Tag: imaginary unit

“Fog everywhere” and Other Confusions

Charles Dickens gives us wonderful sociopolitical insight in his serial Bleak House, with his imagery of impenetrable fog in the second paragraph of chapter 1:

Fog everywhere. Fog up the river, where it flows among green aits and meadows; fog down the river, where it rolls deified among the tiers of shipping and the waterside pollutions of a great (and dirty) city. Fog on the Essex marshes, fog on the Kentish heights. Fog creeping into the cabooses of collier–brigs; fog lying out on the yards and hovering in the rigging of great ships; fog drooping on the gunwales of barges and small boats. Fog in the eyes and throats of ancient Greenwich pensioners, wheezing by the firesides of their wards; fog in the stem and bowl of the afternoon pipe of the wrathful skipper, down in his close cabin; fog cruelly pinching the toes and fingers of his shivering little ’prentice boy on deck. Chance people on the bridges peeping over the parapets into a nether sky of fog, with fog all round them, as if they were up in a balloon and hanging in the misty clouds.

Charles Dickens, Bleak House. Bradbury & Evans, 1852-1853.

Think of the Trump–Epstein cover-up and the machinations of the U.S. government to attempt to conceal everything in a similar fog. We discussed another dimension of our ignorance with Friedrich Nietzsche’s assertion that we “are unknown to ourselves”.

Let’s consider a third level of our confusion and how we attempt to extricate ourselves by expressing the world around us through mathematics. For example, the square root of -1 is i. ii is approximately 0.208. To a student encountering this concept for the first time, it can be inscrutable that an imaginary number to the power of itself results in a real number. To quote Wikipedia:

In electrical engineering, signal processing, and similar fields, signals that vary periodically over time are often described as a combination of sinusoidal functions (see Fourier analysis), and these are more conveniently expressed as the sum of exponential functions with imaginary exponents, using Euler’s formula.

Wikipedia, Euler’s formula [with added links]

The esteemed physicist Roger Penrose has said on multiple occasions that he believes the realm of imaginary numbers or complex analysis will be more informative in physics than real numbers. (See his classic book, The Road to Reality: A Complete Guide to the Laws of the Universe.)

Let’s conclude with our inevitable mortality as living beings as another source of perplexity and confusion. As we grow older, half of our mind is fixated on the enjoyment we get from life, while the other half is focused on the anxiety that it is not forever. We attempt to sidestep this anxiety by clinging to the escapist thought that “Besides, it’s always the others who die.”

Education and “Intuition Pumps”

Professor Daniel Dennett of Tufts uses the word “intuition pumps” in discussing intuitive understanding and its tweaking.

Let’s do a simple example, avoiding as always “rocket science,” where the intricacies weigh you down in advance. We make a U-turn and go back by choice to elementary notions and examples.

Think of the basic statistics curve. It’s called the Bell Curve, the Gaussian, the Normal Curve.

The first name is sort of intuitive based on appearance unless of course it’s shifted or squeezed and then it’s less obvious. The second name must be based on either the discoverer or the “name-giver” or both, if the same person. The third is a bit vague.

Already one’s intuitions and hunches are not fool-proof.

The formula for the Bell Curve is:

\begin{equation} y = \frac{1}{\sqrt{2\pi}}e^{\frac{-x^2}{2}} \end{equation}

We immediately see the two key constants: π (pi) and e. These are: 22/7 and 2.71823 (base of natural logs).

The first captures something about circularity, the second continuous growth as in continuous compounding of interest.

You would not necessarily anticipate seeing these two “irrational numbers” (they “go on” forever) in a statistics graph. Does that mean your intuition is poor or untutored or does it mean that “mathworld” is surprising?

It’s far from obvious.

For openers, why should π (pi) be everywhere in math and physics?

Remember Euler’s identity: eiπ + 1 = 0

That the two key integers (1 and 0) should relate to π (pi), e, and i (-1) is completely unexpected and exotic.

Our relationship to “mathworld” is quite enigmatic and this raises the question whether Professor Max Tegmark of MIT who proposes to explain “ultimate reality” through the “math fabric” of all reality might be combining undoubted brilliance with quixotism. We don’t know.