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 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 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 . 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) = cos (ln i) + i sin (ln i)
By definition: = i
Also: = ln i
Therefore: ln i = i
It should now be obvious to anyone who has taken a basic course in complex variables that multiplying i 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): = cos () + i sin ()
By half angle formulas:
Convert 2nd term i to :
(2) Simplify the 2nd term:
Euler’s cosine identity is: cos θ =
Therefore: cos (iπ) =
(3) Simplifying: cos (iπ) =
Substitute line (3) into line (2) and simplify:
Now it’s just a matter of pulling out an old calculator and punching the keys.
= .043214; = 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 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 — — which equals (1.57078… i ). The ln of — — 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.
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 — . 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 ?
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 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 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.
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 theEditorial 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.
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.
Russian hypersonic ship killing missiles fired from submarines and jets pose an existential threat to the US Navy Fleet. From AP Photo.
UPDATE BY THE EDITORS: 5 AUGUST 2021; The Associated Press reported yesterday the following: A prospective Russian hypersonic missile has been successfully test-fired from a nuclear submarine for the first time, the military said Monday. The Russian Defense Ministry said that the Severodvinsk submarine performed two launches of the Zircon cruise missile at mock targets in the Barents Sea.
Editors Note: The January 16, 2018 edition of the New York Times reports that five Naval officers are being charged with negligent homicide related to the incidents described in this essay where seventeen sailors lost their lives, others were injured, and ships were damaged. The decision to take these officers to trial or court-martial is imminent.
The United States Navy insists that it has 277 ships on active-duty. About 132 are combat surface vessels; 75 or so are submarines. About 70 are logistical craft designed to supply the fleet.
Do the math. 132 surface ships patrol the oceans. Yes, the Navy says they have a fleet of 160 or so non-commissioned ships held in reserve, but they are unavailable and ineffective during first strike scenarios.
The New York Times agrees with the Navy. Everyone agrees — we have 277 ships. I hope the Navy and the New York Times are lying, because if they aren’t, we are in big trouble. We don’t have enough boats.
132 surface ships can’t control the Great Lakes, let alone the world’s seven oceans. Submarines, everyone knows, are almost useless except when used for nuclear deterrence.
The Navy’s Seventh Fleet is headquartered in Yokosuka, Japan. The fleet is responsible to cover 48 million square miles — from Japan to South Korea to Singapore, unless the faraway reaches of the South China Sea are included; then the square miles are too confusing for anyone to compute.
China claims the whole of the South China Sea as its sovereign territory, including all reefs, atolls, and islands.
How many ships bear the awesome duty to keep the sea-lanes open and safe from pirates and hostile powers like North Korea? The Navy says, 70. The USA deploys one aircraft carrier and 69 ships.
Some news outlets have reported that an additional carrier group has been sent into the Sea of Japan to augment the current force configuration. A typical carrier strike group consists of eleven vessels, two of which are submarines. So, the total on the date of this essay might be as high as 81.
Sorry, but someone is ordering our sailors to do an impossible job. The job is too big, the resources are too thin, and guess what?
A flotilla of 81 vessels scurrying about the South China Sea to keep a lid on China, which is seizing islands that belong to Vietnam, Taiwan, and the Philippines as they build and fortify new ones wherever they want is more than enough problems to exhaust any navy. People get tired. Accidents happen.
We have 70 ships in theater, the Navy says. We probably have 81. What does the other side have?
Well, we don’t really know. They lie. So do we. But we and they both watch; and we and they both spy and calculate.
Hillary Clinton — once upon a time (anyone remember?) she had the security-clearance to know — during a 2016 presidential debate let slip that Russian drone submarines are patrolling our coasts. These are cheap subs that sail apparently with no crews on board.
She said it once. Her assertion was never repeated in the press or public media. Everyone pretended they didn’t hear, for good reason. The number and types of ships in the Russian and Chinese fleets that are arrayed against our tiny arsenal of boats are state secrets. It’s all classified — out of reach of everyone except those with a clearance and a need to know.
Chinese frigates like this one often stalk US ships in the South China Sea.
It seems clear to more than a few casual observers of Chinese shipping that the Chinese are building the most high-tech navy the world has ever seen. They have been building it for a few decades now.
The Chinese have practiced their sea-going skills in coordination with the Russian navy since 2012. Last year the Russians and Chinese held joint naval exercises in the South China Sea, of all places. Joint land-based military exercises started in 2007.
China is selling its naval technologies and hardware to smaller countries that don’t normally threaten us. Thailand is buying Chinese subs. With military hi-tech weapons spread among a dozen or more countries in secret alliances with China… well… if it’s happening more than we know, does anyone think it’s good for our side?
But really, what would any reasonable person suspect are the forces arrayed against us? Look around. Hundreds-of-thousands of Russians live on the island of Cuba ninety miles from the United States.
Upscale area southwest of the airport in Havana, Cuba. (From Google Earth. Street View not available.)
Go on Google Earth and look at the Cuban neighborhoods. Some nice ones have Russian street names. It’s true. The Russians have a number of wonderfully designed, modern military bases for both subs and ships; and — oh yeah — they have fighter jets and missiles, as well.
Does anyone disagree? Go look.
Am I trying to scare people?
Doesn’t the public have enough to fear? Isn’t terrorism, immigration, climate change, distant war, disease, and precarious health care that could collapse now that the GOP is in charge enough to worry about? Of course it is.
Secretary of State Rex Tillerson helped Boris Yeltsin and Vladimir Putin build Russia’s oil and gas infrastructure. It is the world’s best. Note from the Editors:Tillerson resigned his post on March 31, 2018.
Besides, our country has thousands of nuclear weapons buried hundreds of feet below the cornfields and deserts of the heartland to extinguish any threats, should we lose our Navy. Until the missiles rot and their plutonium payloads leech into our soils, why worry?
Everyone should sleep well at night, right? Who wants to alarm anyone? I don’t.
Secretary of State Tillerson said we should sleep well, so why not? He knows all about the Russians, having helped them build their oil industry over many decades.
Depending on when anyone takes its measure, Russia’s energy industry is the world’s largest and most productive — bigger than Saudi Arabia’s. People don’t believe it, but it’s true.
Russia is the world’s biggest oil and natural gas producer and exporter. Secretary Tillerson must know what he’s talking about, right?
Well, here is some stuff that is not so comforting. It might scare some people. Between 1975 and 2016 (41 years) our Navy experienced nine accidents, mostly between our own ships. Only two accidents involved the boats of foreign countries. That’s not bad. That’s not the scary part. But hear me out.
The Ehime Maru was on a 74-day voyage to train high school students to become commercial fishermen when it was struck on 9 February 2001 by a US submarine. It sank. Of the 35 on board, nine died, including four teenagers.
In 2001 a Japanese fishing-training boat, the Ehime Maru, with thirty-five Japanese citizens aboard, was obliterated near the Hawaiian island of Oahu when the commanding officer of one of our attack submarines allegedly hot-dogged the craft for civilian joy-riders.
Our new president, George W. Bush, went on national TV to apologize to the Japanese, and the United States paid huge fines and compensation to the Japanese government and the grieving families of the nine who died, which included four high school students.
In 2004, the aircraft carrier U.S.S. John F. Kennedy ran over an Arab sailboat in the Persian Gulf. 15 people died, but the Navy didn’t identify who they were it seems, and no one was compensated, as far as I know. Two jet fighters parked on deck were damaged. The Navy relieved the commanding officer.
USS Belknap, guided-missile cruiser, destroyed in 1975 near Sicily.
The most serious accident was in 1975 when the same U.S.S. John F. Kennedy hit one of our own guided-missile cruisers, the USSBelknap, off the island of Sicily. The Belknap was completely destroyed; seven sailors died.
A fire burned on the Belknap for twenty hours just a few yards from the magazine where Terrier surface-to-air missiles were stored. The ship was constructed with aluminum, which caught fire. The entire above deck structure melted. It took nearly five years to reconstruct the ruined cruiser. In 1995, the Navy struck it from the Naval Registry and began using it for target practice. They sunk it during a live-fire exercise in 1998.
A year after the Belknap accident, the USS John F. Kennedy collided with another ship, this time the aging USS Bordelon destroyer during a refueling. The Navy struck the ship from its registry and sold it to Iran for parts in 1977. No one died.
So, during the forty-one years between 1975 and 2016, the US Navy had nine peacetime accidents, seven of which were friendly-fire and self-inflicted. 24 foreign nationals died; 7 U.S. sailors; 1 U.S. civilian. Ship losses: one cruiser and one obsolete destroyer. Maybe other losses occurred. I haven’t heard about them, if there were any.
The USS John S. McCain collided with a Liberian oil tanker, the Alnic MC, on August 20, 2017 in an early morning incident that killed ten sailors. The crash took the destroyer out of action for at least one year.
And now comes the scary part; hold onto your pants: In the seven months since the inauguration of our comb-over commander-in-chief (and keeper of nuclear codes), the U.S. Navy has suffered four major accidents, which killed 17 sailors and injured scores more.
It’s lost two of its most powerful missile-guided destroyers — the U.S.S. Fitzgerald and the U.S.S. John S. McCain. It might be years before they are back in service. Readers can read about the fates of the USS Antietam and the USS Lake Chaplain in the links below.
At least two dozen sailors and officers have been disciplined, including a Vice-Admiral, a Commander, and a Lieutenant Commander. Admiral John Richardson, chief of Naval operations, has ordered an “operational pause” to all fleet commanders. He’s ordered a months-long review of protocols, because, he says, “there’s something out there that we’re not getting at.”
All this commotion is happening during a time when we’re planning to conduct war games against North Korea and are daily challenging the Chinese in the South China Sea.
Can I put things into perspective? If the accident rate of the past seven months was applied to the past forty-one years, the U.S. Navy would be short another 85 ships and 800 sailors. Thousands more young men and women would be maimed and wounded, and 250 promising Naval careers would be wrecked.
In peacetime, essentially, the U.S. Navy might have lost one-third of its fleet and some of its best sailors and officers at a cost of hundreds of billions of dollars.
The USS Antietam, a guided missile cruiser, grounded itself in Tokyo Bay January 31, 2017. It released over one thousand gallons of toxic hydraulic oil into the bay and damaged both of its propellers and propeller hubs. Repair costs will exceed $4 million.
Here are a few of the headlines from our country’s newspaper of record, the New York Times. All headlines are from articles written in 2017. I’ve read every single one of them. Readers can access their content by clicking on the links. The remainder of my essay is written below these headlines:
*Some readers may have noticed that the last headline seems to have no connection whatsoever to this essay. But they would be wrong.
Recall that the battleship USSIndianapolis was the fiercest war machine we had during World War II in the Pacific. We used that ship to deliver the atomic bomb Little Boy (dropped on Hiroshima) to Tinian Island in the western Pacific Ocean sometime during July, 1945. It would be assembled and delivered to the Japanese people with terrifying effect on August 15.
From Tinian the Navy ordered the Indianapolis to advance to Leyte Island in the Philippines to prepare for an all-out assault and invasion of Japan scheduled to follow the atomic blasts that were soon to occur.
On July 30 the lumbering battleship encountered a Japanese submarine which delivered six torpedoes in the wee hours of the night. The sub commander later said that the clouds parted, which permitted the Indianapolis to be silhouetted by moon-light. It made targeting easy. Two torpedo struck the Indianapolis.
The ammunition on board caught fire and blew-off the front quarter. It took twelve minutes for the battleship to sink below the surface. The ship sucked four hundred men to the bottom and left behind an oil slick that would sicken and blind many of the nearly one-thousand sailors and marines who survived to face death by dehydration, drowning, and sharks.
The USS Indianapolis delivered the atomic bomb to the Air Force in the Pacific before being sunk by a Japanese submarine. The ordeal took the lives of nearly a thousand men during five days in the open sea. Read Devil’s Voyage by Jack L Chalker.
The Navy didn’t notice that their prized battleship was missing. After five days of vomiting, diarrhea, hallucinations, and shark attacks, three hundred men were still alive (some in lifeboats, including the Commanding Officer) when an aircraft on an unrelated mission saw something suspicious and flew down to take a closer look.
Twenty-two men who were pulled from the water remain alive today. The Navy court-martialed the commander, Captain McVay, and convicted him for not zig-zagging as he sailed. The Japanese sub-commander testified that zig-zagging would not have mattered. The Indianapolis was going to the floor of the ocean, in any event, he insisted. Nothing could have stopped what happened.
Losing a ship, even in war, is a big deal in the Navy. It’s not something that anyone takes lightly, even when there are extenuating circumstances and good reasons for failure. Captain McVay committed suicide in 1968—clutching a toy sailor in one hand and his service revolver in the other.
The Navy has a history of not being able to keep track of its ships. The earth’s oceans are vast, and we don’t have that many boats on them. Hiding ships from our enemies means we sometimes hide them from ourselves.
Civilian boats are another matter. Merchant fleets deploy 51,405 ships on our oceans. Most of them are bigger and longer and heavier than our 277 Navy ships. Almost all run on auto-pilot most of the time, especially at night when the crews sleep. If the computer directs the tanker to ram a boat like the Fitzgerald, that’s what is going to happen. In collisions, chances are Navy ships will lose.
Collision avoidance should be easy. Crews need only have situational awareness and the ability to steer the boat. The problem is that to perform these tasks crews rely on a complicated matrix of technologies that always seem to fail in critical situations like combat or rule violations by other boats.
These technologies should be used to confirm human observation and decisions; instead sailors confirm what the technology tells them, but only when something goes wrong, which is almost always too late. An alarm sounds and a glance at a computer screen shows that a tanker is 500 meters to starboard, so a crew member looks out a window to see if it’s there. No! That’s bassackwards and will get someone killed.
Officers might better demonstrate proficiency in the absence of high-tech aids for situational awareness and steering, then add high-tech proficiencies one skill-set at a time. Maybe they wear merit badges to enable COs to tell at a glance who can handle hydraulic controls and who is good at computer-aided navigation, for example.
Every officer doesn’t have to master every skill-set, and the least skilled officer should be able to turn off the high-tech systems they haven’t mastered in order to steer the boat and stop it using the skills they do have, when necessary.
Laser distance finders (like those used by golfers) and wide-field-of-view night vision binoculars should be standard issue. A half dozen or more sailors should be stationed around the perimeter of every boat and be required to report what they see or don’t see every five minutes or so. No snoozing!
Mischief Reef is the site of a Chinese airstrip and military installation built on a contested atoll in the Philippines.
Anyway, one thing about the four accidents this year (January 31, the USS Antietam; May 9, the USS Lake Champlain; June 17, the USS Fitzgerald; August 20, the USS John S. McCain) bothers me: the destroyer McCain was nearly sunk just two weeks after it challenged the Chinese at a contested atoll named Mischief Reef, which the Chinese have in recent years built-up into a military base.
I have a problem with coincidences that turn out bad for our side. Malevolent intent by an adversary is always possible. Every bridge officer should understand the protocols to avoid intentional (or unintentional) collisions initiated by rogue (or wayward) boats.
A Philippine-manned cargo ship, the ACX Crystal, rammed the USS Fitzgerald, a guided-missile destroyer, on June 17, 2017. Seven US sailors died. The night was clear; the seas calm. The commanding officer and another crew member were severely injured. Repairs will cost hundreds of millions and take years to complete.
Our Navy is a mess. Everyone knows it. The optics of powerful warships limping into port under the power of a dozen or so tugboats emboldens our enemies and demoralizes our patriotic fighting men and women.
We have the wrong ships, designed the wrong way, for the wrong wars, for the wrong reasons. And our Navy is overworked to the max. We all know it’s true. It doesn’t have to be. It’s good to have high-tech systems, but they are useless during a crisis. Everyone must be proficient at low-tech and know how to enable it. Seriously.
Politics and corruption, profiteering and greed, laziness and lack of zeal are going to kill us all if we don’t wake up. It’s time for civilians to step up and defend our way of life. It’s time for corporations and billionaires to do what’s right — not what makes them wealthy at the expense of our country’s defense and the prosperity of our citizens and the people of the world who are looking to us for leadership.
We are going to regret privatizing our military and using contractors instead of citizens to fight our battles. We are going to lose our freedoms and our country if we don’t fight for both. Everyone must do their part. Corruption can have no role in the process.
We must use our power to make the world safer, freer, and better for everyone, not just ourselves. People are sick and tired of “America first.” We have so much, already.
It’s time to share our advantages, with love. If we do what’s right, if we embrace public service and reach out to the disadvantaged in the world (the military, after all, doesn’t have the room or the money for every citizen), we won’t need to kill everyone who hates us like we’ve been doing for hundreds of years.
A year or two of public service by every American in impoverished neighborhoods and blighted communities might make a big difference in the why, how, and who we fight.
interesting is the four real numbers it generates. (The numbers are +.2078… , -.2078… , +4.8104… , and -4.8104… .) 
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?
= cos (ln i) + i sin (ln i)
= i
= ln i
= cos (
) + i sin (
:
= .043214; 
= 23.140693. 
who has a simple calculator with a square root key.
?
? 
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.
