In the earliest days, there were only numbers. Counting numbers. Numbers like 1, 2, 3, 4, and so on. They were very useful, for they allowed a person to know how many sheep were let into the fold at night or how many souls had been baptized on Easter Eve.
In those days, of course, there wasn't the distinction between mathematics and theology which we try to impose today.
From these helpful numbers our mathematical ancestors went on to invent computation, through which you could add up the number of sheep in several different sheepfolds without going to the trouble of moving them all to one place and counting the combined flock. I imagine this was especially advantageous with the rise of kings and incas who owned flocks in far distant places. Those same ancestors also invented the reverse operation, by which you could determine how many sheep you had left after giving seven to the tax collector.
At some point, and likely somewhere in India, it was recognized that it would helpful for performing these calculations if there were a specific symbol for the number of sheep left if the tax collector takes all of them. Thus the zero was born and we were able to write an answer for the expression 155 - 155.
Now, mathematicians are an inquisitive lot. It is fairly clear what happens if you have 155 sheep and the tax collector takes away 7. With the zero, we could also say what happens if you had 5 sheep and the tax collector takes all 5. What would happen if you only had 5 sheep and the tax collector took 7? Well, that's impossible! said the scoffers. And so values such as 5 - 7 were known as "impossible" numbers. Impossibility did not deter the mathematician, however. "Suppose there could be such a number," they said. And so they began to write 5 - 7 = 0 - 2 or, as a shorthand, -2.
This invention, impossible as it seemed, became quite useful since at least the time of the Medicis and the negation of your future earnings by reason of debts you owed the bankers. Those "impossible" numbers became quite possible and were quietly renamed as "negative" amounts.
Meanwhile, other mathematicians invented such useful ideas as multiplication and ratios which had plenty of application in building buildings and sailing ships and all manner of other human endeavor. Mathematicians were particularly entranced by repetitious patterns of things. For example, in the area of multiplication, they were especially intrigued by multiplying a number by itself: 2 times 2, 4 times 4, 64 times 64, and the natural extensions such as 2 times 2 times 2 times 2 times 2. And the the reverse: What one number, multiplied times itself, gives 64? All these examples you can figure out rationally, but there are others. What one number when multiplied times itself gives 7? You can't rationalize the answer; it isn't a counting number, it isn't even a ratio.
"Suppose there could be such a number," the mathematicians said once again. If you accept these irrational numbers, you can apply arithmetic to the shapes of plane geometry. For example, if three people stand "in a circle" (as we'd now say), and each of them is 5 feet from the center of the group, then you could compute how far each person is from the other -- the distance would be 5 multiplied by the number which if multiplied times itself would make 3, obviously not a rational result.
As more and more people are converted and join the heavenly circle (or as the shape formed by these people more and more matches a perfectly round circle), the ratio of the distance around the group to the distance to the center of the group approaches a value which is not quite 22/7, not quite 355/113, in fact not quite anything rational. It turns out, in fact, that this ratio transcends even those irrational numbers of triangles and other shapes. Theologically, how wonderful that transcends all reason! Mathematically, we just name it pi and use it in equations.
With all these new numbers being found or (the sceptic might say) invented, perhaps it is not surprising that mathematicians were able to imagine still one more. I've already said that squares and square roots were particularly intriguing to them, and it must have galled the mathematicians that squaring the impossible numbers made them into ordinary, possible, positive values. For example, 5 times 5 is 25, but so is -5 times -5. Shouldn't there be a way to get -25? Just what would the square root of -25 be?
"Imagine that there could be such a number," the mathematicians would say, for in actual reality mathematics is all about imagination. In fact, it doesn't take very much imagining at all; we only need to imagine one new number. After all, -25 can be thought of as -1 times 25 and we know -- those of us who have been playing with square roots for years and years, we know -- that the square root of a product is the same as the product of the square roots of the factors. That is to say, the square root of -25 is the square root of -1 times the square root of 25. The square root of 25 is easy; it's just 5. So the answer is 5 times the square root of -1, if we can imagine such a thing.
But what about this square root of -1? We can imagine that this number exists. We can even give it a name -- i would be a good choice, given that it is imaginary.
Thus the history of mathematics teaches us a very simple lesson:
i am the square root of an impossible number