This essay is one of the shortest on the website, but it might be the most timely during this moment of impeachment inquiries. Questions like the following are being asked on forums around the internet and the world:
Would you rather have Joe Biden or Elizabeth Warren run against President Trump? September 2019 on Quora.com
The problems that President Trump creates are the most serious to face Americans in my lifetime. Three exceptions might be the Cuban Missile Crisis, the Vietnam War, and 911.
As a people, we are at this moment completely dependent on our elected representatives. Nothing any civilian does can make any difference; we are helpless in the face of an alleged sociopath who holds the most power of any human on Earth.
Tony Schwarz wrote Trump’s Art of the Deal and split the profits 50-50. He has said that Trump is a sociopath capable of initiating a nuclear conflagration.
The following is an answer to the Quora question, which I hope people who matter carefully consider. And yes, the answer is somewhat oblique. I don’t think whoever runs against Trump will make much difference in the events that will come after.
Does anyone believe that Trump is the kind of man who relinquishes power?
The surest way to remove Trump is through impeachment, but it’s not clear that he will step down if he is convicted by the Senate. Waiting for clarity is not a valid reason for caution. Inaction will bring catastrophe. Too much is at stake.
I know it’s difficult for many Americans to remember, but Trump insisted that the 2016 election was rigged by Hillary Clinton; many believed he planned not only to challenge the election in the courts but to lead an insurrection or even a revolution if the result turned against him.
Trump is allied with the NRA—an armed, insurrectionary group of lunatics dedicated to revolution should they fail to get their way. The NRA has been swarmed by Russian intelligence. The Russian objective is to destabilize the United States by fanning the flames of paranoia in a politically influential organization known to be susceptible to manipulation by conspiracy theorists.
Southern evangelical Christians are the backbone of the new Confederacy, which Trump leads; he is their modern day Jefferson Davis who plans first to reestablish legalized segregation under the guise of, I don’t know, “right to choose”, maybe.
Other atrocities will follow. The suppression of non-whites (like immigrants) is in full-swing, which white-supremacists, among others, support.
White evangelicals don’t preach against the ethnic-cleansing of America; they endorse it. Jerry Falwell Jr. of Liberty University has encouraged his students for years to train in the use of firearms so that they will be able to kill Muslims if given the opportunity. For years he has advocated for violence against those he hates by inciting unaccountable, non-state actors — like the students who attend his so-called “university”.
NOTE BY EDITORIAL BOARD: On August 25, 2020 Jerry Falwell accepted a $10.5 million severance package from Liberty University to step down. Like Adam in the Garden of Eden, he blamed his wife for the sex scandals that followed him pretty much everywhere during his tenure.
I think most reasonable folks are at a point where they can figure-out that the coming election might amount to no more than a simple diversion — a shiny object to pull people’s attention away from what might turn out to be the only course of action with any chance at all to remove the president — an impeachment in the House of Representatives followed by conviction in the Senate.
Many people are in denial; people withdraw their attention from events that disturb them. Seizure of American institutions has already taken place by people working for the other side. Some people seem to have unrealistic expectations that a media owned by oligarchs like Trump will warn them.
Very few are able to face head-on a reality that includes loss-of-country. People sleep better at night if they pull their heads inside their shells and pretend everything is OK when it isn’t.
Some resort to a version of the Stockholm Syndrome to help them cope with the sickness that tears at their stomachs when they remember that Trump grabbed them by their electoral colleges; they embrace the tormentor to maintain their sanity.
Stockholm Syndrome is at the core of Trump’s power. Trump is right that he can kill, and his supporters will stand with him. He has nothing to fear from so-called “church-goers” who love him because he is bad.
Trump has put into place policies that seem designed to terrorize sick children; ICE puts them in cages where they risk death from neglect and illness; the president’s policies strike fear in families who are running from gangs to search for safety north of the southern-border.
Everyone wants to believe that a shining-city-on-a-hill exists somewhere in the world that they can crawl to during hard times. It’s a myth, unfortunately. Trump has shredded people’s dreams for a better future. Impoverished freedom-seekers are neither welcome nor safe in Trump’s America.
A bad election result is the easiest thing for a thug who already holds power to cast aside. An election can be rigged; it can be ignored; it can be challenged in stacked courts; it can be won in the electoral college.
Trump ought to know; he lost the popular count in 2016 by a record 11 million votes; 3M from Hillary; 8M from third-parties. No president has lost the popular vote by a larger margin. The president knows he will lose the popular vote in 2020; it didn’t stop him in 2016; it won’t stop him next year, either.
Losing the popular vote by a wide margin is not going to be a problem for President Trump. Who doesn’t agree that it’s true?
If Americans wait for the 2020 election to make the change that everyone with sense knows we must make, then our country could be lost for a long time, perhaps forever, at least for us, the little people who depend on it to breathe free.
Billy Lee
Note from the Editorial Board: We recommend that readers view each video in Billy Lee’s essay. The short videos are historical in nature. Current confirming videos can be found on YouTube and other websites. Billy Lee included the older videos to remind people about where they’ve been. He wants to encourage folks to reflect about what they imagined was going on when first they saw them.
For decades now, Nick Bostrom has defended his view that the reality of existence can be described by one of only three possible states:
1 – Life is rare in the universe; what life does exist always perishes before it reaches “technological maturity.”
2 – Life is rare in the universe; some life reaches techno-maturity, but all advanced life decides to avoid the temptations and the consequences presented by its mastery over artificial super-intelligence and other high-technologies.
3 – Life is abundant in the universe, but it is simulated. A few technologically mature civilizations yielded to temptation; their thirst for knowledge and entertainment pulled them into a spawning-orgy to artificially inseminate faked-life within an ever-increasing globe of stars and planets, galaxies and clusters — perhaps throughout all space.
Bioethicist and philosopher of artificial intelligence, Nick Bostrom.
This injection of simulated-life (and the infrastructure to sustain it) serve the research and entertainment needs of the original civilizations who created the simulations and then broadcast them into the cosmos like farmers throwing flower-seeds into empty gardens.
By now, faked life is pervasive. By now — after at least three generations of star formations — the number of simulations is hundreds-of-millions; perhaps billions; perhaps hundreds-of-billions.
Humans forced to wager on the odds that they themselves are artificial — that they are in fact simulated — must place their bets knowing that the odds could be as high as a billion to one, maybe more.
Humans are machines; they aren’t real; they aren’t what emerged from the chemistry of the universe but were instead invented in the imaginations of super-computers programmed by an ancient civilization whose address and time may forever remain unknowable.
If human civilization is faked, so might be its history, what it has been allowed to know, and what technologies it feels compelled to develop. A simulated, artificial civilization won’t necessarily know what is the time or era or eon in the real universe that exists beyond its view. It might never be able to understand how large or old is the real universe where its creators live and play.
Simulations might be created by life-forms curious about how certain scenarios they can imagine play out. Simulation might be a mature-tech version of television where advanced life-forms unveil the vagaries of their visions for the entertainment of vast audiences.
Some simulations could be simple games designed for small children to entertain themselves while mommy does laundry and dad mows the lawn.
Simulations might be simple algorithms developed by quantum super-computers to test the limits of their power.
Even the rules-of-play embedded in some releases might be undiscoverable — hidden by super-intelligent gamers who perhaps don’t really care about us; they are sure one day to lose interest and unplug the simulation.
What are the odds?
People who think like Nick believe the odds make countless simulations a near-certainty. If it were not so, then it is equally certain that human civilization will implode like all those civilizations that came before; humans will become the victim of their own technological march into the high-risk skill sets that lead inexorably to oblivion.
It’s what the Fermi Paradox is all about, right? Astronomers assume that life is common — pervasive perhaps — but they search the universe in vain for our companions.
Where is everybody?
Is it possible that the evidence will forever be that in this cosmos humans are alone and on their own?
If life — pervasive intelligent life — makes itself known, can anyone be sure that it will be authentic and not a simulation created by a life-form they will never meet? Can anyone trust that this newly encountered life is conscious?
Or is consciousness simulated so that no one real can discern who is genuine, what is authentic, who is faking true love, or what might behave in blind obedience to rules that render impossible any prejudices against faked life, which does not care?
What are the odds?
During his #1350 podcast of 11 September 2019 Joe Rogan asked Nick Bostrom why it cannot be true that we humans are the first to broach the limits of the technologies that are spread before us. Why cannot humans know for sure that they are real, not simulated?
Why is it not realistic to assume that human civilization is on the cusp of becoming the first creator of simulations and artificial life instead of being itself one more simulation added to a long line of simulations that have spanned the cosmos during the past billions of years?
In a simulation, which of its fake creatures is able to determine how old is the authentic universe it will never see? What avatar is able to determine how long or short-lived will be the simulation where it is trapped?
Why can it not be more certain that the civilization that survives and prevails will be ourselves, the species human, who will be first to spawn false populations and fake technologies to coat with lies the cosmos whose lifespan is likely to last trillions of years?
Nick Bostrom went through the numbers with Joe. He described how the probabilities of his ideas are constructed; he explained that if humans are not fake; not simulated; not artificially created, it is more likely — much more likely — that homo-sapiens won’t make it into the future.
We will suffer the extinction of every advanced civilization that went before, no matter where in space and time they were once located — if ever there were any.
Neither Nick nor Joe seemed interested to discuss more than perfunctorily the validity of the second listed possibility — the existence of technologically mature civilizations who refuse to extend their capabilities to logical conclusions.
The idea that civilizations might forego the use of artificial super-intelligence to secure their grip on the universe seemed a boring and unrealistic option. Nick included the possibility on the list of three only because it is possible to imagine that dozens of civilizations might decide, perhaps independently, to lay down their powers for some higher, universal moral-order.
Such a scenario defies common sense, does it not?
So the choices seem to have collapsed from three to two: self-annihilation or a successful breach of the barrier that enables breachers to create new, simulated worlds — to raise their status, finally and forever, to the heights of what the ancient-world called “gods”, the creators of worlds.
Is the ancient-world even real? — or is it another fabrication by simulators?
What do we know and when did we know it?
Why does science and history make no sense?
What are the odds?
Every theoretical physicist seems to be saying that quantum mechanics and general relativity cannot be fundamental. A reality underlies these systems of physics that seems to lie beyond our reach.
Physicists today admit that at least for now they are stuck on stupid. They wait for la seconde venue d’Einstein — Albert Einstein, part two.
How smart and creative can a simulation be when it can’t answer basic questions like:
What time is it?
Where am I ?
Is anyone in charge?
Why do the simplest things make no sense?
Are simulated life-forms — necessarily separated by their natures from reality and truth — always insane?
Are simulations evil when they challenge the authentic life that created them? Is authentic-life virtuous when it destroys the faked-lives of its troublesome simulations?
Which deserve to feel the emotions of existence more intensely — real-life? or artificial super-intelligence? or the billions-of-simulations, which Bostrom’s probabilities argue flow from them both?
Billy Lee
Warning from the EDITORIAL BOARD: Billy Lee sometimes “pontificates” to try out ideas, which to our minds are absurd.
For one thing, Billy Lee seems to imply in FAKED LIFE that the absence of evidence for intelligent life in the Universe is exactly what a civilization locked inside a simulation would experience.
Simulated life hidden behind the walls of a game constructed by “Super-Intelligence” will inevitably come to believe it is alone and dependent on a Supreme Being who loves only the “simulants”, because no one else is “out there” for God to love.
Billy Lee postulates in FAKED LIFE that the universe and its history make no sense, because it isn’t real. Maybe Billy Lee is dumb; maybe he doesn’t get things, because he can’t. Did the possibility that he is stupid ever cross his mind?
Perhaps “pity” is what Billy Lee deserves.
Billy Lee has written in the past that desperate folks might want to trust God to explain things they don’t understand; otherwise, they will miss chances to make the world a fairer and more loving place for the billions of people who live in misery — the weak and impoverished, right?
According to Billy Lee, the Bible says that God created people; God is love; He promised to never abandon the poor, the sad, the humble, the strivers for what is right, the merciful, the pure-hearted, the peacemakers, the persecuted.
God’s power is that He cannot change.
So, according to the Bible, it’s all true. We are simulants who will never be unplugged. The rules of the game are simple: love God who gave us our lives; love each other as much as we can.
Who will do it?
We are as real as our Creator made us to be. God decides who is real and who is fake, who lives and who dies. In His eyes, we aren’t fake — even though some of us say He is.
Some readers may know that I post on Quora about all kinds of subjects.
I don’t hold back. I write a lot of random stuff that I would never publish on my blog.
Well, not always.
People want to know — What’s up with all the reports about ALIENS on the internet?
I recently answered a question about “space-invaders” that 10,000 viewers chose to read during the first hour of posting.
People like to read about aliens — but only if they come from deep space.
I took the time to work out the answers to several questions.
All true facts. No fake news. No messing with heads.
This stuff is real, people.
Read, learn, and be afraid.
Why do people think NASA found an alien base on Jupiter’s moon?
You are referring of course to the moon Europa, which is one of 79 moons that orbit the massive planet Jupiter.
Europa is a perfect place for a base, because the escape velocity is less than 185 mph—easily attainable by small jet-planes and other craft. Europa’s icy surface is smooth enough to land and take off with specially equipped seaplanes.
Plenty of water, plus gravity that is 1/8 of Earth’s, and abundant food resources in the under-ice oceans—Europa is a paradise compared to other moons in the Jovian system.
The Japanese established several bases on Europa in 2007. President Bush — preoccupied by the coming financial collapse of 2008 — decided against a challenge to Japan’s hegemony.
Obama didn’t want to deal with the hysteria that would follow an official announcement; it would interfere with his passage of the Affordable Care Act.
Trump decided that the Japanese are “very fine people”. He has fallen “in love” with Europa’s Commander, Admiral Yoko Oh No (the Sumo-wrestler “snowboarder” who poses in the pic below).
The “best people” on the president’s advisory council decided that a “premature” announcement of Japanese sovereignty over an “alien world” would likely undermine the orange man’s reputation for stability and genius—qualities of character they are unwilling to sacrifice to those who might accuse the Donald of being a “nut-case” should he disclose what he knows to a public suffering from TDS (Trump Derangement Syndrome).
So, permit me to answer your question.
Ham radio operators listen in to the communications (in Japanese) between Europa and Earth all the time, but government officials won’t confirm what they have known for nearly 12 years.
Six families own all the media in the world. They won’t report the story, so it remains untold.
In the era of fake news, who will believe truth when they hear it?
Admiral Yoko Oh No — Japanese Commandant, Europa, Jupiter. Reduced gravity makes snowboarding easy for the 400-lb. Sumo-wrestling enthusiast.
Why are aliens so interested in our planet?
Aliens swarmed Earth in the 1950s.
Everyone who doesn’t live under a rock knows it’s true.
The problem for the aliens is they didn’t bring weapons. Many hundreds of the captured are kept in cages in Area 51. Surely, someone “out there” knows about Area 51 and what goes on, right?
Does anyone believe that the computer and iPhone are the product of human ingenuity? Has anyone anywhere ever met a human who is capable of inventing these technologies and building them?
Of course not.
All this “high tech” stuff is always built in faraway countries inside factories no one has ever seen or can ever visit, right?
Has anyone considered that there might be good reasons why no one understands the technologies they are using?
Knowledge is unattainable. Resistance is futile. We all know why.
The orange man puts aliens in cages for a reason. He sucks brains to increase his power.
Challenge the orange man, anyone who is foolish enough to try. They will find themselves “throttled” by Quora, Facebook, and Twitter among other “collaborators.”
Those who don’t believe, look at the view statistics for this post. Though hundreds-of-thousands will see, the “official” view count will never exceed a few dozen.
Upvotes? Forget it.
“They” don’t want civilians to know the nature of the new Earth where “orange man” forces homo-sapiens to languish — suffocated and imprisoned by confusion; overwhelmed by terrors unleashed by ruthless captors of alien hoards.
Another alien swarm from “out there” is making plans to rescue their friends. This time they won’t forget weapons. Meanwhile, they burn forests to deprive Earthlings of oxygen. Once their sapient tormentors are debilitated, the aliens will unload.
After the fluids of humans settle finally into the warm dirt of Earth, it is the human captors — the survivors — who will live in cages.
The aliens plan to convert Area 51 into a zoo where tourists from other worlds can visit to learn about evil humanoids and their orange leader.
Aliens will place the chosen one, with all the reverence due him, into the largest cage of all.
All organisms on Earth are carbon based. Do you think there are silicon based life forms out there?
Silicon based life forms are called “rocks”.
Four phylum of rocks exist: pebble, stone, boulder, and sand.
These life forms move around, reproduce, and sometimes present an aggressive demeanor that can threaten humans.
How many times have we read accounts of boulders cascading down mountains to crush folks who travel unawares on our national highways?
It is almost impossible to eradicate boulders. Even an explosion of many tons of TNT creates instead tens-of-thousands of stones and pebbles. These silicate life-forms are particularly annoying, especially when humans walk barefoot on dirt.
Silicon-based life-forms are an urgent problem that demands to be solved.
Are we watched by aliens?
Extra-terrestrials made their first visits to Earth about 5,000 years ago according to accounts in the Bible and other sources from that era.
They left after exploring nearby planets and moons. They stayed to explore for about a 1,000 years give or take.
In the 1950s extra-terrestrials returned in force. Tens of thousands of UFO sighting became a matter of public record. The Air Force and CIA spent considerable resources to understand the threat UFOs posed to humans.
What they discovered was a blunder by aliens that no one anticipated.
The aliens came unarmed.
When aliens finished their initial explorations 4,000 or so years ago, they reported to their leaders that life on Earth posed no threat. The need to bring weapons on a second voyage was considered an unnecessary expense, energy depleting, and useless. Leaving weapons behind would save billions of units of whatever currency they were using at the time.
Since the 1950s, a coalition of nations have learned to capture and successfully detain aliens. Many aliens have escaped, but thousands who weren’t so lucky now live in cages in the area numbered 51, which some readers may have read about.
These aliens are essentially prisoners. Some have been trained to communicate with humans.
Information gleaned from interrogations has yielded a treasure trove of new technologies, which have been introduced as rapidly as possible during the past two decades.
It is humans who now observe extra-terrestrials.
The current situation is this: Aliens are preparing to swarm Earth in a “third wave” to rescue their compatriots. This time they will bring weapons.
Trump is currently negotiating with an advanced guard who his advisors refer to as the “Deep State”. The future of Earth depends on the president’s ability to negotiate and to act the “stable genius” people everywhere have learned to believe-in and love.
President Trump is the only one who can save humankind from the alien hordes, who intend to alter forever Earth as we know it through terra-forming.
What makes you sure there are/are not aliens visiting planet Earth?
Aliens swarmed Earth in the 1950s and never left. They occupy many positions of power and influence. Their base of operations is near Lake Vostok in Antarctica.
How can humans identify space-aliens?
The identification process is easy but it takes a little work. Many so-called news celebrities are aliens. Listen carefully for mispronounced words and unusual turn-of-phrases. These are dead giveaways, which are ignored by most people.
Also, search for signs of Botox poisoning. Alien skin sags in Earth’s highly oxygenated atmosphere. Almost all aliens use Botox to prevent the reptilian appearance that frightens so many children. The overuse of Botox can make aliens look like “cat people.” Again, they aren’t difficult to spot if one remains vigilant.
Another identifier is reliance on teleprompters. Most aliens never achieve true fluency in human language, preferring to chirp or hum loudly when not reading from prepared text. Heavy reliance on teleprompters by powerful aliens is one more sure sign that humans should never ignore.
Aliens tend to rule from the right, because conservatives are the easiest people to outsmart. Trump switched from Democrat to Republican to take advantage of GOP stupidity to win the nomination in 2016.
Aliens immediately endorsed him.
What is the likelihood that aliens would be able to decipher the messages on the gold plate on the Voyager spacecraft?
The message on the gold plate is simple: life exists somewhere else.
Aliens who discover the craft will use artificial super-intelligence to decipher its information — if not immediately, then eventually as they develop the capability.
Once the message is unraveled, aliens will know our location and weaknesses. The streaks in the image on the lower left of the gold disc reveal Earth’s location relative to 14 pulsars, which are GPS beacons to any civilization as advanced as ours.
Click pic to enlarge in new window.
Aliens will determine from the materials that make the craft what resources exist on Earth and whether these resources are worth pursuing.
Should aliens decide to visit, they are likely to bring weapons to overwhelm us; they will take whatever they want.
What might they take?
Look at any picture of Earth from space. Isn’t it obvious that Earth is a paradise where water exists in three phases—liquid, solid, and gas (oceans, ice, and clouds)? What alien wouldn’t want to live here to research and perhaps rule to satisfy its idiosyncratic desires?
We’re vulnerable.
How do we defend Earth against aliens who know all about us and our capabilities while we know nothing of theirs?
Thank you, NASA.
Thank you to all the eggheads who may have given our location away to monsters they’ve never met—maybe not now but someday.
Good luck, planet Earth.
Is Russia really larger than Pluto?
It depends on what the definition of “larger” is.
When it comes to demographics, Russia’s population is thought to be much larger but a census of the dwarf planet has yet to be released to the public.
When it comes to nuclear warheads, well… Pluto takes its name from the chemical element plutonium, right? The Plutocrats who run the place keep the number of warheads a tightly-guarded secret but NSA analysis of data gleaned during the recent “fly-by” speculates that the white-planetoid might harbor both the capability and the desire to destroy the solar system.
During recent tests of stockpiles, Plutonians blew up the Kuiper Belt. It’s not something military planners are likely to ignore.
Economy?
Isn’t the answer obvious?
Everyone who’s been there says Pluto boasts the best ski-resorts anywhere.
The Japanese, for some reason, disagree. They ski indoors; the temperature is warmer, which means humans don’t have to “layer-up” or even wear gloves.
Plutonians scoff at amenities like comfortable temperature, oxygen canisters, and helicopter rescue teams who “stand-by” to retrieve tourists who downhill into craters.
In some respects the argument about who is larger, Pluto or Russia, has faded to irrelevance. Revenues from ski-tourism enabled Pluto to buy Russia from the Communists decades ago. Today, Pluto-installed oligarchs have hand-picked the “Orange-One” to tighten Pluto’s grip on power over everything human.
The future of Earth is secure; it’s as solid as ice; it’s as safe as the Kuiper Belt where resistance from Kuiper terrorists evaporated during the plutonium accidents of yesteryear.
Did the USA and Russia find an alien presence on the moon and is that the reason they have not revisited?
The first astronauts who landed on the moon communicated off-line for almost an hour with NASA about structures on the horizon that appeared to be unnatural.
On a subsequent mission, astronauts used a dune buggy to travel to the structures to explore them.
Inside structure two, they found a photograph of a large lizard in a space suit. Other pics showed the lizard naked — surrounded by five other lizards of various sizes.
The surprising part was the conclusion by NSA analysts that the lizards seemed to be frolicking in a water park of some kind.
Subsequent radioactive dating revealed that the age of the structures and photos was 137 million years; everything was preserved in pristine condition — due possibly to the moon’s sterile environment.
No astronaut who was involved in the discovery or the coverup that followed is currently employed by NASA.
Retired astronaut Guy Gizzard works at a zoo near Orlando where he cleans lizard cages. He doesn’t do interviews according to the New York Times.
Why astronauts no longer travel to the moon remains a mystery.
As civilization advances, so does arrogance. Instead of colonizing Mars, why not spend the money to save Earth?
All civilizations approach an asymptotic limit to knowledge and technology that dramatically reduces their odds of survival.
No evidence has yet been found to show that any intelligent civilizations have survived anywhere in the universe — at least in those areas of space where humankind has been able to look.
Should people survive for a few thousand years more, will the evidence for human isolation change?
Maybe not, but it would be good to be proved wrong.
Billy Lee
Disclaimer by the Editorial Board: Despite what Billy Lee claims, we warn readers that no one has been able to verify anything he wrote.
Added May 15, 2026:Sample problems with solutions provided at end of essay.The Editors
Can anyone calculate by hand (without a calculator) the square root of 5.71 ? How about the two-dimensional complex number (4 + 2.53 i ) ?
Of course not. Normal people who are not mathematicians punch these numbers into calculators or math apps on their iPads and computers to calculate the answers.
Without iterating — that is: guessing, deriving a result, and then zeroing in with better guesses) — finding the square root of 5.71 requires knowledge of some arcane mathematics. No one labors by hand to find the answer, which is 2.38956… . It’s the principal square root, of course.
How does anyone iterate to derive the square root of the complex number (4 + 2.53 i ), which happens to be (2.0896… + .606375…i ) ? It is also a principal square root. What are the others? Is there more than two? People use calculators and pewters to find out; there is no easier way.
In high school and basic college math courses, people typically learn to solve algebraic equations. A typical algebraic equation looks like
…right?
They have polynomials with integer coefficients. The solution is , which in this case is an algebraic irrational number. Equations like trig and log functions that transcend algebra (called transcendental equations) are taught maybe to engineers and science majors; math majors, of course, don’t struggle with this stuff. It’s why they are math majors.
Several categories of transcendental equations are commonly encountered in the sciences. Many simple problems can be solved by Newton’s Method, which is taught in basic calculus. I won’t explain the method in this essay. Folks can click on links to learn more if they want.
A category of transcendental equations that can get complicated is of the general form
The biggest problems arise when “y” is known, but “x” isn’t. How to solve for “x”?
Any transcendental equation that is able to be transformed into the form xex can be solved for “x” using the Lambert W function. The equation can be inverted into the form, x = W(y). People are going to have to take my word, for now.
The math behind the Lambert function is mind-bogglingly complicated to most people. The function can sometimes require unusual and involved “series expansions” and transcendental-styled integrals that are not possible to solve easily or quickly without a computer.
The Lambert W function (sometimes called the omega (ω) function or the product-logarithm function) is not a key or button that can be pushed on most calculators. However, math apps like Wolfram Alpha and Mathematica use it, sometimes to solve transcendental problems in the background when equations that need solving aren’t so easy.
Omega functions are almost always many valued, because once an equation is inverted there are usually many paths that lead back to the original equation. The useful solution requires picking the principal solution, which is the number identified by subscript 0.
I ran across a transcendental equation on the web that is perfectly suited to teach the “ω” method. Here it is:
I want to solve it to demonstrate how to use the ω method for transcendental functions that aren’t otherwise so easy to work out. I challenge anyone to solve this equation using Newton’s Method or other iterative techniques. Most will struggle to the point of pulling out their hair, probably. And they will waste time. Yes, it can be solved by those techniques.
We will solve this equation step by step using the Lambert method shortly. Meanwhile, here is the strategy:
1. Substitute an exponent function (et) for “x” everywhere in the expression.
2. Manipulate the equation into the form:
3. Invert the equation to introduce the ω function.
4. Use the ω function to solve for “t”.
5. Write out x = et using the expression derived for “t”.
6. Solve ω(y) using WolframAlpha or any other app with the capability.
7. Use the value of ω(y) to solve for “x”.
Each step of the strategy will be identified by numbers 1-7 in the solution below.
Here’s the thing:
In this problem it turns out that there are four ω values of y, which will generate two real solutions and two complex solutions. These omega values are:
ω0(y)
ω-1(y)
ω–2(y)
ω1(y)
Inverting the equation to introduce the omega function creates solution branches, some real, some complex, which are layered sequentially and labeled by integers from minus ∞ to plus ∞. In the example, real solutions lie on grey and yellow spiral surfaces numbered 0 and -1. Complex solutions lie on red and green spiral surfaces numbered -2 and 1. Omega subscripts in the list above this graphic identify the spiral layer where the solution for its omega value is located.
WolframAlpha will generate all the solutions automatically; no need for the user to understand anything. People can punch in the original equation and trust the answers the app returns.
But the solution steps that follow are fashioned to demonstrate how the problem is solved when all anyone has is an algorithm to generate the values of the omega functions. Omega functions are difficult to solve without using certain algorithms involving integrals and expansion series on robust computers.
The process that surrounds the computation of omega values, which permit the working out of the appropriate values (the right answers) to the kind of equation I will soon solve is interesting and enlightening, at least for me, and hopefully for certain readers.
Some folks will appreciate the insights this exercise provides.
Having knowledge will make the Lambert process that is used to solve certain transcendental functions less mysterious. Of course, one can always take the time to learn the expansion series and integrals. In some cases, Newton’s Method can generate the values.
Unless humankind loses the technology of computers, I don’t think it is a good use of time and resources to learn the series, integrals, and algorithms that generate omega values.
Let’s face an unpleasant fact: most of us aren’t going to live more than 80 years or so. We don’t have time to waste. For some folks, knowing how to use and apply the functionality that surrounds the Lambert function to give it power is enough to make life worth living. Count me in.
No one needs to wade through the jungles of series expansions and transcendental integrals. Let math apps do the tedious work, knowing full well that any interested person can master whatever they choose if necessary, but someone already did the work. Why duplicate the effort?
I want to solve novel functions — complicated formulas that transcend algebra. Understanding the process that solves these equations is fascinating. It’s not as rewarding to tread over mathematically esoteric ground already mapped by experts who are far more able than people who spend most of their time working in other fields.
Here is the solution process:
What we know:
IF y = f(x) = xex THEN x = ω(y) [where “ω” is the Lambert W function]
Solve:
LET x = et
(1) THEN
(2)
Referring to “what we know“, the equation is now in the desired form y = xex
where “y” is equal to and “x” is equal to “-3t “, right?
We are now free to use the omega operator to “invert” the equation into the following form: x = ω(y)
(3) (-3t) = ω
(4) ∴ t =
Notice that we have worked through step (4) of the strategy. I don’t like the way the formula generator writes the Greek letter omega (ω), because it’s hard to read. From here on, I will sometimes use “W” instead of “ω” for readability. It shouldn’t confuse anyone. In this essay, consider W and ω the same symbol, please.
On to step (5).
SINCE x = et
(5) THEN
I need to know what -ω0(-1/8) equals so that I can use it to compute one of the values of “x”. As mentioned above, three more omegas with three other subscripts (-1, -2, and 1) are needed to compute all four of the solutions to this equation.
How does anyone know how many solutions the original function has? How does anyone know what subscripts are required?
This is where someone who doesn’t have a masters degree in mathematics needs a math app like Wolfram Alpha or its cousin, Mathematica. Otherwise, they have to work series expansions or difficult integrals to derive the omega values associated with (-1/8). Who wants that? Not me.
Here’s the series expansion for ω0(-1/8) according to Wolfram Alpha. Who wants to compute it?Here are two integrals for ω0(-1/8). My advice is to use the second integral, anyone who has the guts.
OK. In WolframAlpha, you get the omega value ω0 for (-1/8) by writing the expression -W[0,-1/8] in the input line at the top of the page. It shoots out the answer and links to its derivation.
It’s so simple. Other math apps might use different notation. I don’t know, because I don’t use other apps.
Inside the brackets, the “0” is the subscript on ω, and the “-1/8” is the “y” value, right? So, in addition to -W[0,-1/8] it is necessary to input:
-W[-1,-1/8]
-W[-2,-1/8]
-W[1,-1/8]
to obtain the three other omega values, right?
The omega values returned are the following:
1.4442135…
3.2616856…
4.21446… + 7.33231…i
4.21446… – 7.33231…i
The ω function values for -1/8 are two real numbers and two complex numbers. I am going to solve the original equation for only the first real number omega value to demonstrate the method.
Here it is:
INPUT -W(-1/8) or -W[0,-1/8], both work for ω0
(6) OUTPUT +0.14442135…
COMPUTE
t = = .04814…
SINCE x = et
THEN x = e.04814…
(7) SOLUTION x = 1.04931755…
CHECKING
BY SUBSTITUTION
VERIFICATION .04814… – .04814… = 0
CONCLUSION: The transcendental equation which is the focus of this essay can easily be solved and verified by simply punching the equation into the input field of a math app like Wolfram Alpha or Mathematica and reading off the answers.
We didn’t perform the simple procedure, because I wanted to share how the Lambert W function fits into the solution process for solving equations.
In truth, all four ω values must be gathered so that the three other “x” values of the original equation can be derived.
In this example, one of the other solutions will be real; the other two, complex. The screenshot below from Wolfram Alpha demonstrates how these four values are displayed. Of course, by clicking links the app will reveal much more.
Wolfram Alpha enables users to input transcendental equations and quickly view answer-sets and methods of computation.
I would be remiss to not mention a famous formula for calculating to what number a fraction raised to successive powers of the same fraction converges.
(The range of numbers where this formula actually works is between e−e and e1/e, that is, between .065988… and 1.444667861… .)
Take a number like ½ (0.5). Raise it to the 0.5 power; raise it again and again to the same power over and over an infinite number of times; the number will converge to a specific value.
What number? How in the world could anyone figure it out without repeating the power-raising process an annoying number of times?
It turns out that a formula involving the Lambert W Function yields up the answer easily.
The formula is:
# =
Put the following expression into the INPUT line of WolframAlpha:
-W[0,-ln(.5)] / [ln(.5)]
Click ” = ” — or hit “ENTER”.
The OUTPUT is: 0.6411857445049859844862…
Compare this result by taking the exponent (0.5) of 0.5 twenty times by hand (on a calculator). The answers will agree to 7 decimal places. Fifty “tetrations” will bring greater agreement if your calculator can parse the answer.
Who has the time?
Billy Lee
NOTE from the EDITORIAL BOARD:Billy Lee was unable to find an appropriate video about the Lambert function on YouTube, or we would have posted it. Most folks capitalize the Greek letter omega (Ω), but in this essay, Billy Lee didn’t, preferring instead to use little (ω), because it looks more like (W).
Who on the BOARD would dare argue?
Apparently, no one.
Another reason is that Ω is sometimes given the value 0.567143… , which is known as the omega constant. Why confuse things?
The video above starts a discussion of the Ω function at 16:30. The Lambert W function is derived for ΩeΩ = 1 at 18:00. The first sixteen minutes and thirty seconds show how to use Newton’s Method to solve the equation. Some readers might want to skip the first 16 minutes; others will enjoy them.
Who knows?
ADDED April 25, 2026
Recently Facebook, in its feed, published math puzzles like this one:
xx = 9
What does “x” equal?
X is exponent in transcendental equations like these, which are easily solved using Lambert.
xx = 9
x ln x = ln 9
LET x = et
SUBSTITUTING et ln et = ln 9 {~2.1972, right?}
THEREFORE et t = 2.1972
OR t et = 2.1972
LAMBERT SAYS t = W [0, 2.1972]
ENTER W [0, 2.1972] into Wolfram Alpha to get ~.8965
t = .8965, right?
BUT x = et
THEREFORE x = e.8965 = ~2.7183.8965 = 2.451
ANSWER x = 2.451
CHECK xx = 2.4512.451 = 9
QED
Here’s another cool problem. What does k equal?
49k = k14
k ln 49 = 14 ln k
k ln 72 = 14 ln k
2k ln 7 = 14 ln k
= ln k
= ln k
LET k = ex
ex = ln ex
ex = x
ex = x
1 = x / ex
-1 = -x e-x
= -x e-x
LET -x = t
= t et
equation is now in correct form.
t = W
PUT W[0, ] INTO WOLFRAM ALPHA
t = -.42536…
BUT x = -t
x = .42536…
BUT k = ex
THEREFORE k = e .42536
ANSWER k = 1.53014
CHECK 49 1.53014 = 1.53014 14
385.688… = 385.688…
QED
Should be mentioned that there are many solutions to this equation as anyone would assume after reading essay above. Wolfram will provide them. However, an integer answer exists that can be solved by inspection.
k ln 49 = 14 ln k
Make the base 49 to make ln 49 disappear.
k ln49 49 = 14 ln49 k
k = 14 ln49 k
Since by definition logs are exponents acting on a base (in this case 49), a sharp-eyed reader might notice that base 49 is the square of the number 7. The square root of a number is its 1/2 power, right?