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



 

EYE TO EYE

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

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 e 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