Mathematical Constants

It is not just glass, it is true for all media though which light can travel. If water [raindrops] did not, then you would never have a rainbow! The refractive index of a medium is a measure of the average refraction - as measured for white light. In fact, the angle of refraction depends on the wavelength of the light.

If the two interfaces between the two media (air to glass and glass to air, for example) are parallel then the different refraction at the first interface is exactly offset what happens at the second interface. So a beam of white light gets split up into the spectrum at the first and recombines into a beam of white light at the second interface. However, if the two interfaces are not parallel - as in a prism, or a raindrop, the result will be white light being split into the spectral components.

The very simplified answer is that imaginary numbers put together with real numbers (to make a complex number) can describe the timing of voltage relative to current, or current relative to voltage, in an AC circuit. Let's say that we're driving an AC electrical circuit with an oscillating current source, and measuring a resulting oscillating voltage. Here's the rub:

Purely Real: If you put a resistor in the circuit and measure the voltage oscillations across it, the voltage will be a purely real number. This means that the timing of the voltage peaks will match the timing of the current peaks exactly.

Purely Positive Imaginary: Now, put an inductor in the circuit instead of a resistor and measure the voltage oscillations. It will be a purely positive imaginary voltage. This does not mean that the voltage is non-existent (as many people think)! It simply means that the voltage peaks will be one quarter cycle ahead of the current peaks, or 90 degrees ahead. The voltage has physical value. If you were to touch the ends of the inductor, you would still get shocked! The imaginary property just tells you that the timing is ahead by a quarter cycle, that's all--nothing esoteric or "complicated." A good analogy to this would be if you were riding your bicycle side by side with your friend, and you were pedaling at the same rate, BUT your pedal was consistently a quarter turn ahead of his.. Your timing could be considered purely imaginary relative to him (or her).

Purely Negative Imaginary: Now, put a capacitor in the circuit and measure the voltage oscillations. It will be a purely negative imaginary voltage, which simply means that the voltage peaks will be one quarter cycle behind of the current peaks, or 90 degrees lagging.

Complex: By putting a combination of resistors, inductors, and capacitors in the circuit together, you get a complex voltage, allowing you to get "in between" values. For example, you could carefully size a resistor and inductor, put them in series, and force the voltage peaks to be 45 degrees ahead.

Hope this is clear. If it's still cloudy, I'll paste a link in the web link area that has a site out there with an interactive explanation showing how imaginary numbers can be used with complex numbers to represent both size and timing (it's actually my site, but for educational purposes only).

While these answers mainly deal with electric power [alternating current], the same concepts apply to waves in general which have a phase difference [difference in timing of peaks and valleys of the waves].

Please see the below link for a graph of the fields around current carrying

conductors by the formula: w=(z-1)/(z+1), z=x + iy.

The theory is simple: the answer for dividing by zero is NaN, "not a number". Zero cannot divide into any number, including itself. Conceptually, there is no correct finite answer because you can always take away more zeros from the numerator. To get this answer in mathematical terms, we instead look at what is called a limit as a function approaches a point that is NaN. For example, take the function f(x)=1/(x^2). This function has no value at x=0. However, if we look at the value of the function at x values very, very, very (actually, infinitely) close to x=0, we can see a trend and use that to understand what happens at x=0. In the case of 1/(x^2) the function approaches infinity at very small negative and very small positive values, so we conclude that the limit as x approaches 0 of 1/(x^2) is infinite, even though the function has no value at x=0.

The above applies to many, but not all, algebraic structures. Specifically, because it models the world we live in, it's true for fields. The axioms of a field require that any number multiplied by zero is equal to zero and that any number multiplied by its inverse is equal to one. Therefore, 0*0-1 equals both 1 and 0 according to these rules which is a contradiction, therefore making division by zero in general undefinable.

We reach a delema in our understanding of what the results are saying. If we look at the question as we approach zero from a positive side we see our limit is positive infinity and if we approach it for a negative direction we reach a limit of negative infinity. Here we throw up our hands and say this can not be and dismiss the answers as impossibles. What if our understanding of infinity is wrong as in thinking infinity must be a number and thus only one answer is allowed. What if infinity is an area of understanding that we do not yet possess. What if division by zero is just what the answers say. the only understanding we can derive from both being correct is that it represents a joining point of negative and positive numbers.

Let us look at Einstin's theory

100

The correct spelling is "googol" which is a one followed by one hundred zeros.

Interestingly, Larry Page and Sergey Brin, the founders of Google, made the same spelling error when naming their company. The rest is history!