What is e exp (-i π) ?

What is e^{-i\pi} ?

I posted a long answer on Quora.com where it sort of didn’t do well. Answers given by others were much shorter but they seemed, at least to me, to lack geometric insights. After two days my answer was ranked as the most read, but for some reason no one upvoted it. It did receive a few positive replies, though.

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

I added some graphics here that wouldn’t post on Quora. The site lacks graphics functionality, apparently. Either that or I’m too dense to figure out how to insert something.


NOTE: 24 Oct 2017: Today Billy Lee finally figured out how to post media on Quora.com. After admonishment and chastisement by the staff, he added a pic and a GIF to his “answer.” Unfortunately, Billy Lee never did get the GIF to run right, so he took it down. Otherwise, Billy Lee is doing good. He really is. THE EDITORIAL BOARD


Anyway, this pic and a working GIF below make a big difference in understanding, I hope. And anyone who doesn’t understand something can always click on a link for more information. (No one ever clicks on links, but I spend a lot of time adding them—maybe so I can click on them myself during times when remembering my name or where I live seems to lie just outside my intellectual skill-set.)

Here is the drawing I added and the answer:

This diagram is excellent but contains a mystery point not on the unit circle—{i^i}. The point is shown at .2078… on the real number line. It has 3 other values on the horizontal line that are not shown: -.2078… and +/- 4.8104… . Its solution can be reviewed in this video link, which Billy Lee has carefully reviewed and confirmed to be sound. An imaginary number raised to the power of an imaginary number yields a result that is four real numbers. How can that be? It’s something to ponder; something to think about. The Editorial Board 

What is e^{-i\pi} ?

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

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

“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.

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 conveniently lie on the edge (or perimeter) of a circle of radius one. This happy fact makes understanding the expressions easy.

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

The number next to the letter “i” is simply the angle in radians where the answer lies on the circle. Draw a line from the center of the circle at the angle 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 180 degrees, right? The minus sign is simply a direction indicator that says to move clockwise around the unit circle—instead of counter-clockwise if the sign was 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 degrees 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. The value you land on is (-1 + 0i), which is -1. The answer is minus one.

Imagine that the number next to “i” is (π/2) radians. That’s 90 degrees, agreed? The sign is positive, so trace the circle 90 degrees 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 the number is written (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 is 60 degrees, right? The cosine of 60 degrees is 0.5 and the sine of 60 degrees 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 answer is always found on the circle between 0 and 2π radians (or 0 and 360 degrees) 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. “r” equals “one” in a unit circle. The Editorial Board

Some readers might wonder about what radians are. 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) and will completely span the perimeter (or circumference) of a circle.

A radian is about 57.3 degrees of arc. Multiply 3.1416 by 57.3 to see how close to 180 degrees it is. I get 180.01… . The result is really close to 180 degrees 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 degrees. The number “i” is always located at 90 degrees on the unit circle by definition, right? By the rule, multiplying “i” by “i” rotates it another 90 degrees counter-clockwise, which moves it to 180 degrees on the circle.

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

So yes, absolutely, “i” multiplied by “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 concept of two-dimensional numbers was too radical at the time for anyone to believe that numbers could exist on a plane as naturally as they do on a simple number line of one dimension.

Visit my website for an illustrated and concise explanation of these topics titled, “What is Math?” It is easily found by making a simple entry into the search box on theBillyLeePontificator.com .

Comments
J.G.
Thanks, I was confused about the -i part, so it’s not – sqrt(-1)? My calculator gives me an answer of -1.

S.X.
Good answer, I approve 🙂

B.L.
Thanks!

Billy Lee

TRUTH

Truth 3

Is it possible for humans to tell the truth always; to never lie?  Psychologists say it is not possible; most reasonably informed people agree. It’s a trait that can distinguish humans from some forms of artificial intelligence, which engineers at Google and other companies are working furiously to bring on-line.

People’s ideas—their belief systems—are inconsistent, incomplete, and almost always driven by logically unreliable, emotionally-laden content, which is grounded in their particular life experiences and even trauma. Does anyone disagree? 

Cognitive dissonance is the term psychologists use to describe the painful condition of the mind that results when people are unable to achieve consistency and completeness in their thinking. Every person suffers from it, to one degree or another. An unhealthy avoidance of cognitive dissonance can drive people into rigid patterns of thought. Political and religious extremists are examples of people who probably have a low tolerance for it.  

Kurt Godel
Kurt Friedrich Gödel (1906-1978) mathematician, logician, philosopher. Kurt trusted no one but his wife to feed him; not even himself. He never ate another meal after his wife died. He starved to death.

Decades ago, mathematicians like Kurt Gödel proved that any math-based logic-system that is consistent can never be complete; it always contains truthful assertions—including but not limited to foundational truths, called axioms—which are impossible to prove.

Whenever an idea or a conjecture that seems to be self-evident can’t be proved, it seems reasonable, at least to me, to assume that some people might feel compelled to disbelieve it; they might even believe they are trapped in what could turn out to be a lie, because no one should be expected to embrace a set of unprovable truths, right?  Axioms that can’t be proved are nothing more than assertions, right? Certainly any theorems built-up from unprovable assertions (axioms) must carry some inherent risk of falsifiability, shouldn’t they?  

Someone unable to convince themselves that an assertion or axiom they believe is true actually is true might necessarily feel uncomfortable; even incomplete. Folks often teach themselves to not examine too closely those things they believe to be true, which they can’t prove. It helps them avoid cognitive dissonance.

I’m not referring to science by the way. No easy way exists for non-technical folks to confirm claims by scientists that the earth is round, for example. The earth looks flat to most people, but scientists who have the right tools and techniques can reach beyond the grasp of non-scientists to prove to themselves that the earth is round.

Reasonable people agree that the truth of science is discoverable to any group of humans who have the resources and training to explore it. Most agree that those who are scientifically well-qualified are fully capable of passing the torch of scientific truth to the rest of humankind.

But this essay isn’t really about science; it’s about truth itself—a concept far more mysterious and elusive than any particular assertion a scientist might make that the Earth is not the center of the Universe, or that the Moon is not made of cheese.

All logically consistent ways of reasoning that we know about have been invented—some say, discovered—by human beings who live on planet Earth. Humans can and often have argued that the unprovable assertions which form the basis of any consistent way of thinking are an Achilles heel, which can be attacked to bring down whatever logical structure has been erected.

But it isn’t only the few foundational axioms of any mathematically logical system that are by definition true, but unprovable. Very complicated conjectures about the nature of numbers, for example, are always being discovered, which everyone believes they know to be true, but will never be proved, because they can’t be.

Freeman_Dyson
Freeman Dyson, British mathematician and physicist (born Dec 15, 1923)

Freeman Dyson—one of the longest-lived and most influential physicists and mathematicians of all time—argues that it is impossible to find a whole (or exact) number that is a power of two where someone can reverse its digits to create a whole number that becomes a power of five.

In other words 2^{11} = 2048, right?  Get out the calculator, those who don’t believe it. Reversing the digits to make 8402 does not result in an exact number that is a power of five. In this particular case, 8402^{1/5} = 6.09363... plus a lot more decimals. It’s not a whole (or exact) number. Not only that, no matter how many decimal places anyone rounds-off 6.09363... to, the number raised to the power of five won’t return 8402 exactly. A calculator will confirm it.

Dyson asserts that no number which is a power of two can ever be manipulated in this way to yield an exact number that is a power of five, no matter how large or unlikely the number might be. Freeman Dyson and all other super-intelligent beings—perhaps aliens living in faraway galaxies—will never be able to prove this conjecture, even though they all know for certain inside their own logical brains that this particular statement must be true.

All logically consistent methods of reasoning which can be modeled by simple (or not so simple) mathematics have these Achilles heels. Gödel proved this truth beyond all doubt; he proved it using a method he invented, which allowed him to circumvent the dilemmas posed by the unprovable truths of the system of thinking he contrived to demonstrate his discoveries.

I’m not going to get into the details of Gödel’s Incompleteness Theorems; books have been written about them; most people don’t have the temperament to wade through the structures he built to make his point. He basically assigned simple numbers to logical statements—some being very complex statements encoded by very long strings of numbers—so that he could perform garantuan operations of logic using rules of simple arithmetic on ordinary whole numbers. Take my word, his method requires traveling over unfamiliar mathematical roads; it takes getting used to.  

But it should amaze non-mathematicians that truths abound in mathematics, which not only haven’t been proved; they never will be proved, because no proof is possible. A logical path to the truth of these statements does not exist; indeed, it cannot exist. But it is useful and necessary to believe or at least accept these statements to make progress in mathematics.

Capture
Paul Joseph Cohen (1937-2007) Stanford mathematician

The late mathematician Paul Cohen—at one time a friend to Gödel—said that Gödel once told him he wondered if it might be true that any and all conjectures in mathematics could be solved if only the right set of axioms could be collected to construct the proofs. Cohen is best known, perhaps, for showing that indeed—in the case of the Continuum Hypothesis, at least—he could collect two reasonable, self-evident, and distinct sets of axioms that led to logically consistent and useful proofs. One small problem, though—the proofs were completely contradictory. One proved the conjecture was true; the other proved it was false.

His result is often explained this way: the consistency of any system of mathematical reasoning cannot be proved by its foundational axioms alone. If it can, the system must necessarily be incomplete; its conjectures—many of them—undecidable.

Cohen showed that a consistent and sound axiomatization of all statements about natural numbers is unachievable. Many such statements, in his view could be true, but not provable. Cohen introduced the concept that all systems of logic built on numbers have embedded within them some combination of ambiguity, un-decidability, inconsistency, and incompleteness.

People who want their thinking to be consistent must believe things that cannot be proved. But believing logical statements that can’t be proved always renders their thinking incomplete, even when it might be flawlessly consistent. What they believe to be true depends fundamentally on what they believe to be self-evident: it depends on statements no one can prove: on axioms, and a little bit more.

For those who decide to believe and accept only statements that can be proved, their thinking will necessarily unravel to become inconsistent or incomplete; most likely both. Their assertions become undecidable. It can’t be any other way, according to Gödel, whose proof has withstood the test of eighty years of intense scrutiny by the smartest people who have ever lived.

Paul Cohen jumped onto the dilemma-pile by showing that the incompleteness made necessary by a particular choice of axioms can turn a logically consistent proof to rubble when a mathematician tampers-with or swaps-out the foundational axioms. A sufficiently clever mathematician can prove that black is white; and vice-versa.

Some might be tempted to say that Gödel‘s Incompleteness Theorems apply only to formal, math-based logic-structures—not the minds of human beings, because minds and the way they operate when analyzed are always found to be inconsistent and incomplete. But such talk makes the point. Think about it.  

paradox

So again, we ask: what is truth?  How do folks determine that a particular statement is true, if it happens to be one of those assertions that lies beyond the reach of logic, which no one—no matter how smart—will ever be able to prove? 

What good do collections of so-called self-evident axioms serve, if different collections can lead to contradictions in theorems?

Most important: how do we avoid believing lies?

Billy Lee

Post Script: This post is one of those living-essays Billy Lee writes that is likely to be modified and added to as he thinks about the subject and learns more. Check back for updates from time to time.

Click on this link to view a short movie clip, where Jesus, played by Robert Powell, is asked by Pontius Pilate, What is truth?  The Editorial Board

[added April 3, 2016]  I found a 2013 essay by Derek Abbott, the Australian Electrical Engineer and Physicist, who argued that our mathematics is invented, not discovered; anthropological, not universal. It enables us to simplify truth to enable our limited minds to find ways to do and understand simple things. Click this link for a good read. 

I don’t know if Derek’s view is correct, but offer it as fodder for readers who are interested in why Truth and mathematics seem connected somehow in the minds of so many thinkers like Plato, for example; and why these thinkers could be dead wrong, at least in Derek Abbott’s opinion. He offers Clifford’s Geometric Algebra as a practical example of a useful system of mathematics used by robotics engineers, which is built very differently than the mathematics most people use.  

[added February 20, 2017] I will offer this opinion: if mathematics is anthropological; if math is just another way our minds work and not the golden key to a deeper reality beyond ourselves, then it can tell us nothing new about the mysteries of existence; we are not going to be able to calculate our way along a path to truth; we will have to do the hard physical experiments to figure out just what the heck is going on.

Based on what the smartest humans know to be true today, we can’t build the kind of instruments required to answer the mysteries of the very large and the very small. Getting answers will take detectors the size of galaxies; it will demand the energy supply of thousands of suns.

If mathematics lacks a symbiotic connection to realty; if God is not a mathematician, if He doesn’t play dice after all, as Einstein insisted, well, we won’t get to a deeper understanding of how the universe works—or why it exists—through clever use of mathematics. It just isn’t going to happen—not now; not anytime soon; not ever.

The 18th century German playwright and philosopher, Friedrich Schiller, wrote …truth lies in the abyss.  

Pray that he’s wrong.

Billy Lee

Microwave Coffee

coffee 2
I drink the high octane stuff.

My wife and I are coffee drinkers. Beverly Mae used to drink a dozen cups of fully caffeinated coffee everyday.  By her third cup she could boogie with the best of them.  But that was a long time ago, before we got old. Now she drinks about two cups, and it’s decaffeinated. Me, on the other hand, well, I still imbibe the high-octane stuff.  I love it.

69050_Coffee Maker.37173901_std
My wife and I have one coffee maker between us. It’s not enough.

 

If your marriage is anything like ours, you probably own one coffee-maker, but you and your spouse drink different brands, flavors or styles of coffee.  For us, it means we have to store the contents of at least one of our coffee-pots off-site, away from the coffee-maker, in containers and carafes; perhaps cups or bowls or glasses or whatever is handy.  The coffee gets cold.

We have two microwaves. It is one of the very few things which seem to have made a difference in our marriage.
Two microwaves: they can sometimes save a marriage.

Only one of us at a time can store coffee in the coffee-pot.  But since we have two microwave-ovens, we don’t really need to keep our coffee hot.  We can turn off the coffee-maker after we brew each pot to save energy.  And so, everyday, we reheat our coffees in the microwaves, more than once, and thanks to having two of them, we don’t wait in line.

Beverly Mae and I have an eclectic collection of coffee mugs gathered together over decades of marriage. I’ve often wondered: how is it that no matter how big the coffee cup or how tiny; how robust the mug or how dainty; how full or empty we fill—or even which microwave we choose—my wife and I almost never set the timer to reheat our coffee more than once? We always seem to set the microwave to exactly the right number of minutes and seconds to heat our coffee to exactly the right temperature.

coffee 4
                She got the calculation right.

Think about it. Can there be any doubt that the mathematics required to accurately set the timer must be beyond the capabilities of 99% of the people who set these timers to reheat their coffee everyday?

The size of the cup, its thickness and material; the amount of coffee in the cup—these are important variables that are required to be taken into account when setting the timer. Not only these variables, but there is the subjective calculation: how hot do I want this coffee to be today? Real hot? Tepid? Mildly warm? There are many tricky variables to track and put into an equation. And, if we are reheating coffee for our spouses, we have to anticipate their calculation of what the best temperature is for their mood and state of mind.

differential equations EngMath_DifferentialEq_Terminology_02
It takes a matrix of differential and difference equations plus a super computer to get microwave coffee right.

It really takes a sophisticated matrix populated with complex differential and difference equations to work out what the results might be under all the possible scenarios. And it might require a government super-computer to crunch the numbers. Of course, I’ve never done the actual work of creating the matrix or the equations. Even if I had, I would have faced the daunting task of isolating all the relevant variables, the tedium of tracking all the units to make sure I ended up with seconds only in the answer, and the exhaustive testing of results to see if they match up with my expectations and experience. Successful creation and application of a workable calculus might involve a lot of tweaking.

drinking coffee
He didn’t bother with the calculation.

Come to think of it, why would I do that? Guessing seems to work better and it’s a lot faster. But I wonder. Am I really guessing? Or is my brain, somehow, doing the math in some far away place inside my brain, behind the scenes and beyond my conscious scrutiny? It’s kind of mysterious, being right all the time, about something as complicated as getting the number of minutes and seconds correct when setting the timer to reheat my coffee.

And my wife, who knows no math, is just as good at the mental calculation as me. Go figure.

Billy Lee

P.S.  Readers take note: on Valentines Day, 2015, Billy Lee bought a second coffee-maker for his wife, Beverly Mae. Why it took so long for Billy Lee to solve his coffee-problem is a mystery even skilled mathematicians won’t solve.  The Editorial Board.

microwave coffee math graphic 6
Don’t try these calculations at home. Seriously. This man tried; his face froze—for two hours.

What is Math?

math with color

People seem to think mathematics is something special—a kind of magic language that, when tinkered with properly, makes it possible for mortals to stumble upon answers to all kinds of mysteries about the Universe hitherto known only to God.

I see it differently. Mathematics isn’t a language per se. Although mathematics can be (and is) explained by language, math itself is a collection of rules and symbols that makes it possible to avoid the encumbrances, flourishes, and ambiguities of language. It accomplishes this feat by defining things and their relationships in strictly limited—but important—ways.

euler formula hatEuler Identity – Khan Academy

Math involves symbols and rules that aren’t explained inside the equations. It is the lack of words that gives math its mysterious and magical reputation. But once everything is defined and understood, applying the contrived but logical rules of mathematics enables folks to manipulate equations to uncover previously hidden and non-intuitive relationships among the things they have defined.

What am I saying exactly? I am saying that it is possible to use words alone to describe the process of solving and manipulating an equation, which can lead to insights into the relationship of the things in the equations. But these words will make the process of computation cumbersome, impractical, and confusing.

Spoken language contains noise and nuances that interfere with the manipulation of carefully defined relationships between narrowly defined variables. Yes, the no-nonsense logic and bare bones precision of mathematics, as well as the reduction of things to a few carefully chosen attributes, enables mathematicians to apply rules to discover consequences that might otherwise remain undiscovered.

But the tightness of mathematical construction makes it a tool which is almost useless for describing and analyzing many subtle yet vivid experiences of a conscious mind—like beauty, the feel of an orgasm, or the experience of grief. For these realities of conscious experience, mathematics has a reputation for being irrelevant.

euler ring     Euler Identity – Wikipedia

Spoken language gives conscious humans the messy modeling mechanism they need to connect with each other to share and understand the more nuanced experiences of life. The messiness and ambiguity of spoken language makes the unique intimacies of human communication possible. Mathematics, despite its elegance, doesn’t do intimacy well.

The Euler Identity, illustrated above, is sometimes presented as an example of the mysterious power of mathematics. But if you think about it, what does the equation say?  It says that minus one plus one equals zero.

Complex PlaneThe explanation is easy.   -1 can be rewritten as e raised to the power of i times π because of simple rules, which place on a circle of radius 1 all the values of e raised to the ith power times anything.  

The number that sits next to i is the angle in radians where the result lies. In this case, an angle of π radians (180°) takes us from the value 1 (at 0°, or 0 radians) to half-way around the circle to the value -1. 

Despite the reputation of equations for precision, it turns out that physicists and other scientists struggle to make mathematics match the results of real world measurements. The non-technical public is unaware, for the most part, that astronomical observations involving the movement of stars, planets, and other celestial bodies—or the results of observations made of the subatomic world (no matter how carefully contrived)—as often as not fail to provide results sufficiently in agreement with mathematics to be of any practical use.

Fudge factors are a big component of doing real science. People have won Nobel prizes for inventing fudge factors to fix things. It’s true.

Renormalization, perturbation theory (for phenomenon both small and large), Green’s functions, propagators, Feynman diagrams, and many other adjustments and tweaks make up the contortions and modifications that scientists overlay onto their beautiful equations to make them work.

They say they have good reasons for all the tinkering; it’s complicated down there among the quarks or up there, among the quasars; there are nuances and messiness and ambiguity in the underlying reality of nature that no one can see or fully understand—not now; not anytime soon; perhaps not ever.

At subatomic scales, a tangled mess of virtual particles which come into and out of existence more or less spontaneously often gets the blame for the mismatch between mathematical elegance and the cold reality of experimental results. On the scale of the universe, dark matter and energy (which can’t be detected) are sometimes blamed for anomalies. Click on the links in this paragraph to learn more.

It’s possible that no system involving mathematics can be contrived by humans to bring the satisfaction of knowing everything for certain; nothing we are able to invent will bring a tranquil end to the pain of cognitive dissonance that seems to drive our species to wonder and explore.

The more deeply people travel into the complexities of mathematics and science the more elusive truth becomes; perhaps God is not a mathematician; maybe Einstein was right when he said, God does not play at dice. A die is cast into the lap, yes, but its decision is from the LORD, according to an old proverb of Solomon.

Can it really be true that understanding the world is beyond the limitations of all life on the earth—beyond the abilities of the most brilliant minds that have lived or ever will live?

If so, it’s time to kneel.

Billy Lee