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47 TONS

Update: March 25, 2021
Today, Japanese public broadcaster NHK News (the Japanese Broadcasting Corporation) reported that the intruder detection sensors at the Kashiwazaki-Karina Nuclear Power Plant have been broken for at least one year; inspectors and regulators rank it a terrorism risk  at the “most serious level”.

The plant has not operated since the Tsunami of 2011, which damaged and closed many of Japan’s nuclear power plants including the catastrophic meltdown and fires at Fukishima — which remain on-going. Sensor readings at the complex suggest that one of the reactors could re-ignite. The area of concern is inaccessible. 

Update: May 16, 2020  
Today, NHK News announced that Japan has officially launched a Space Defense Unit, otherwise called the Space Operations Squadron (SOS). It’s mission is to protect Japan’s many satellites from debris and other space hazards and threats. A spokesman said that communication protocols between Japan and the US are yet to be worked out. 

Update: May 11, 2018
In April Congress approved Japanese American Paul M. Nakasone to lead the National Security Agency, the Central Security Service, and the U.S. Cyber Command. Paul’s grandparents were Japanese citizens born in Japan. His father worked for Army intelligence services during World War II when Japanese Americans, many of them, were rounded up and detained in USA internment camps. Nakasone is an Army General appointed to lead an agency (the NSA) that has been led by a rotation of Navy Admirals, Air Force and Army Generals. 
The Editorial Board

47 TONS

While the Little Rocket Man and North Korea capture the world’s attention, our president is in Tokyo to deal with a threat that dwarfs anything we have faced since the Japanese attack on Pearl Harbor 76 years ago.

The surprise attack against our Navy on Sunday morning, 7 December 1941, started a cascade of retaliation against the Japanese that three-and-a-half years later resulted in 67 Japanese cities burnt to ashes during a few months of sustained “fire-jelly” attacks by hundreds of Boeing-29 Superfortress bombers and other aircraft. After napalming the cities to ashes and dust, the United States followed the horror with a “preemptive” nuclear strike against the cities of Nagasaki and Hiroshima.

Nine million Japanese civilians were left homeless. The death toll has never been definitively calculated, but two million souls is a reasonable guess.

(Click map to enlarge.) The distance from Pyongyang, North Korea to Tokyo, Japan is 800 miles; Hawaii, 4,400 miles; Los Angeles, 6,000 miles; Seattle, 5,150 miles; Alaska, 3,200 miles. The border of North Korea meanders 25 to 50 miles from downtown Seoul, South Korea.

A few years after the stalemate of the Korean War, General Curtis LeMay — head of Strategic Air Command — claimed that his pilots had killed a similar number of Koreans by aerial bombardment — 20% of the population.

The United States killed an estimated million Iraqi civilians in the more recent wars in the Middle East, which included the Gulf War and the Iraq War.

It killed 2 million Vietnamese civilians during the Vietnam genocide of the 1960s and 70s. In the region of southeast Asia, the USA killed 3.4 million souls, military and civilian, by the war’s end.  

In August 1945, USA bombers killed hundreds of thousands of Japanese civilians in atomic bomb attacks on Hiroshima and Nagasaki. Millions were burned alive in fire-jelly (napalm) bombings, which decimated 67 cities over several months. Evacuation of cities helped to reduce casualties. 

Why am I bringing up a bunch of disturbing statistics? What’s the point? Why not leave unpleasant memories forgotten in a distant past where they can’t impact the happy lives we live now, not then?

What possible benefit can remembering the past confer upon our contented present? Why bother puking up a sour history that only the old-timers among us experienced?

May I ask one more question? Maybe thinking about the answer will help some to make sense of current events that seem to have no rhyme or reason.

Of the fifty countries against which we have directed our military wrath since World War II, which among them has a right to the biggest grievance? Who did we hurt the most?

Which country has been forced to endure the shame of a military occupation that never ends? Ok, maybe it sounds like more than one question. Deal with it.

America fights secret and not so secret wars against communist, Islamic, western hemispheric, and, it turns out, African countries all the time. We have conducted strategic operations against friend and foe alike since World War II.

We have meddled in the internal politics of super powers like Russia and China. The Dalai Lama of Tibet wrote in his book Freedom in Exile that the United States gave him millions of dollars to incite violence against China, for example.

The USA has attacked militarily one in four of the 190 countries on the earth during the modern era. Which country is the one most likely to harbor a secret ambition for revenge?

America keeps itself in a state of perpetual war to feed the appetites of voracious weapons manufacturers whose stockholders are among the world’s most affluent. The AUMF (Authorization for the Use of Military Force against terrorists) passed Congress, and President Bush signed the bill in August 2001 for a reason — to fuel the ambition of arms dealers by freeing them from the inconvenience of securing approval of Congress to declare wars — which the Constitution demands. Only California’s Barbara Lee (no relation to Billy Lee) voted against it.

Since 1991 Congress has passed and the president signed four AUMFs, mostly to cut down on the amount of work and delays that are inevitable when large elected groups of representatives are compelled to go on record for or against any particular conflict.

We in America live under a lot of illusions. We tell ourselves a lot of lies about how wonderful we are and how everyone wants to be like us. Our enemies who fear us the most insist — some of them, anyway — that they love us; they want to live with us and be like us, and we believe them.

No one tells a command officer who is carrying an automatic assault rifle that he is a pig; the term “butt-wipe” is never used. No one wants to die for a no good reason like name calling. Our subjugates place flowers in the barrels of our guns and tell us they love us.

Everyone who has been shamed and humiliated prays for their day of liberation; the day of their revenge; the day the world is finally set right. It’s human nature. The desire to settle scores crosses cultural, religious, and geographical boundaries.

Few countries that have suffered cremation by fire of millions of their citizens forget. They don’t forgive. Not really. Think long and hard.

It’s true.

For almost a year Billy Lee lived where he could view Mount Fuji from his bedroom window during his two-year stay in Japan. The Editorial Board

The situation in Japan is dire; it is. The United States for some insane and goofy reason permitted the Japanese over the past 30 years to build the most sophisticated nuclear power grid the world has ever seen.

The USA sold the Japanese uranium-impregnated fuel rods. A by-product of their use, which is to produce the intense heat required to generate electricity, is plutonium. Instead of collecting and disposing the spent fuel rods, the Japanese built facilities to extract the plutonium. They promised to use the plutonium for fuel in advanced power generators called “fast reactors.”

Fast reactors are in principle cheaper and less complicated; they are also more volatile; more dangerous to operate.

After the earthquakes and tsunami of 2011, the Japanese abandoned “fast reactors”. They discovered during audits following the disasters at Fukushima and other facilities that their fast reactor safety records bordered on terrifying. They stopped using plutonium for fuel. With no place to “burn” what they continued to harvest, plutonium began to accumulate, bigly.

In the entire universe plutonium is found above trace amounts at one location and one location only: planet Earth. Plutonium went extinct due to radioactive decay billions of years ago.

Plutonium can be created during rare cosmic events, but the bomb-making kind — Pu 239 — is a manufactured element that does not occur in nature. It is a byproduct of nuclear fission reactions that hides within the matrix of poisons that make the remnants of spent fuel rods.

Plutonium is among the most poisonous substances known. The speck of plutonium dust that kills you, you will likely never see. Some scientists today have downplayed the lethality of plutonium 239. My advice is to be skeptical whenever vast amounts of money and power fuel a controversy.

Regardless of its lethality as a poison, no one argues that fourteen pounds is enough plutonium to make an atomic bomb of a construction so unsophisticated that a high schooler could fashion the necessary components in shop class. Sophisticated bombs require even less plutonium — a mere nine pounds.

This is what plutonium powder looks like. Japan has 94,000 pounds of it. 14 pounds are required for an unsophisticated bomb; 9 pounds for a sophisticated version.

Japan has harvested 47 tons (94,000 pounds) of high-grade plutonium from its nearly one-hundred or so nuclear power and processing plants, which include power plants, research reactors, fast reactors, reprocessing installations, and recently decommissioned facilities — decommissioned due mostly to safety concerns.

Japan’s production schedule is running at a frenetic pace — adding eight tons of surplus plutonium to its stockpile every year into the foreseeable future unless the United States is able to shut down Japan’s reprocessing installations with an agreement scheduled for negotiation in 2018. Our new president has said the old agreements won’t be changed.

By this time next year the Japanese will have accumulated enough high-grade Pu 239 to make as many as 12,000 atomic bombs. Should it ever make that choice, Japan will possess the world’s largest nuclear arsenal almost overnight. 

Plutonium is heavy. Fourteen pounds of Pu 239 is the size of a softball. Nine pounds is the size of a baseball. It is exactly the right amount to construct the 4.38″ diameter ring used in a typical bomb. By this time next year Japan will possess enough plutonium to make 12,000. Depending on how the bombs are tampered, configured, and initiated such bombs can release an equivalent explosive power in a range from 20,000 to 100,000 TONs of TNT. The bombs can be launched by artillery, dropped from planes, or delivered by missiles, drone subs and boats.

What follows next in this essay is the scary part. Some readers might want to bail and maybe find a good comic book to occupy their imaginations.

Despite agreements with the United States that followed World War II, Japan has one of the largest military budgets in the world. The country spends 42 billion dollars per year on its military. This expenditure does not include its civilian nuclear power system or its civilian space exploration programs.

The Japanese consolidated three civilian rocket launching companies into one (named JAXA) in 2003. They are launching rockets into space all the time. JAXA designed, built, launched, and maintains the largest module on the International Space Station. The Japanese have spacecraft in the asteroid belt and spy satellites in Earth orbit. These are civilian programs.

The Japanese have, over the past fifty years, resurrected RIKEN, the laboratory complex used in their atomic bomb program during WW2. The lab employed two cyclotons, which the United States destroyed in a bombing run in 1945.

Today RIKEN labs sprawl across the islands of Japan in dozens of complexes that are impossible to police. The labs are involved in so many areas of research that no one can keep track of it all even if they wanted to. Some of the research is diabolical. One administrator was found hanging by the neck in his office in 2014. Enough said.

Although the military budget of the United States seems huge, people might want to consider that the USA spends one-third of its military dollars on salaries and pensions. No other military spends as much. It maintains 800 military bases in 70 countries at an expense of $200 billion — an expense that other militaries simply don’t have. Japan spends about the same amount on defense as England, France, and Germany. A controversial argument can be made that the combined military might of Russia, China, Japan, and North Korea exceeds that of the United States. It is an argument that is hard to prove, because countries lie about their military expenditures, war-fighting readiness, and technical capabilities. The chart above is misleading in another important way, because it doesn’t include expenditures on nuclear weapons — their production, maintenance, and modernization — which are state secrets in all the countries that possess them.  

The Japanese don’t have to make bombs from their plutonium stores to wreak havoc on an adversary. They can pulverize the metal into aerosols and release plutonium dust into the air over cities.

They can load plutonium into drone subs like rumors say the Russians have done and set hundreds of them in the coastal waters of our country. The subs can lie in ocean sand and silt for decades before releasing their poisons, should it ever become necessary.

Their advanced missile technology might enable Japan to overwhelm our defenses by launching multiple warhead missiles over our homeland. It might take a few months, but poisoned populations would eventually succumb to the release of toxic dust.

And, should they choose to make bombs, well, any country with the resources of a country as sophisticated as Japan can turn high-grade plutonium into bombs in a few days; they can possess the capability to create hell on Earth in the blink of an eye, anytime they choose. With the right (or wrong) leadership they can unleash a nightmare of suffering far worse than the inferno we inflicted on them 72 years ago.

This plant is the place where the Japanese extract plutonium from spent nuclear fuel rods. The Japanese have admitted on NHK television that they have 94,000 pounds of plutonium, which they have no use for nor any place to safely store.

Plutonium is an artificially produced killing material that no human being, company, or country should ever be allowed to possess or use. It is a forbidden apple of physics that can only bring anguish to whoever uses or shares it with others.

Japan has the potential to threaten the world with the same level of terror as the United States, Russia, China, Pakistan, India, Britain, France, Germany, Israel, and who knows what other countries. Many countries are conducting (in secret) diabolical engineering even now and will continue to do so into the foreseeable future.

What could be worse? Believe it or not, our predicament might already be much worse than anyone in the USA is willing to think about or imagine.

What about the possibility that North Korea and China are playing a game of good cop / bad cop with our military planners? What if Japan is toying with the idea of leading an unholy alliance? Behind our backs? Do we really have enough Japanese-speaking spies to keep track of all the secret Samurai cults that might be conspiring at the highest levels of government?

Do we?

What if Vladimir Putin thinks: The United States lied to me. I helped to elect an American president who is ineffective — a buffoon who can’t help me the way he promised. Let’s get ’em!

Imagine an alliance of China, North Korea, Russia, and Japan; an alliance led by the one country that has the greatest lust for payback; the strongest ache to settle scores once and for all.

A Hunkpapa Lakota holy man, Sitting Bull, had the vision that led to the defeat of the USA’s 7th Calvary Regiment on June 26, 1876 — one week before the USA’s 100th anniversary. Five of seven battalions were decimated — one led by Civil War hero George Armstrong Custer. Sitting Bull became a celebrity who worked in Buffalo Bill’s Wild West Show. Later in life he became a leader of the Ghost Dance movement, which terrified whites, because it prophesied the exodus of white people from native American lands. Ten days before Christmas — on December 15, 1890 during an arrest by police on reservation property — Indian Affairs agents shot Sitting Bull in the head and chest in front of his family and friends. Agents removed his body to Fort Yates, where they buried him in a makeshift coffin.  

A surprise attack by such an alliance would be nation ending. It might end like the Battle of the Little Bighorn. We don’t have enough soldiers or missiles or ships to fight a gathering of tribes who possess tens of thousands of nuclear weapons.

The USA has the power to destroy the whole world if we must, but we can’t save ourselves; we can’t save our country; we can’t save the planet.

In the conflagration that took the hyper-alert Lieutenant Colonel George Armstrong Custer by surprise, all his ribbons and medals; all his accolades; all his friends in high places couldn’t save him, his men, or even his horses. The battle of the Little Bighorn was a massacre that dwarfed Custer’s reputation for being a really good person; a hero of the Civil War loved by every patriotic American.

To those who say, Billy Lee, you’ve gone paranoid on us… the Japanese would never organize an attack against America unawares… not a nuclear attack… they know how bad it would be… they suffered through one… they know better than anyone… and look at them, how they smile when we tell bad jokes. The last thing on their minds is revenge. The very last thing!

I say, you are so right!  The Japanese would never hurt us. I lived in Japan for two years after the war. The Japanese have their quirks, yes, but most of them are not cruel or insensitive. They don’t enjoy watching torture videos for entertainment, most of them. Tying up women and twisting their bodies to prepare them for rape is not something most Japanese men would have any part in. Am I right? Of course I am.

The Japanese are not monsters. They are a kind and gentle people who don’t farm or ranch or mine, because they are resource impoverished. When I lived there our Japanese house-maids and yard-boys were as sweet as they could be. They meant us no harm. I see that now.

How on Earth are the Japanese going to get rid of the 47 tons of plutonium poison they have produced? And how will they dispose of the eight tons they plan to produce each year into perpetuity — plutonium which they admit has no longer any peacetime applications whatsoever?

Hayabusa2 Rocket on pad at the Tanegashima Island Space Center off the southern tip of Japan. The rocket landed a  probe on the asteroid Ryugu on 4 October 2018. The Editorial Board 

Everybody knows plutonium has a radioactive half-life of 24,000 years. It’s never going to go away. Someday, through inattention or from whatever other cause, plutonium containment structures are going to rot, and the poison will leach into the soils, the oceans, and the atmosphere to kill all living things. It is Earth’s best case scenario — the scenario where nuclear war never happens, the world disarms, and plutonium is tucked away out of reach and out of sight of war makers and other terrorists.

The process that will sterilize the planet of all life is already well underway and cannot be stopped — not over a period of tens of thousands of years. Read the essay, RISK, elsewhere on this site. Humans are likely to be extinct by the time the unnatural poisons of war and opulence first make their advance against the innocent, less intelligent life-forms that we will leave behind — like chipmunks and kittens, for example — who will never be able to understand what is killing them or why.

Our new president is in Tokyo as I’m writing this essay. Anyone who asks him will learn — because he’s not afraid to say it — he is really smart and bigly educated. He understands people and how best to manipulate them to maximize his advantages and get what he wants. You don’t believe it?  Ask him — for the love of God — ask him.

Maybe we should help the Japanese store their plutonium in a safe place — a place much safer than their earthquake tormented islands that float within the largest fisheries of the Pacific Ocean. We could store the plutonium perhaps deep in a cave somewhere. Maybe we could store it beneath the volcanic calderas southeast of Yosemite — or some other remote location, like a trench astride the San Andreas fault.

Yeah, that sounds good. Let’s do that.

If we talk nicely, will the Japanese listen? Maybe they will, if our new president has the sense to ask. Does anyone have a better idea? For the love of God, tell someone.

Billy Lee

 

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What are complex numbers? What does “i” mean, anyway?  How can a number be “imaginary“?   What does it mean to multiply “i” exactly “i” times?  Why is math hard?

For me, math is difficult because it’s interesting.  I learn things from equations that aren’t obvious when I think about the world using words and images. Some things can’t be put into words. Some things can’t be pictured.

It’s true.

What makes i^i interesting is the four real numbers it generates. (The numbers are +.2078… , -.2078… , +4.8104… , and -4.8104… .)

Can anyone give a geometric reason why an imaginary number raised to the power of an imaginary number generates four real numbers and no imaginary ones?

What does \sqrt{-1}^{\sqrt{-1}} even mean? Is there anyone who can visualize a reason why the answers make sense? Are all the answers even correct? Or is only one correct, as any calculator that can do the calculation will tell?

Complex numbers are two-dimensional numbers that are made by raising the number ”e” to the power of an imaginary number — called ”i” — times an angle in radians. Complex numbers lie on a circle in the complex number plane. Unless ”e” is preceded by a number that stretches or shrinks it, the numbers always lie on a unit circle like the one in the picture. Recall that ”i” is the square root of minus one. When ”i” is raised to the power of ”i”, the result collapses onto the real number line — in one of four possible places. Which one? The numbers don’t land on the unit circle. The process can be demonstrated mathematically, but any physical intuition about why imaginary numbers with imaginary exponents behave the way they do can be elusive.

Abstract math that hides no model that anyone can visualize makes results startling, even unnerving. It’s a lot like the quantum mechanics of entanglement or the physical meaning of gravity. They can be mathematically described and their effects accurately predicted, but no one can explain why.   

Mathematics alone can sometimes describe (or at least approximate) realities of the universe and how it seems to work, but as often as not when humans dive deep into the abyss of ultimate knowledge, math is unable to provide a picture that anyone can understand. 

How can that be? Things seem to happen that cannot be thought about except by playing around with numbers and being taken by surprise. Intuition is difficult, if not impossible.

Here is the solution of  i^i. Perhaps clues exist in the math that I’ve overlooked. If a model exists in the mind of a reader somewhere, I hope they will share it with me.

(1)       i^i = e^{\ln(i^i)} = e^{i\ln(i)} = cos (ln i) + i sin (ln i)

By definition:  e^{i\frac{\pi}{2}} = i

Also:  ln {(e^{i\frac{\pi}{2}})} = ln i

Therefore:     ln i =  i (\frac{\pi}{2})

It should now be obvious to anyone who has taken a basic course in complex variables that multiplying i (\frac{\pi}{2}) by i equals the exponent on e in line (1).

Right?

It’s a real number that returns a real result when used as the exponent of e and plugged into a calculator.  The answer is completely abstract, though. We might learn more if we take a different path to the result.   

By substitution into line (1):    i^i = cos (i\frac{\pi}{2}) + i sin (i\frac{\pi}{2})

By half angle formulas:             i^i = (\sqrt\frac{1 + cos (i\pi)}{2}) + i (\sqrt\frac{1 - cos (i\pi)}{2})

Convert 2nd term i to  \sqrt -1 :

i^i = (\sqrt\frac{1 + cos (i\pi)}{2}) + \sqrt -1 (\sqrt\frac{1 - cos (i\pi)}{2})

(2)     Simplify the 2nd term:     i^i = (\sqrt\frac{1 + cos (i\pi)}{2}) + (\sqrt\frac{cos (i\pi)-1}{2})

Euler’s cosine identity is:   cos θ =  \frac{e^{i\theta} + e^{-i\theta}}{2}

Therefore:                          cos (iπ) =  \frac{e^{i(i\pi)} + e^{-i(i\pi)}}{2}

(3)     Simplifying:               cos (iπ) =  \frac{e^{-\pi} + e^{\pi}}{2}

Substitute line (3) into line (2) and simplify:

i^i = \sqrt{{\frac{1}{2} + \frac{e^{-\pi} + e^{\pi}}{4}}} + \sqrt{{\frac{e^{-\pi} + e^{\pi}}{4}} - \frac{1}{2}}

Now it’s just a matter of pulling out an old calculator and punching the keys.

e^{-\pi} = .043214;  e^{\pi} = 23.140693. 

I rounded off both numbers, because they seem to go on forever like π and “e”; they prolly are irrational, because they don’t seem able to be formed from ratios of whole numbers. [In fact, they are transcendental numbers, because they transcend algebra. In addition to being irrational, they are not roots of any finite degree polynomial with rational coefficients. Take my word.] Using these numbers will enable anyone to compute {i^i} who has a simple calculator with a square root key.

When square roots are calculated the answers can be positive or negative. Two negatives make a positive, right? So do two positives. So doing the math gives four numbers. See if your numbers match mine: .2078… , -.2078… , 4.1084… , and -4.1084… .

I don’t know why. The answers aren’t intuitive. Who would guess that imaginary numbers raised to powers of imaginary numbers yield real numbers? — not a solitary number like anyone might expect, but four. Pick one. In nature a unique answer can be arbitrary — determined by chance, most likely.

In this case, no.

It feels to me like the imaginary fairies flying around in complex space are destined to collapse onto the real number line for no good reason, except that the math says they must collapse (maybe from exhaustion?) in at least one of four places. Can anyone make sense of it?

The ln i is well known. It is —   i\frac{\pi}{2}  — which equals (1.57078… i ). The ln of —   i^i   — can be rewritten by the rules of logarithms as i ln i, which is i times (1.57078…i ), which equals -1.57078… (a real number).  Right? The ln of the correct answer must equal this number. Only one of the four results listed above has the right ln value: .2078… .

It seems odd that a set of equations I know to be sound should return a set of results from which only one can be validated by back-checking. Maybe there is something esoteric and arcane in the mathematics of logarithms that I missed during my education along the way.

Then again square roots can be messy; there are two square roots in the final equation, each of which can be evaluated as positive or negative. Together they produce four possible answers, but just one result seems to be the right one.

Adding the four numbers is kind of interesting. They sum to zero. That is so like the way the universe seems to work, isn’t it? When everything is added up, physicists like Stephen Hawking claim, there’s really nothing here. Everything is imaginary. Some philosophers agree: everything that is real is at its core imaginary.

Are there clues in the pictures and models of complex number space that would ever make anyone think? Sure, I totally get it. Yeah, I’ve got this. Real numbers cascading out of imaginary powers of imaginary numbers make perfect sense — like snowflakes falling from a dark sky.

A mathematician told me, Rotating and scaling is all it is. The base must be the imaginary ”i” alone; ”i” is the key that unlocks everything. The power of the key can be any imaginary number at all; ”i” is why the result of every imaginary power of ”i” becomes real. 

The explanation calms me; but it seems somehow incomplete; it’s missing something; in my gut I feel like it can’t be entirely right, though it purports to persuade what the math insists is truth. 

Believe, for now at least, and move on.

Billy Lee

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WHAT IS e exp (-i π) ?

What is e^{-i\pi} ?

I posted a long answer on Quora.com where it sort of didn’t do well.

Answers given by others were much shorter but they seemed, at least to me, to lack geometric insights. After two days my answer was ranked as the most read, but for some reason no one upvoted it. It did receive a few positive replies though.

I can’t help but believe that there must be nerds in cyberspace who might enjoy my answer. Why not post it on my blog? Maybe someday one of my grandkids will get interested in math and read it.

Who knows?

Anyway, below is a pic and a working GIF, which should help folks understand better. Anyone who doesn’t understand something can always click on a link for more information. 

Here is the drawing I added and the answer:

This diagram is excellent but contains a mystery point not on the unit circle — {i^i}. The point is shown at .2078… on the real number line.  An imaginary number raised to the power of an imaginary number yields a result that is a real number. How can that be? It’s something to ponder; something to think about. The Editorial Board 

What is e^{-i\pi} ?

The expression evaluates to minus one; the answer is (-1). Why?

Numbers like these are called complex numbers. They are two-dimensional numbers that can be drawn on graph-paper instead of on a one-dimensional number line, like the counting numbers. They are used to analyze wave functions — i.e. phenomenon that are repetitive — like alternating current in the field of electrical engineering, for example.

A simplified explanation of {i^i} starts at 02:30.

“e” is a number that cannot be written as a fraction (or a ratio of whole numbers). It is an irrational number (like π, for instance). It can be approximated by adding up an arbitrary number of terms in a certain infinite series to reach whatever level of precision one wants. To work with “e” in practical problems, it must be rounded off to some convenient number of decimal places.

Punch “e” into a calculator and it returns the value 2.7182…. The beauty of working with “e” is that derivatives and integrals of functions based on exponential powers of “e” are easy to calculate. Both the integral and the derivative of ex  is ex  — a happy circumstance that makes the number “e” unusually curious and extraordinarily useful in every discipline where calculus is necessary for analysis. 

What is “e” raised to the power of (-iπ) ?

A wonderful feature of the mathematics of complex numbers is that all the values of expressions that involve the number “e” raised to the power of “i” times anything lie on the edge (or perimeter) of a circle of radius 1. This feature makes understanding the expressions easy.

I should mention that any point in the complex plane can be reached by adding a number in front of e^{i\theta} to stretch or shrink the unit circle of values. We aren’t going to go there. In this essay “e” is always preceded by the number “one“, which by convention is never shown.

The number next to the letter “i” is simply the angle in radians where the answer lies on the circle. What is a radian?  It’s the radius of the circle, of course, which in a unit circle is always “one”, right? 

Wrap that distance around the circle starting at the right and working counter-clockwise to the left. Draw a line from the center of the circle at the angle (the number of radius pieces) specified in the exponent of “e” and it will intersect the circle at the value of the expression. What could be easier?

For the particular question we are struggling to answer, the number in the exponent next to “i” is (-π), correct?

“π radians” is 3.14159… radius pieces — or 180° — right? The minus sign is simply a direction indicator that in this case tells us to move clockwise around the unit circle — instead of counter-clockwise were the sign positive.

After drawing a unit circle on graph paper, place your pencil at (1 + 0i)—located at zero radians (or zero degrees) — and trace 180° clockwise around the circle. Remember that the circle’s radius is one and its center is located at zero, which in two dimensional, complex space is (0 + 0i). You will end up at the value (-1 + 0i) on the opposite side of the circle, which is the answer, by the way.

[Trace the diagram several paragraphs above with your finger if you don’t have graph paper and a pencil. No worries.]

Notice that +π radians takes you to the same place as -π radians, right?  Counter clockwise or clockwise, the value you will land on is (-1 + 0i), which is -1. The answer is minus one.

Imagine that the number next to “i” is (π/2) radians (1.57… radius pieces). That’s 90°, agreed? The sign is positive, so trace the circle 90° counter-clockwise. You end at (0 + i), which is straight up. “i” in this case is a distance of one unit upward from the horizontal number line, so write the number as (0 + i) — zero distance in the horizontal direction and “plus one” distance in the “i” (or vertical) direction.

So, the “i” in the exponent of “e” says to “look here” to find the angle where the value of the answer lies on the unit circle; on the other hand, the “i” in the rectangular coordinates of a two-dimensional number like (0 + i) says “look here” to find the vertical distance above or below the horizontal number line.

When evaluating “e” raised to the power of “i” times anything, the angle next to “i”—call it “θ”—can be transformed into rectangular coordinates by using this expression: [cos(θ) + i sin(θ)].

For example: say that the exponent of “e” is i(π/3).  (π/3) radians (1.047… radius pieces) wraps around the circumference to 60°, right? The cosine of 60° is 0.5 and the sine of 60° is .866….

So the value of “e” raised to the power of i(π/3) is by substitution (0.5 + .866… i ). It is a two-dimensional number. And it lies on the unit circle.

The bigger the exponent on “e” the more times someone will have to trace around the circle to land at the answer. But they never leave the circle. The result is always found on the circle between 0 and 2π radians (or 0° and 360°) no matter how large the exponent.

It’s why these expressions involving “e” and “i” are ideal for working with repetitive, sinusoidal (wave-like) phenomenon.

In this essay Billy Lee uses θ in place of the Greek letter φ shown in this GIF.  Remember that ”r” equals ”one” in a unit circle, so it’s typically not shown. The Editorial Board

In case some readers are still wondering about what radians are, let’s review:

A radian is the radius of a circle, which can be lifted and bent to fit perfectly on the edge of the circle. It takes a little more than three radius pieces (3.14159… to be more precise) to wrap from zero degrees to half-way around any circle of any size. This number — 3.14159… — is the number called “π”.   2π radians are a little bit more than six-and-a-quarter radians (radius pieces), which will completely span the perimeter (or circumference) of a circle.

A radian is about 57.3° of arc. Multiply 3.1416 by 57.3° to see how close to 180° it is. I get 180.01… . The result is really close to 180° considering that both numbers are irrational and rounded off to only a few decimal places.

One of the rules of working with complex numbers is this: multiplying any number by “i” rotates that number by 90°. The number “i” is always located at 90° on the unit circle by definition, right? By the rule, multiplying “i” by “i” rotates it another 90° counter-clockwise, which moves it to 180° on the circle.

180° on the unit circle is the point (-1 + 0i), which is minus one, right?

So yes, absolutely, “i” times “i” is equal to -1.  It follows that the square root of minus one must be “i”. Thought of in this way, the square root of a minus one isn’t mysterious.

It is helpful to think of complex numbers as two dimensional numbers with real and imaginary components. There is nothing imaginary, though, about the vertical component of a two-dimensional number.

The people who came up with these numbers thought they were imagining things. The idea that two-dimensional numbers can exist on a plane was too radical at the time for anyone to believe.  Numbers, they believed, only existed on a one-dimensional number line of one dimension and no place else.

Of course they were mistaken.  Numbers can live in two, three, or even more dimensions. They can be as multi-dimensional as needed to solve whatever the mysteries of mathematical analysis might require.

Click the link, “What is Math?” for another explanation.

Billy Lee

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ILLUSIONS

I read a report about the lunatic in Las Vegas; he was a multi-millionaire who owned nearly 50 high-powered guns plus a lot of other scary stuff. His dad graced the FBI’s ten most wanted list—ten years a fugitive.

These guns, which many civilians now own, were designed to shatter the bones and scramble the internal organs of victims—violating the spirit of international norms, agreements, and treaties agreed to by all countries before and after World War II. 

These Geneva Convention prohibitions (and others) were crafted to make hollow-point style ammunition illegal. To evade these restrictions, US gun-makers designed weapons to fire high-velocity bullets that tumbled—to inflict crippling injuries with more ferocity than the banned hollow-points.

Billy Lee has never visited Las Vegas nor does he plan to. It was built as a stopover during WWII for GIs on route to the west coast, where they boarded ships to fight their way to Japan. According to legend, criminal syndicates built “Sin City”. People say the bad guys moved out. The president owns a hotel there. The Editorial Board 

During combat officer training in the Vietnam era, I fired one of these weapons (an M16 rifle) at a bucket of water. The bullet went in clean but blew out the back. Shards of metal and water flew everywhere. The container exploded, basically.

Every massacre involving these weapons reaps what we sowed. The USA violated both the spirit of the international consensus and basic common sense nearly six decades ago. Our country put the lethality of heavy weapons into rifles that handled like toys. Weapon manufacturers created bone smashing ammo.

People shot by these guns don’t recover. Survivors carry their wounds to the grave.

Modern high-tech guns and ammunition are inhumane, lethal, and crippling. The military shouldn’t use them; neither should civilians; especially civilians who aren’t properly trained or supervised; some civilian gun owners have an unhealthy obsession with these kill-sticks; some are lunatics.

Flags are set at half-mast across the USA to honor the fallen in the Las Vegas attack. This pic was taken by Billy Lee in a Belle Tire parking lot. The Editorial Board

As for hollow-point ammo, police inside the United States ignore the international prohibitions. Many agencies use black-talon style hollow-points to reduce the penetrating power of tumbling slugs (that can kill bystanders) while dramatically increasing debilitation to the person shot.

Misunderstanding of the second amendment has put four million tumbling-slug killing tools into the hands of ordinary people who have no accountability and who are in some cases insane.

After all these years only God knows where these weapons are. What could possibly go wrong?

Despite being a pontificator who by definition lacks expertise, I don’t generally speculate about things I know nothing about. I really don’t. I try to not think about the hundreds of mass shootings that have taken place during past decades, because it is depressing and demoralizing (and scary) to believe that going to public venues is dangerous.

It’s hard to say who is worse off during these mass-casualty events, the dead or the wounded or those who witness the violence up close and personal. So many people are traumatized for no good reason. I suspect that even viewers of television coverage get a sick feeling in their stomachs when these horrors occur. I know I do.

The recent attack in Las Vegas was strange. A number of active duty U.S. military personnel — recently returned from Afghanistan, plus their wives and partners — attended the Route 91 Harvest country music festival.  Daesh — called ISIS or ISIL in the U.S. — claimed that the shooter, Stephen Paddock, was a contractor who worked for them. He was a kind of highly-paid sleeper mercenary; he did what he was told when his time came apparently. 

His handlers—who may have helped to set up the killing zone—occupied the hotel suite alongside him during the attack. They might have killed him to make it look like suicide and exited the building via a service elevator disguised as hotel workers—maybe. It’s possible.

Another disturbing possibility is that they let the shooter live expecting that he would escape and join them in another attack. He might have been disabled by gas — perhaps injected under the door by police. If so, he is now in custody.

Anything is possible, when conspiracy theories start percolating. The shooter might have been a kind of patsy, like Lee Harvey Oswald claimed to be (for those readers who have convinced themselves that Oswald did not conspire alone).

If the Las Vegas massacre was an ISIS attack (as ISIS claims) it’s not likely that the United States will give the group the satisfaction of an acknowledgment. Disclosure would undermine confidence in law enforcement’s capability to protect the public from terrorist attacks. [Editors Note: As of January 2018 the number of casualties stands at 58 killed and over 500 wounded.]

Agencies will instead work behind the scenes to uncover, debrief, and terminate with extreme prejudice all the players. Justice will be served. It will be methodical and relentless. It could take time — months or even years.

This vehicle is being tested for battle-worthiness.

Most Americans never fully understood the Iraq War that spawned ISIS and filled its ranks with experienced and ruthless fighters; nor have they grasped how powerful is Saddam Hussein’s family, his friends, and his army — once one of the world’s largest and most formidable. I’ve heard people say some dumb things about what all that fighting in the Middle East was about those many years ago.

Before the Iraq War Saddam’s family was one of the world’s wealthiest; they owned a lot of stuff — popular media and franchises, magazines and food networks, even sophisticated enterprises, some with international reach.

Izzat Ibrahim al-Douri, second in command in Saddam’s Iraq and founder of ISIS.

Saddam’s closest advisor and deputy — who President Bush called the King of Clubs — was never apprehended. His name is Izzat Ibrahim al-Douri. Some believe he is the mastermind behind the formation of ISIS.

A few years ago reports appeared in the press that Ibrahim died during an attack on his security detail. But no one saw him die. No one attended his funeral. His body (including DNA evidence of his death) is missing.

Some say that ISIS agents planted these stories of al-Douri’s demise. Ibrahim faked his own death. Saddam’s allies own a significant chunk of the world’s media.

Who knows?

Anyway, my understanding is that ISIS was formed by members of Saddam’s family and loyal remnants of his army who are trying to take back what they lost during the Bush presidency. That’s how many see it, including reporters at Haaretz, an Israeli news organization.

When the USA conquered Iraq, Saddam’s army (and leadership) melted away, but they had billions of dollars stashed in banks, the walls of buildings, and in holes underground. They have not been afraid to spend it.

ISIS travels first class. It has the best of everything, including trucks, cars, weapons, and drones. It captured an astounding amount of USA war fighting machinery in fights with Iraqi Shiites (ISIS is Sunni) after the USA exited Iraq.

Some of the captured equipment included MRAPs (see earlier illustration). It seems unbelievable, but its true. (Note from the Editorial Board: Billy Lee helped design the run-flat wheel that permitted fighting vehicles in the Gulf wars to stay mobile after their tires were shredded, punctured, or shot through.)

Billy Lee helped develop for Chrysler the run-flat technology used by combat vehicles like this one. The Editorial Board

The way I understand the conflict, the Sunnis of Iraq could reasonably be compared in some ways to the southern whites who served the confederacy during America’s Civil War. The Shiites in this analogy would be the negro slaves.

Think about it. After the Iraq War, the downtrodden Shiites (with help from the USA) took control of Iraq from the entitled Sunnis, much like blacks took control of the southern states after the Civil War with help from the north’s military occupation (called Reconstruction).

Southern whites eventually wrested control from their former slaves, but it took twenty years of terrorism by the Ku Klux Klan to make it happen. A similar dynamic is underway in the Middle East today, it seems. The Sunnis are reestablishing their control through terror, for the most part.

Eventually the Sunnis might win — like the Ku Klux Klan won their fight. In the meantime, a lot of innocents are getting hurt and worse.

The territory that ISIS controls in the Middle East is vast. It is comparable in size to the country of Egypt. Yes, USA backed forces have recaptured some ISIS cities and towns in recent months, but the fights are costly in lives and treasure; the victories do not seem to have turned the tide of the war, at least not yet.

Every time the USA hits ISIS hard, as it has in recent months, ISIS seems to find a way to hit back. In the meantime, we destroy towns and cities in order to “save” them. The cities aren’t coming back — not for a long time. 

It’s difficult (some would say, impossible) to defeat a determined foe in their own country. We learned this lesson in Korea, Vietnam, and Cuba. The fight against ISIS is going to be a long one. Our country might go bankrupt before victory comes. It’s possible.

No one wants to admit it, but the USA is teetering on the edge of financial collapse right now, as this essay is being written. Our last president, Barack Hussein Obama, (now called Barry Obama by friends) worked out some fixes to stave off an economic crisis, but the current president seems hell-bent on bringing our country to ruin.

The president and his wealthy friends seem to want to eliminate the estate tax so that they can leave thousands of millions of dollars (they call them billions) to their crazy kids who can flee with our national treasure to whatever solvent country will welcome them after the dust settles.

I’m told that the people around the president are Christian patriots who are determined to prevent really big screw-ups from being implemented. The country is safe.

More importantly, Christians don’t do genocides. They don’t do mass killings of civilians like that lunatic in Las Vegas. Yes, Hitler said he was Christian, but history has judged him differently.

The Christian patriots in the White House won’t permit the president to first-strike North Korea, for example, with nuclear weapons. They won’t kill ten to twenty million people over a few missile tests, which many countries conduct without threat of retaliation, including the United States.

Atomic bomb test at Bikini Atoll. Hydrogen bombs are much larger.

The USA dropped dozens of hydrogen bombs in the Pacific Ocean, remember, and no one did anything about it. Countries around the world have conducted 520 atmospheric nuclear explosions and 1,352 underground detonations. We aren’t going to exterminate an entire country like North Korea over a few low-level, underground atomic tests. No rational, humanity-loving civilization would even contemplate such an atrocity.

So far, so good, I guess. Yes, we are in good hands, I’m told. No one is insane — not here; not there.

In time, no one will remember the killings in Las Vegas, anyway. No illusions. Everyone knows the truth when they see it, right?

When given a choice, decent people do what’s right, don’t they? Of course, they do. They show mercy; it’s what the Bible says God wants.

Billy Lee

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BEGGING FOR IT

Nikki Haley, UN Ambassador, told the world that North Korea is begging for war. Her statement reminds me of the rape defense that bad men used in decades past at trial before court—she was begging for it, they would always insist.

The victim—always a woman—exercised her constitutional right to dress any way she chose. A prowling lunatic observed her and believed instead that she was advertising a willingness to copulate with a grease-ball, so he attacked.

In today’s United States the grease-ball defense no longer works. Women (and countries) have rights. The courts protect them, most of the time. When they don’t, women have recourse to civil suits.

Nikki Haley is the daughter of Sikh immigrants from Punjab, India. Her mother once owned a clothing company, Exotica International, where Nikki worked as a bookkeeper starting at age 12. She served as governor of South Carolina until she became UN Ambassador in the Trump Administration. Note from the Editorial Board: Haley resigned her UN post on December 31, 2018.

All countries (and women) have the right to defend themselves. No one should have to say it. But the United States has mucked-up the waters of international affairs by waging preemptive wars against communist and socialist countries for the past seventy-five years. It seems like these wars will never end.

The policy of the United States is to undermine any country that espouses communism; it attacks using whatever methods are devised and recommended by the brilliant, depraved minds of our intelligence community. Despite all the evidence, despite all the complaints lodged against the United States before international courts and the United Nations, most Americans refuse to believe it.

The USA birthed its war-policy against collectivism in the early 1900s when the first communist revolutions undermined the noble classes of eastern Europe. The well-connected saw what happened in Russia and Hungary and decided to do something about it. Writers like Aldous Huxley (Brave New World), George Orwell (1984), and Ayn Rand (We the Living) wrote short, easy to read tracts that portrayed life under communism as a personality-destroying nightmare. The public ate it up. 

After World War II, the job of suppression was handed over to our newly formed intelligence agencies who gave the fight its name: the Cold War. With the recent CIA assisted transformations of the Soviet Union and Red China into oligarchies (marking a putative end to the biggest conflicts) other terms have emerged from inside the intelligence community to describe how the USA is continuing to war against what few holdouts remain. The current buzz phrase is strategic strangulation.

Billionaires don’t look at the world in the same way as the disenfranchised and exploited. In the neglected and overlooked far reaches of the world where billionaires don’t live, common people sometimes try to organize themselves by producing wealth cooperatively and sharing it as best they can. Once they attract the attention of the powerful, the powerful send in missionaries and agricultural-aid workers to undermine and disenfranchise their leaders.

If God and food don’t work, they send in assassins. It’s true. Since WWII, the United States has gone to war with one out of four countries. It has overthrown (or co-opted) many states including Iran, Guatemala, Chile, Poland, Iraq, Afghanistan and dozens more.

Eventually, the USA sends in Coca-Cola and other first-line-of-attack companies to overwhelm the nascent economies of the sharers and cooperators. The wealthy move in to buy the arable land and lock up the country; billionaire power-brokers make it their business to bury every way of governance that might threaten their right to administer an economic empire built for one purpose only: to empower their own families and the families of their friends to rule unimpeded into perpetuity.

This billboard appears all over the island of Cuba. It says: BLOCKADE: THE LONGEST GENOCIDE IN HISTORY. It shows a Klansman’s noose strangling the island. Inside the USA intelligence community, it’s called strategic strangulation. Our leaders have enforced the embargo of Cuba for 55 years. Odds are it will never end.

The USA is good at undermining socialist countries. It’s why socialism doesn’t do well. When a communist country like Cuba manages to hold on somehow, they are tormented by embargos, infiltration by agents, sabotage, disinformation, slander, assassinations and so on. Radio Free Whatever is beamed into the targeted country to undermine morale and brainwash listeners with the most seductive psychological warfare techniques in our arsenal. It’s why communist countries seal their borders and jam our broadcasters.

The United States makes sure that no administrator can relax; none can kick back without constantly looking over their shoulder and keeping guard. As long as the privately-owned United States exists, countries of the opposite sex — that is, socialist countries — are going to get blamed for however the rich and powerful choose to hurt them. The better these socialist countries do for their people the bigger the target they become. Their success is their mini-skirt; their low-cut blouse. Obviously, they are begging for war as Madame Haley so eloquently put it.

Blaming the victim is as old as Cain and Abel. It’s as old as history. The bullies win. They put on coats and tails, send their kids to exclusive schools, learn to speak with a different accent, and set themselves apart. They tell wonderful stories about themselves on TV, movies, radio, books, magazines, and the internet until people start to actually believe deep in their hearts that wealthy people are more wonderfully made than themselves or their kids.

The poor and disadvantaged admire their employers, because no one is left to help them understand that they are slaves — fools before a noble class that thinks of them as having no more value than farm animals. They live in the cow-pastures outside the gates of the wealthy. A few unlucky cows get a glimpse from time to time of what has been stolen from them; most of the time they are too incredulous before the view to believe their own eyes.

Today, billionaires are planning as they have for the past seventy years to obliterate the North Koreans. Tens of millions of Koreans will die in the first hours. The North Koreans and their Chinese allies were the first countries to challenge America after WWII. Apparently, our arsenal of atomic weapons didn’t intimidate them. The USA has a long memory.

If we use nuclear weapons in a first strike, no one will live on the peninsula again. That this threat has been floated by our new president (fire and fury like the world has never known) makes it easy to understand what the USA is all about.

Despite denials in the press, America may in fact have neutron bombs in its arsenal which kill people but leave infrastructure intact outside the immediate impact zone. Only bunkers constructed from heavy concrete impregnated with barium sulfate can shield against both neutrons and the gamma ray by-products of collisions between neutrons and the proton-rich substrate of heavy concrete.

High level command bunkers are likely to be built with these materials. Otherwise elements like lead and heavy metals are the barriers of choice against most frequencies of nuclear radiation. But heavy metals tend to be transparent and ineffective against neutron bombardment.

Neutron bombs emit little residual radiation so they don’t contaminate the attack zone. An occupying force can be inserted with little risk of radiation poisoning. It’s possible the USA will use neutron bombs against civilians for the first time if the Koreans continue begging our country to kill them. A follow-on insertion of Marine and Airborne divisions would locate and destroy any surviving command bunkers.

The United States always seems to be first to try diabolical things. We used anthrax against Chinese troops during the Korean war back in the 1950s. Everyone knows about it but us of course. We have short memories when it comes to remembering our own sins like the genocidal Vietnam War and other cruelties such as the war against Iraq and the destabilization of the Middle East.

And no one seems to remember that it was a CIA agent (yes, he was disgruntled and no longer employed, the agency insists) who blew up the first commercial airliner and wiped out the Cuban Olympic fencing team. It was an act of terrorism that didn’t happen when you ask most Americans. Lee Harvey Oswald was also disgruntled and no longer employed, come to think of it.

Speaking of begging for war, is anyone out there begging for peace?  The USA killed two million North Koreans during aerial bombardments approved by President Truman. What’s stopping us from doing it again?

During the first days of the Korean War, the South rounded up 100,000 of its own citizens and summarily executed them, believing that each of their victims might be a leftist sympathizer of the North. The Korean War was an atrocity — which I won’t devote blog space to explain or explore. Click on links, those who want to know more.

South Korea from North Korea’s point of view. A massive China has their back. South Korea’s capital is close to their border — a tempting target. (Click on map to produce full-size view in a new window.)

The point is, we aren’t listening to the North or South Koreans. Most want the peninsula unified. We don’t. Most folks want to live in a safe and fair place where billionaires don’t reduce the average person to poverty. South Korea doesn’t want us to sacrifice millions of its own citizens in a military strike against the North — possibly triggering a nuclear conflagration — to make a point. It’s simple, really.

The only sensible solution is to leave Korea and to assist reunification as best as we can if the sides invite us. We aren’t leaving obviously, because our policy is to topple socialism wherever it takes root.

In the past the USA demonized so many of its enemies that it has become more confusing for most folks to know who the bad guys are this time around. Immigrants to the Americas once labeled native Americans with the pejorative, savages. We exterminated their food supply (buffalos), then their men, women and children.

Right now, no one knows who to trust. With our new president we will never know what the facts are. We will never know for sure who is begging for war, us or them.

Billy Lee

From the EDITORIAL BOARD:

Billy Lee supports private and public ownership of property (a mixed economy) but advocates for internationally enforced limits on personal incomes and estate sizes to reduce the temptations that drive the wealthy to burn down democracy on altars of greed. Read Billy Lee’s essay Capitalism and Income Inequality.

Billy Lee opposes war on humanitarian grounds. During war people get killed and are maimed in huge numbers. It sometimes takes generations for both winners and losers to emerge from the trauma and horrors.

Billy Lee supports the United States during war but in peacetime reserves his constitutional right to pontificate freely.

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