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:
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).
Often called the unit of the “imaginary numbers,” is equal to the square root of negative one (). 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 , 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 represents in our equation.
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, 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…
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 (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
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…
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?
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!