About that equation…

In my post of February 17, 2007, I mentioned my favorite equation and said that I hoped to get around to explaining it “next week.”  Well, apparently “next week” was a mistranslation and I meant “next month.”  Seeing how today, 3/14, is Pi Day (and thanks for the card I have received!  You know who you are!), it seems a fitting day to follow up on that task.  For those of you who find things mathematical distasteful, I give you full permission to ignore this post today.  (Though, please do not take offense if I skip your post on the history of French cinema, as well.)

Here is the equation under discussion:

e^{i\pi}+1=0

So, what’s so special about it and why am I so mushily fond of it?  Well, let me summarize: this one, simple equation contains what are arguably the five most important numbers in mathematics (representing in an ambassadorial sort of manner four fundamental branches of mathematics), the three most important operations, and the single most important relation AND it contains nothing else — just those nine elements mixed together beautifully and elegantly.  Here’s a break down, beginning with the numbers:

1 & 0

These numbers, one and zero, are most likely the most familiar to us (if not, then what planet did you grow up on?).  They represent Arithmetic, and are arguably the most important numbers in that branch of mathematics — fundamentally necessary, yet sufficient (in fact, the binary number system on which our digital civilization is founded ultimately rests on the back of the abundant use of these two digits and none others).

i

Often called the unit of the “imaginary numbers,” i is equal to the square root of negative one (\sqrt{-1}).  The term imaginary comes from the fact that all “real” numbers produce positive results when you square them, due to the fact that not only is it true that “a positive times a positive is a positive” (e.g., 4 × 4 = 16), but also “a negative times a negative is a positive” (e.g., (-4) × (-4) = 16).  [Many math teachers, to be sure, hate this “rule” because they don’t know how to explain why it is true, but it doesn’t change the fact that it is, indeed, true!  Perhaps I will explain why it is true in another post, in the event some of you are having a hard time falling asleep…]  Anyway, if all the “real” numbers only produce positive numbers when you square them, then it must that only “imaginary” numbers produce negatives when you square them.  Thus, since by definition i^2=-1, i and its cohorts have been forever labeled as “imaginary.”

Such a moniker gives a false impression (Gauss suggested replacing “imaginary” with “lateral,” and I have used the same mental shift — thinking of a difference of direction as opposed to “reality” — to help make imaginary numbers more real to my kiddos).  “Imaginary” numbers are just as much a part of our world as “real” numbers, though you often don’t become aware of their “reality” until you get into some higher physics.  Until that time, they simply represent one more thing to make high school algebra more complicated, and it is this branch of mathematics, Algebra, that the number i represents in our equation.

\pi

This number, pi, is likely familiar to the majority of us, even if it does bring back fearsome flashbacks of high school geometry and “proving stuff.”

Unlike the “imaginary” number discussed previously, \pi probably needs very little explanation.  It is, very simply, the number obtained when one divides the length around a circle by the length of its diameter.  Doing this with any circle whatsoever results in the exact same number, whether big or small and regardless of what unit you measured in (inches, centimeters, light years, etc.).

Often “memorized” as 3.14 or 22/7, these expressions are only approximations, since the true value of pi cannot be written as a fraction and cannot be written out using our happy collection of ten digits.  It’s decimal expression goes on forever, beginning with 3.14159265358979…, never ending and patternless.

Its importance to mathematics — and to humanity, in general — is unquestioned, and it stands in our equation as an ambassador for Geometry: that branch of mathematics dealing with spatial relationships.

(As a side note, the St. Louis Post-Dispatch mentioned, today, that pi is approximated in the Bible as 3.  However, they say that this is done in “God’s instructions to Noah about how to build the arc (sic)”.  For one, this is incorrect — it is in the description of the Temple’s “big tub” in 1 Kings 7:23 (width is given as 10 cubits and distance around is given as 30 cubits: 30/10 = 3), and not in God’s instructions to Noah (unless I missed that part).  Secondly, the misspelling of “ark” is funny, since if God really had told Noah to build an arc — which is a section of a circle — then He might, indeed, have provided an estimate of pi. 🙂 )

(And, as a side note to my side note, there are a number of reasons why the approximation of pi in 1 Kings 7:23 is of such limited precision (e.g., 3, instead of, say, 3.1 or 3.14).  However, it is interesting to note that the three letter Hebrew word translated “line” in 1 Kings 7:23, QVH, is normally accompanied in Hebrew texts with a parenthetical or marginal reading of the word that contains only two letters: QV.  When a ratio is created by using the numerical values of each of these words (since all Hebrew letters have numerical values) we get 111/106, which when applied as an adjustment factor to the pi value of “three” that the text seems to imply, we get a much better approximation of pi: 333/106 = 3.1415…  This value is MUCH more accurate than any other ancient civilization ever had (including those smartypants Greeks).  Coincidence?  I’ll leave that to you to decide.  There is a nice paper on the matter here, for those not already bored out of their skulls…)

Finally, my favorite number…

e

The number e is, like pi, a number that cannot be written like a fraction and has a decimal expression that continues forever, without pattern.  It begins 2.718281828459045…  (My kids understand that if you had e dollars, you’d have more than $2.71, but less than $2.72 🙂 )

It is hard for me to explain the significance of this number in the time (and blog space) I wish to allot myself, today, but it is huge.  We are surrounded by this number, though most of us do not recognize it.  Were there justice in the world, we would be as familiar with e as we are with pi, but pi has a much better Public Relations staff, apparently.  (Although some have tried to make up for this, and there is a great book you can read on the matter by Eli Maor titled e: The Story of a Number, and it is more entertaining to read than the title might lead you to believe.)

e shows up in the application of compound interest, the graceful curve of a hanging chain, the familiar spiral of a nautilus shell, and in the area of the space under a hyperbola.  For the calculus-minded, the function e^{x} (and its multiples) is the only function in mathematics that is its own derivative.  Really, I cannot do it justice here, and I may try one day when I have more time (and energy).  Until then, those whose curiosity is strong enough to propel them forward (but not enough to motivate them to check out Maor’s excellent book) are invited to look at the Wikipedia entry on e, here, or to the Math Forum FAQ entry, here, or to whatever resources you can dig up yourself.  [I do love this number!  And I may write on it more one day, but today’s not that day…]

As it is in calculus and the branch of mathematics known as Analysis that the number e really shines and displays its glories and its fundamental significance to mathematics (and, therefore, the universe), it is Analysis that e represents in our equation.

And that’s the numbers.  Concerning whether or not these five constants truly are the five most important numbers in mathematics, as Dr. Jerry P. King suggests in The Art of Mathematics, “[t]here is no doubt of this; just stop any 100 mathematicians and ask them.” 🙂

With the numbers out of the way, the last two points should go more quickly…

  • The three most important numerical operations in mathematics which just so happen to appear in the equation are these: addition, multiplication, and exponentiation (raising to a power).  You might wonder where subtraction, division, and taking roots are in that list, but they are there, too, since they are merely the evil twins of addition, multiplication, and exponentiation.  The equation displays all three of these, in that i is multiplied times pi, e is raised to the power of that product, and the result is added to 1.
  • The most important relation in mathematics? Equality: that happy little equals sign, “=”.  Where would we be without it?  We certainly wouldn’t have any equations to talk about, which I suppose is immediately obvious…

And all of these things are, in this one equation, united beautifully and simply.  That there is nothing in addition to these things only adds to the elegance — as if you had not only created the most beautiful quilt ever crafted, but as you turn around to clean up your mess you find that you had used every scrap of material you had, with nary a thread left lying.  You needed nothing more, and you used nothing less, as if the quilt was somehow meant to be.  How remarkable.

To me, the seemingly disparate natures of the numbers, too, add to the wonder of it all.  I mean, looking at 1 & 0, e, i, and pi there is, on the surface, nothing at all to suggest that they relate to each other in any special way at all.  And yet they are intimately related.  As if you plucked four or five individuals randomly out of the vast mass of mankind scattered all over the face of the earth, only to find that they were all siblings, separated at birth long ago by forces of time, circumstance, and culture.  Would you believe that your selection of these people was a coincidence?  I think not.

Lastly, this equation is my favorite because it is the discovery of my favorite mathematician, Leonhard Euler.  Explaining why he is my favorite mathematician would be another blog entry in itself, but suffice it to say that he was remarkable, and one of the greatest mathematicians the world has seen in the world of men.  It was he who discovered the relationship of e to the trigonometric functions and the imaginary unit in the equation

e^{ix}=\cos\theta+i\sin\theta

which, after replacing θ with pi and doing a bit of aesthetic rearranging, results in the equation I have been blathering on about.

Let’s look at it one more time…

e^{i\pi}+1=0

Isn’t it pretty?

Okay, you can wake up now!  I return you to your regularly scheduled (less mathy) surfing.  (And I am not responsible for any accumulation of “nap drool” in your keyboard…)

** EDIT, 3/15/2007, 1:47am:  I can’t believe I didn’t mention this!  e is also related to the famous St. Louis Gateway Arch, which my family and I drove by a few hours ago.  The arch is a caternary, which is the shape of a hyperbolic cosine curve (y = cosh x).  And what is a hyperbolic cosine?

\cosh x = \dfrac {e^x + e^{-x}}{2}

So, if you’re crossing the Mississippi and admiring the engineering behind the St. Louis Arch, you can also think to yourself, “Wowie… Wow… e…”  OK, that might be weird.  But still: you can remember that e is in there somewhere!

13 thoughts on “About that equation…

  1. Tim Wilson

    Mr. Smith,

    Thanks for the math blog. You and my wife have inspired me to write about it too. My wife (the former Jessie Ostrom), has the Euler equation on a homemade bookmark with the quote from Benjamin Peirce concerning it:

    “Gentlemen, that is surely true, it is absolutely paradoxical; we cannot understand it, and we don’t know what it means. But we have proved it, and therefore we know it must be the truth.”

    The quote always bugged me. I was taught to look at complex numbers as coordinates on perpendicular axes. So, 5 would be five units on the horizontal axis, 4i would be four units on the vertical axis, and i*pi would be 3.14 (or so) units on the vertical axis. 5 + 4i would be five on the horizontal and 4 on the vertical.

    I was also taught to look at complex numbers as a vector rotated around counterclockwise from the real axis. So 4i would be vector of length 4 and rotated 90 degrees. 1 + i would be a vector of length one and rotated 45 degrees. cos(theta) + i*sin(theta) would be a vector of length one rotated theta degrees. Any number multiplied by i would amount to a counterclockwise rotation by 90 degrees.

    The derivative of cos(theta) + i*sin(theta) is -sin(theta) + i*cos(theta). Or in other words, the derivative of that combination amounts to a counterclockwise rotation of 90 degrees. Also, the derivative of e^ix = i*e^ix. So, e^ix is the same as cos(x) + i*sin(x).

  2. Immensely cool, Sir.

    But wait: “*e* shows us…in the spiral of a nautilus shell…” That’s (1 + 5^-2)/2 or phi (1.6180339…), isn’t it? That number shows up all over the place too. If memory serves there is even an elegant little equation in which e and phi are related. (Don’t quote me on that; having lost so many of my paperbacks while moving here, I may never find it. Not even through Google have I found it yet.)

    Here is the Wikipedia article on phi. It cites that book I mentioned earlier, among much else:

    http://en.wikipedia.org/wiki/Golden_ratio

    Phi has the honor of being the most irrational number, despite its being so prominent in nature and art (go figure).

    Wikipedia’s page on e is interesting too, and unsurprisingly it confirms what you say here:

    http://en.wikipedia.org/wiki/E_%28mathematical_constant%29

    > The number e is one of the most important numbers in mathematics, alongside the additive and multiplicative identities 0 and 1, the imaginary unit i, and π, the circumference to diameter ratio for any circle.

    But the article is almost impossible for me to understand, because I don’t know calculus.

    I notice that in various Web sites, pi, phi and e all have their partisans. Here is one of the more intense partisans of phi:

    http://goldennumber.net/

  3. Michael

    I wouldn’t call it pretty. “Gorgeous” comes to mind, though. And thank you for the link to the paper on pi and the Bible. I was asked about this a few months ago, and only was able to come up with an attempt to show that the approximation is well within measurement uncertainty for the units given (and besides, if the Bible wrote an exact answer, it’d be three times longer… though we would have saved many years of computation time). I also have to agree with your comment on Euler. Working with the Bernoulli family, Euler laid the foundation for fluid dynamics (and where would I be without that??).

  4. Messrs Wilson, Wheeler, and Dorothy —

    Thanks so much for your comments! I wasn;t sure what kind of traffic a post like this might get, but I am *very* glad that you enjoyed it. Let me respond to each of your comments just a bit:

    Mr. Wilson — Thanks for your discussion of how you understand i. I agree that thinking of i as a unit on a vertical axis (as opposed to 1 being a unit on the horizontal axis) is a great way to get away from the feeling that i is purely “imaginary” and without ground in reality, and that is the way I have approached it with my children. You did make one little mistake, which I have often made myself: the vector 1 + i has a length of 1.414… (the square root of 2) instead of 1. Still, your examples were great — thanks for writing!

    Mr. Wheeler — Yes, I knew you were quite a phi fan! (Or should I say phi phan?)

    Concerning the nautilus shell, let me say a couple of things that might clear things up: (1) The golden spiral (based on φ) is a specific kind of logarithmic spiral and is, thus, also based on e in an elementary way*. (2) A nautilus shell, however, while generally following a logarithmic spiral, is not always a golden spiral (just as not all rectangles are squares). A reference on this would be the last paragraph on the Wikipedia entry on “Golden Spiral”: http://en.wikipedia.org/wiki/Golden_spiral.

    *[All logarithmic spirals fit the polar coordinate format r = ae^{bθ}, where a & b are curve-specific constants. In the case of a golden spiral, b = ln{φ × 2/π},which would mean the golden spiral’s equation simplifies to r = aφ^{2θ/π}.]

    Mr. Dorothy — Yes, those Bernoullis… What a wacky couple of kids! There is a quote that I will dig up one day that says something to the effect that if convention dictated that mathematical results and theorems must be named after the mathematicians who actually discovered them, then 1/3 of all theorems would be named after Euler. I really will have to write about that fellow some day, and I very much look forward to meeting him in the resurrection. (Actually, I might include something about the Bernoullis, too. One of my all time favorite quotes comes from one of them (I forget which brother) concerning Isaac Newton.)

    Really, thanks guys for writing in. I appreciate it!

    Best regards,
    Wallace Smith

  5. I don’t know about all this “proof” stuff. I entered -1 on my calculator and hit the Sqrt button and all I got was an error. 🙂

  6. Doug Young

    John Wheeler, with his Hebrew knowledge, could confirm this. Chuck Missler says that using the numeric values of the Hebrew letters in Genesis 1:1, take the product of the letters and divide by the product of the words, you get pi to 4 decimal places. When you go to the other creation verse, John 1:1 and use the numeric values of the Greek letters, take the product of the letters and divide by the product of the words, you get e to 4 decimal places. If true, it is stunning.

  7. One thing Mr. Missler wrote about pi relates to another verse entirely, 1 Kings 7:23:

    http://www.khouse.org/articles/1998/158/

    Mr. Smith has already mentioned this particular point somewhere.

    OK, Mr. Smith has made us dizzy with mathematics; now it’s my turn as a Hebraist. The written text has (וקוה) — this is the *ktiv reading. In Masoretic Hebrew this would normally be pronounced *weqaweh, “and a line”. However, the marginal note gives a *qre reading, telling the reader to pronounce the word as if respelled (וקו) and pronounced *weqaw, “and a line of”. But there is a further complication in that the *ktiv may simply reflect an archaic spelling pronounced as *weqawo, “and its line”. In any case these various readings have nothing provable to do with math (let alone a latter-day idea of *remez or “something deeper”), but only with orthographic convention and considerations of grammar.

    The real solution to the “contradiction” (as scholars have acknowledged before now) has to do with the fact that the wall of the basin was a handbreath thick (verse 26). A handbreath was one-fifth of a cubit. There would’ve been two measurable radii for the sea of bronze, the inner one being 4.8 cubits and the outer being 5 cubits (the Bible gives the second radius). Apparently the circumference was measured at the cup-like brim, at the inner radius (which makes sense given the volume of “two thousand baths” given in the same context). 2 * pi * 4.8 = 30 cubits to two significant digits.

    It seems to me both unnecessary and unlikely that the Bible, in its *ktiv and *qre readings here, would have encoded a value for pi with a margin of error “15 times better than the 22/7 estimate that we were accustomed to using in school”. It is so easy to come up with “significant” results when assigning numerical values to the Hebrew letters that I am extremely skeptical of *any* claims in that direction. (I wonder if Mr. Smith has heard of the statistical fallacy of “the enumeration of favorable circumstances”? I suspect it applies here.)

    And then there is another page, on which Mr. Missler addresses pi and e:

    http://www.khouse.org/articles/2003/482/

    The formula he applies in both Genesis 1:1 and John 1:1 is:

    The number of letters x the product of the letters
    ——————————————————
    The number of words x the product of the words

    Thankfully, he gives charts so I don’t have to work this out by hand.

    For Genesis 1:1 (he says),
    > You get 3.1416 x 10^17. The value of p to four decimal places! Hmm.

    For John 1:1 (he says),
    > You now get 2.7183 x 1065, the value of e. Curious!

    > Each of these is another of those puzzling ostensible “coincidences” that are too astonishing to dismiss, and yet present challenges in suggesting any real significance. And taken together, they do evoke some conjectures. There are, however, at least two problems: why just four decimal places (they both deviate from the fifth place onwards) and what do you do with all the “extra zeroes”?

    > I frankly don’t know. Nevertheless, I thought it would be an excellent conversation piece as we return to our academic schedules this month. The rabbis would suggest that each of these may simply be a remez, a hint of something deeper.

    > Let me know if you have any suggestions. Meanwhile, let’s continue to praise our Creator-Savior for His marvelous Word!

    Indeed…and thanks to Mr. Missler for being humble enough to admit what he doesn’t understand.

  8. Howdy, Mr. Wheeler —

    I feel compelled to say that when I gave a sermonette once on the topic of “Pi, the Big Tub, and the Bible” I did refer to the handbreadth explanation and demonstrated that this could easily reconcile the figures.

    However, the odd coincedence that within a verse that is actively discussing a circumference and a diameter there should be such a simple means of calculating one of the great and rational approximations of pi is startling. Whether startling in the manner of finding something wonderful that has been purposefully placed to be found or startling in the manner of encountering such an improbable-yet-still-random occurance is for others to determine.

    The question at hand is whether or not such a meeting of pi, circle, and diameter in 1 Kings 7:23 is a coincedence is a “planned placement” is not something easily decided beyond a shadow of a doubt. I would say that since there are grammatical reasons for including the variant reading, the probability of “planned placement” is reduced, yet it cannot be eliminated by such a consideration.

    The “pi in Genesis 1:1” and “e in John 1:1” seem (to me) to fall quite quite clearly into a category where one can dispute their significance. Perhaps if the powers of 10 had been identical, it might be more convincing, or if numerous verses relating to the creation had been found and every single one produced such fundamental universal constants upon performing the same calculations. Even then, though, the case could not be considered “air tight” though it would be more convincing. The fallacy you mentioned (yes, I am familiar with it, thank you very much) is certainly at work in such cases, and had e *not* been “found” in John 1:1, surely its absence would not have been mentioned.

    I find the “pi + circle + diameter” of 1 Kings 7:23 more intriguing because of the curious confluence of elements. Do I find it completely and unavoidably compelling? No, I don’t — I see no way of proving that a highly accurate approximation of pi in that verse has been purposefully placed. I am quite a skeptic when it comes to such things, and I place the bar quite high (or at least I like to think I do) for such proof. If the probability of such a confluence could be calculated accurately with reasonably solid assumptions, I would be interested in seeing it — however, I think such a calculation would be hideously error prone and the proper foundational assumptions would be almost impossible to agree upon. (The “Jesus Tomb” people might want to take a crack at it, as they seem a little more willing than I am to make unfounded assumptions.)

    This is why I introduced the “side note to my side note” with the phrase “it is interesting to note” — because it is just that. Whether it is actually “significant” and not just simply “interesting” is not for me to decide, and — as always — I appreciate your informed opinion, Mr. Wheeler. I think the paper to which I link in that section of the entry above does a rather good job of presenting the situation as well as drawing some good conclusions concerning whether or not it is a coincedence, and I would refer anyone trying to decide for themselves to that paper.

    Now, I wonder if the Bible verses about charging interest can be used to approximate e…

    Best regards,
    Wallace Smith

  9. Dear Mr. Smith,

    Thanks for your lengthy response!

    I’m not at all surprised that you know of “the enumeration of favorable circumstances”, and maybe my rhetorical question should’ve reflected my confidence that you do more clearly. Sorry about that. Likewise with the proper solution to the “contradiction” in 1 Kings 7:23 — it was directed not to you (for I would be very surprised if you had NOT worked this out or found someone who had), but to one of my fellow respondents.

    The basic problem with claiming that 1 Kings 7:23 contains a “code” enshrining a very accurate value for *pi* is that the set of *ktiv/*qre readings overall is younger than the Hebrew Masoretic Text itself, let alone its forebears of biblical times. The *qre readings were inserted in the margins of the Masoretic Text in the Middle Ages; we find no such readings in the Dead Sea Scrolls, and no reference to anything like them before Talmudic times. Mr. Missler’s argument makes the hidden assumption that both the *ktiv and the *qre readings are of biblical age. (Probably he has been reading too many rabbinic apologetics.) This assumption is certainly not testable, and it is almost certainly false. The only way that could be possible would be if two variant readings having the same meaning were transmitted in separate manuscript traditions and then combined in medieval times. We know this has happened in other cases; we have the “paper trail”, documented in the critical printed editions and manuscripts. But does it make sense that a “code” would be passed down by separate transmission lines and then “created” by a combination made long after the Bible was completed? I think not. As remarkable as the coincidencal value seems, it’s just that — coincidence. Such very interesting (but ultimately meaningless) coincidences are appalingly easy to find in biblical Hebrew, because of the peculiar nature of the language.

    I’ve done some more checking — in the New BDBG Lexicon, p. 874a, and in the Keil & Delitzsch Commentary on this verse. The *ktiv/*qre readings *weqaweh and *weqaw are said to mean the same thing, “and a line of”, so I correct myself there. Strong’s Dictionary confirms this. But the former *ktiv is found in three places in the Bible (Zechariah 1:16; this verse; and Jeremiah 31:39, Hebrew versification), while the other is much more common. In all three places the *ktiv is *weqaweh and the *qre is *weqav. Are we meant to conclude that all three places enshrine a very accurate value of pi? I think not. Rather, an archaic or rarer form is being updated with a more recent or more common form — nothing more. This again would’ve been done by the medieval scribes and rabbis, for purely lingustic reasons, for no connection with circles is found in the other verses.

    The only alternative I can see offhand is that the medieval scribes and rabbis themselves realized that the combined *qre/ *ktiv reading enshrined a very accurate value of *pi, and they happened to place that combination in this verse. But then, we should find elsewhere in rabbinical mathematics that same very precise value of *pi expressed, and we do not.

    Thanks again, and bye for now!

  10. Howdy, and thanks, again, Mr. Wheeler —

    Your comments about later additions & notations to the Hebrew text is to be noted, and I am always a bit leery about claims that such things are inspired, as should we all so be.

    I note that the mathematical paper to which I refer actually only briefly refers to the alternate reading and, instead, focuses on the different words used in 1 Kings 7:23 and 2 Chronicles 4:2 (which is the same pair of words). They, too, make a good case for the “neat coincedence” hypothesis, and they mention, as well, the other locations in which this pairing is made but there is no mention of circumference or diameter. It really does seem to give a nice, concise treatment of the matter, and I heartily recommend it. For my part, I will continue to mention it as an “interesting observation.”

    But if Simcha Jacobovici and company ever discover a “Lost Gospel of Euclid” I might have to reconsider my stance on such things…

    Warm regards,
    Wallace Smith

  11. Pingback: Did… Did you REALLY just ask that? SWEET!!! « Thoughts En Route

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