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Imaginary Numbers and the Real World

Part I: An Imaginary Parable

This is a fairy tale, but it is not far short of the truth. The way mathematicians played with words and meanings is true enough, only the names and circumstance have been changed to keep the story moving.

Many years ago, early communities took to farming and to trading in markets. They began to record the numbers of things they owned and traded on knotted string, others cut notches in wooden sticks, yet others scratched their tallies into clay tablets. Presently, somebody got the bright idea of creating symbols for each number so you didn't have to count and remember long tallies. It all seemed very natural.

But something was missing. Presently money appeared and moneylenders set up in the market too. One day a lender ran short of cash and, rather than send his client to a competitor, he borrowed some more himself from another moneylender and lent that to his client. He looked at his tablets and wondered how to record it. He scratched a little − next to the number he owed and called it a "negative number" because he lacked it. Soon, another client paid him back just what he had borrowed but not the interest on the loan. The account balanced neatly but still had to be kept open. If he left its entry blank it would look as if he had not done his sums yet, so he scratched a little circle in there to show that there was no net cash in his purse yet. The number zero was born. His colleagues looked over his shoulder and were disgusted.

"You shouldn't be writing those silly minus and zero numbers down," they complained, "it's not natural."

"What do you mean?"

"Well, numbers naturally start at one and work up. You can't go the other way, you can't have a minus sheep, and if you haven't got any sheep then you haven't got any, it's absurd to say that you have a flock of "zero" sheep in your field. I've got a "zero" empire too but that doesn't make me an emperor."

After much discussion, and not a few black eyes and broken drinking-vessels, everybody agreed to say that there were two kinds of numbers, natural and unnatural. But the unnatural ones proved so useful that everybody started using them.

And something else was still missing. many years later a baker was having trouble. She had thought up the idea of baking one big loaf and then selling off pieces to hungry customers. It was much quicker and easier than making lots of little bread-cakes, and not so easy for the street urchins to steal either.

Then one customer tried to be sneaky. He paid for a single piece and then tried to take the whole loaf.

"Oy! Stop that! You paid for a quarter of that loaf and that's what you get!"

"No," he wheedled, "I paid for my lump of bread. That's the lump you have got, so that's what you sold me."

"Don't be cheeky, I sold you a quarter of it." She cut the piece off and gave it to him, "There, that's your piece."

"No, that's a quarter of a piece, I paid for a whole piece."

He had probably drunk a bit too much or something, anyway in the end they had to call on the village Elders to make peace.

"I bought a whole piece" declared the hungry man, "you can't walk off with a fraction of a thing, you have either got something or you haven't."

"It wasn't whole, said the baker crossly, it was a quarter he paid for and a quarter he got."

The Elders saw immediately that the baker was right, but the customer was big, strong and a little drunk. They didn't want the black eyes that the moneylenders had given each other. After a brief, muttered conversation the Chief declared that there were two kinds of number. Whole numbers were the normal order of things, it was absurd to suggest otherwise. The big man smirked and reached for the remaining loaf. But, the Chief continued as he pushed his staff firmly in the man's way, broken fragments yielded useful fragmentary numbers too, which may be called fractions for short. If somebody sold a quarter fraction then that was what they had sold. The big man scowled, opened his mouth to retort, saw the Chief raise his staff meaningfully and shut it again. The Elders had spoken and that was that.

One of the Elders had long been staring at the night sky, watching the moon and planets in their slow dances. They noticed how the stars circled at a fixed pace and how the sun, moon and planets each followed their own sometimes more erratic paths round and around. Periodically some objects would line up then drift apart again. The baker's invention of fractions gave him an idea and he began whittling gear wheels, like the ones they sometimes used for winding up buckets of water and things, to try and make a model of these cycles. The number of teeth for each gear was crucial and, patiently overt he years, he and his descendants refined and refined the toy. His trick was, he argued, perfectly reasonable and he called the relative number of teeth on each gear the ratio (which is Latin for "reason") between them.

It was not until several generations after that, when the village had grown to a town and established a university, that the next squabble arose and yet more missing numbers were found. A young geometer had sketched a diagram which he claimed showed that the diagonal of a square could not be a fractional number, it led to a contradiction. The Dean's first instinct was to put him to death, but his friends persuaded the Dean that he would have to be proved wrong first. It was all to do with something called "the square root of two". The young man was locked in a cellar anyway and everybody pored over his drawings. It all went terribly quiet. Nobody wanted to be the first to say, "He's right, you know," in case the Dean cut his head off too.

"Well?" cried the Dean after a day and a half. "Numbers are the foundation of reason, nothing if not numbers is entirely rational. This is the very origin of the word 'ratio'. Where there is no ratio there is no reason. Can nobody disprove this irrational nonsense?"

After two whole days, the Dean wrote an edict that all numbers were fractional ratios, it was only rational that they be so, and any other claim was wholly irrational and without foundation. But by then the other mathematicians had studied the idea so hard they were beginning to catch on to it. One proved that pi was irrational, another that all square roots of prime numbers were. "It may not be rational but it certainly helps" became a popular catchphrase. Nevertheless, "irrational" these numbers remained.

But the question of "the square root of minus two" remained a mystery. In fact all the square roots of minus numbers were very obviously missing. "What about the square root of minus one?" insisted a particularly precocious student several centuries later. "It doesn't exist", replied his teacher. He sulked for weeks after that, muttering to himself, "Why can't it exist? It ought to exist, mathematics is broken and incomplete without it. I shall make it exist!" He started doing some sums with it and quickly found that it gave rise to a whole new set of numbers alongside the old ones. If you mixed the two kinds of numbers together it all got a bit complex, but it was also immensely powerful. He showed his Professor.

"But they all have your funny square root of minus one in there, he grumbled. "You know very well that doesn't exist and you are just imagining it, none of them are real numbers that you can do real mathematics with. It's very elegant I'm sure my dear boy, but quite imaginary. Look here, if you use the letter i for "imaginary" then we will all know what you are talking about. But please don't bother me with your complex numbers, they give me a headache."

It was indeed elegant, one of his equations has been voted the most beautiful in all of mathematics, ever. And it was so powerful that at long last the family of all numbers was proved complete.

Part II: An Imaginary Experiment

This part is true, it is imaginary only in terms of the numbers used in the maths. I was lucky my manager didn't walk in while I was playing.

My day job was once electromagnetic engineering. One quiet afternoon in the lab left me at a loose end with nothing to do, so I dug out some big lumpy bits and built what is called an L-C resonant oscillator or Tank circuit, with as slow a resonant frequency as I could make it. Using mains power transformers as inductors and lashing together several of the big capacitors that usually smooth out the supply, I got it down to two or three cycles per second, around the rate at which a grandmother clock ticks, which in audio terms would be several octaves lower than anybody can hear. Then I added some old-fashioned moving-pointer meters to show the current and voltage in the circuit.

A funny thing about such "reactive" circuits is that their equations give the current through them as an imaginary number. The Smith chart, a circular graph paper with one diameter marked real and the one at right angles marked imaginary, is familiar to anybody who designs radio equipment.

Back in the lab, I switched on my giant resonator. The pointers sprang to life. The frequency was so low that the pointers swung madly to and fro, following the instantaneous voltage and current as the oscillator shuttled its energy back and forth between the magnetic field of the transformers and the electric field of the capacitors.

Now, the imaginary depiction of the current is used to keep track of something called phase. If the current is neatly in phase with the voltage, as happens with say an electric toaster, then the current is mathematically real. But if you hook the supply up to something reactive like an electric motor then the current starts to get behind. On a Smith chart, we see it angling off from the real axis and this is called its phase angle. Mathematically we give it an imaginary component, and the bigger the phase angle the more imaginary it gets. In a Tank circuit, the phase reaches ninety degrees, a full right angle, and the current becomes wholly imaginary.

That was exactly what I now saw in my pointers. As the voltage swung to a peak, the current was rushing past zero. As the voltage swung back down past zero to become negative, the current hovered at its peak, and so on. The two signals were exactly ninety degrees out of phase. The "imaginary" current was visible on a very real pointer.


The lesson from my experiment is simple. In no way was that current imaginary in the usual sense of the word, it was right in front of me and perfectly real. As my parable illustrates, mathematicians have a historical reason for once thinking such numbers to be imaginary, but they soon came to their senses and realised that that was plain silly and, well, unimaginative. Still, the name has stuck.

I can think of no better illustration than my swinging pointers to show that "imaginary" numbers, and the things they describe, are just as real to us as any other numbers.

And that is the secret of imaginary numbers. They may be called unnatural, fractional (broken), irrational or imaginary, but that is only because mathematicians are as quarrelsome and unimaginative as everybody else. All numbers are, in the ordinary way, just as natural, unified, rational and real as any other number.

Next time you read Stephen Hawking's explanation of "imaginary time", remember that it is no more imaginary than anything else in his book, it's just a reminder to mathematicians to wrap that curious i up in it.

Updated 5 Feb 2018