Friday, January 11, 2013, 5 PM - 7 PM in Room 3, Upper Level, San Diego Convention Center.  All mathematical poets and those interested in mathematical poetry are invited.