Imagination
Back in the early days of the middle ages, mathematicians were thinking about square roots. They had long realized that when squared, a number must be positive. That is, 2 x 2 = (-2) x (-2) = 4 and so a square root could exist only for positive numbers. For example, the square root of 4 is equal to +2 or -2, but the square root of -4 makes no sense.

Then a mathematician realized that it might be fun to see what the properties were of an impossible, or imaginary, number whose square was negative. So the imaginary number i was born and was defined as:

Well, it turned out that i could be put to some very interesting uses. It could be combined with a normal number and then used to define a plane in space. It also gave unexpected results when multiplied by itself a number of times. The real surprize came, however, when a mathematician named Euler (pronounced "oiler") combined it with another unusual number, called e. The number e is similar to the number (pi) in that it cannot be expressed as the solution to an algebraic equation (like, x² = 2), and its decimal digits repeat without pattern and go forever. Numbers like these are called "transcendental."  , as you remember, is the ratio of the circumference of a circle to its diameter. No matter how big or little a circle is, its circumference divided by its diameter is always equal to . The number of 's decimal places go to infinity and yet there is no repeating pattern.

Like , the transcendental number e appears in many places but most often in the branch of mathematics called differential calculus. Euler took e and raised it to an imaginary power, i, that is, e^ix, where x is any normal number you'd like. Euler thus found one of the most astonishing and beautiful results in mathematics: this strange combination of numbers was equal to the sum of a cosine and an imaginary sine:

The number e turns out to be closely related to trig functions, but only if it is raised to an imaginary power!! WOW!!

= , then what do you get? 

If you haven't studied calculus and trig, you might be completely bored by this story so far. But if you have, then as I did, you must be running around the room and slapping your head with your hand, saying, for this to be right, there HAVE to be layers of relationships we don't yet understand! I mean, this is deep.

Well, as beautiful as this is, it did not seem to have a use for any physical process they knew of at the time. But eventually, mathematicians found that e^ix could be useful in solving very complicated problems, known as differential equations. They would use the properties of e^ix to solve an equation, and then the imaginary part of it (the part that is multiplied by i) would be thrown away. So e^ix turned out to be a really wonderful trick to solve these special problems.

e^ix to the Rescue!
A century or more passed. By the end of the 19th century, physics was in trouble. Light sometimes acted like a particle yet at other times like a wave. Matter itself was showing that on the subatomic scale it also could be a wave, and no matter how physicists thought about the problem, their models, derived from Newton's laws, could not make accurate predictions about the behavior of subatomic particles. In 1900 a physicist, Max Planck, used what he thought was a mathematical trick to solve one of these experimental problems of the day — he made restrictions on the energies particles could have. In other words, he "quantized" energy. Eventually, physicists began to think that Planck's mathematical trick was not a trick at all but was telling us that on the subatomic scale energy comes in packets and does not have values inbetween. Finally Schrodinger decided, as had Planck, to toss caution to the wind and to put together an equation not by deriving it from basic principles, but by accepting that light and matter had both a particle and wave nature and that energy comes in discrete packets. Thus Schrodinger's equation was born:

Do you see that i there on the right side? When this equation is solved by normal, everyday methods, a real solution comes out, but that little i also provides an imaginary part to the solution. And here, unlike the situation when physicists could throw away the imaginary part, we cannot do that with quantum mechanics! The imaginary part of the solution is physically meaningful!! This is no trick! This is WEIRD! How is it that a crazy number like i could have meaning in the physical world? What does this say about the nature of Nature?

How Does this Relate to Your World?
Had it not been for the imaginary and otherwise silly concept of i, quantum mechanics would not yet have been constructed. The computer you are reading this on works because of quantum mechanics. Had it not been for an abstract mathematical concept from the middle ages, you would not be taking a course on the internet, because our computer technology is based on quantum mechanics.

Do you see what I mean about the math seeming to be unrelated to the physical universe, a fantasy in someone's head, but yet here it is providing the structure on which physical phenomena are described! Go figure!

If you don't have a headache by now, here are some discussion questions you might want to think about ==> ?

The third example is a rare case of a physical theory coming along without a mathematical structure to frame it in. Solution? Newton invented a new branch of mathematics to suit!