Rule One in hooking the reader with a query headline is: Don’t answer the question immediately. Let’s break that one. Yes, cyber warfare is imaginary, but only for a certain kind of “imaginary.” What kind is that, you ask. AaaHAH!

It all has to do with number lines. If the early Greek theoreticians had been in charge, the only numbers in the Universe would have been the integers: 1, 2, 3,…. Life is simple when your only calculating tool is an abacus without a decimal point. Zero hadn’t been invented in their day, nor had negative numbers.

Then Pythagoras did his experiments with harmony and harp strings, and the Greeks had to admit that ratios of integers are *rational*.

More trouble from Pythagoras: his *a ^{2}+b^{2}=c^{2}* equation naturally led to

*c=√(a*Unfortunately, for most integer values of

^{2}+b^{2}).*a*and

*b*,

*c*can’t be expressed as either an integer or a ratio of integers. The Greeks labeled such numbers (including π) as

*irrational*and tried to ignore them.

Move ahead to the Middle Ages, after Europe had imported zero and the decimal point from Brahmagupta’s work in India, and after the post-Medieval rise of trade spawned bookkeepers who had to cope with debt. At that point we had a continuous number line running from “minus a whole lot” to “plus you couldn’t believe” (infinity wasn’t seriously considered in Western math until the 17th century).

By then European mathematicians had started playing around with algebraic equations and had stumbled into a problem. They had Brahmagupta’s quadratic formula (you know, that [*-b*±√(*b ^{2}-4a·c*)]/2

*a*thing we all sang-memorized in high school). What do you do when

*b*

^{2}is less than 4

*a·c*and you’re looking at the square root of a negative number?

Back in high school they told us, “Well, that means there’s no solution,” but that wasn’t good enough for Renaissance Italy. Rafael Bombelli realized there’s simply no room for weird quadratic solutions on the conventional number line. He *made* room by building a new number line perpendicular to it. The new line is just like the old one, except everything on it is multiplied by ** i**=√(-1).

(Bombelli used words rather than symbols, calling his creation “plus of minus.” Eighty years later, René Descartes derisively called Bombelli’s numbers “imaginary,” as opposed to “real” numbers, and pasted them with that letter * i*. Those labels have stuck for 380 years. Except for electricity theoreticians who use

*instead because*

**j***is for current.)*

**i**Suppose you had a graph with one axis for counting animal things and another for counting vegetable things. Animals added to animals makes more animals; vegetables added to vegetables makes more vegetables. If you’ve got a chicken, two potatoes and an onion, and you share with your buddy who has a couple of carrots, some green beans and another onion, you’re on your way to a nice chicken stew.

Needs salt, but that’s on yet another axis.

Bombelli’s rules for doing arithmetic on two perpendicular number lines work pretty much the same. Real numbers added to reals make reals, imaginaries added to imaginaries make more imaginaries. If you’ve got numbers like *x+ i·y* that are part real and part imaginary, the separate parts each follow their own rule. Multiplication and division work, too, but I’ll let you figure those out.

The important point is that what happens on each number line can be specified independently of what happens on the other, just like the *x* and *y* axes in Descartes’ charts. Together, Bombelli’s and Descartes’ concepts constitute a nutritious dish for physicists and mathematicians.

Scientists love to plot different experimental results against each other to see if there’s an interesting relationship in play. For certain problems, for example, it’s useful to plot real-number energy of motion (*kinetic energy*) against some other variable on the * i*-axis.

Two-time Defense Secretary Donald Rumsfeld used to speak of “kinetic warfare,” where people get killed, as opposed to the “non-kinetic” kind. Apparently, he would have visualized cyber somewhere up near the * i*-axis. In that scheme, cyber warriors with their ones and zeros are Bombelli-imaginary even if they’re real.

~~ Rich Olcott