A quibble about an item from the Pi Day site, which I’ve seen quoted in a few places:
Pi is an irrational and transcendental number meaning it will continue infinitely without repeating.
That pi is an irrational number does mean it will continue infinitely without repeating. For all rational numbers, and only rational numbers, the decimal representation of the number will at some point start repeating and keep repeating, e.g., 3.1415926926926926926…. (If you’re wondering about a number such as 2.5, it’s actually 2.50000000…., or for that matter 2.49999999….)
But that pi is a transcendental number doesn’t quite “mean[] it will continue infinitely without repeating.” It’s true that all transcendental numbers are irrational, and therefore will indeed continue infinitely without repeating. But not all irrational numbers are transcendental, so a number’s being transcendental means something more than that it continues infinitely.
An irrational number (say, the square root of 2) is merely a number that can’t be represented as a ratio of two integers. But a transcendental number is a number that can’t be represented as a solution of any polynomial with integer coefficients: Square root of 2 is thus irrational but not transcendental, because it is a solution of the polynomial x^2-2=0. Pi is indeed transcendental, as is its soulmate e; but it would have been more precise to say,
Pi is an irrational number, meaning it will continue infinitely without repeating, and is also transcendental [possibly followed by a definition].
UPDATE: The same quibble applies to the CNN story, which says “Mathematicians know that pi is irrational — it cannot be represented as one number divided by another — and transcendental, meaning it is not algebraic. That means, theoretically, that its digits will continue on indefinitely without ending in repetition — in other words, the digits won’t suddenly continue infinitely as 5s after 3 trillion digits ….” The “That means” is precise as to the “irrational” part, but I think it doesn’t adequately capture the meaning of “transcendental, meaning it is not algebraic.”
Anonsters says:
Let me be the first to leave a random, unrelated comment, from Friedrich Nietzsche, Twilight of the Idols (“Maxims and Arrows,” § 22):
March 14, 2010, 10:23 pmOperationCounterstrike says:
I object to calling e a “soulmate” of pi. What do they have in common, other than being transcendental and irrational and important? Other than these three incidental properties, nothing. Nothing intrinsic in common.
March 14, 2010, 10:46 pmOperationCounterstrike says:
I once read that it had been proven that there is no sequence of seven sevens in an infinite decimal writing of pi. Anyone know if this is true?
March 14, 2010, 10:47 pmMax Power says:
@Operation Counterstrike: Both pi and e appear together in Euler’s Formula, perhaps the most elegant and deeply “intrinsic” identity in mathematics: e^(pi * i) + 1 = 0. So I think they are soulmates in that sense.
March 14, 2010, 11:02 pmMax Power says:
Oh, and the string 7777777 occurs at position 3,346,228 of pi, counting from the first digit after the decimal point. See the pi decimal expansion search page at http://www.angio.net/pi/bigpi.cgi
March 14, 2010, 11:05 pmmls says:
And I thought I was going to get some pie.
March 14, 2010, 11:09 pmJaredS says:
OperationCounterstrike: It has been neither proven nor disproven that pi is normal. A proof of the nonexistence of any particular finite sequence of digits in its decimal expansion would have as a corollary that pi is not normal.
March 14, 2010, 11:22 pmSteve says:
I’ve always wondered: how can it be proven that pi doesn’t repeat at some point?
March 14, 2010, 11:34 pmathEIst says:
In an infinite number sequence every sequence would have to appear and appear an infinite number of times. PowerMax, did you know that i to the ith power(whatever that can possibly mean) is a real number! If it is irrational or transcental I don’t know, but it is equal to 5.5————
March 14, 2010, 11:35 pmAnonsters says:
Me too, but I fear the answer.
I managed to get through school without taking geometry, for god’s sake. :P
March 14, 2010, 11:44 pmreadery says:
Does the Due Process Clause “rational basis” doctrine prohibit government from basing laws on this number?
March 14, 2010, 11:44 pmD.R.M. says:
No, it wouldn’t. The decimal representation of 10/7 is infinite, yet no cases of 0s 3s, 6s, or 9s appear in that infinity, nor do any sequences the same digit repeating twice in a row.
March 14, 2010, 11:45 pmGeorge says:
did you know that i to the ith power(whatever that can possibly mean) is a real number!
Since e^(i*(0.5*pi + 2*pi*n)) = i, for any integer n, we can calculate
i^i = e^(i*log i) = e^-(0.5*pi 2*pi*n).
The principal value is usually taken to be e^(-pi/2).
March 14, 2010, 11:49 pmGeorge says:
Also, Ivan Niven published “A simple proof that $\pi$ is irrational” in the Bulletin of the American Mathematical Society, Volume 53, Number 6 (1947), p. 509.
March 14, 2010, 11:55 pmEugene Volokh says:
OperationCounterstrike: As Max Power points out, e and pi are indeed related in an important way.
March 14, 2010, 11:57 pmGeorge says:
Oh, and Niven’s article is nicely unpacked on The Math Less Traveled.
March 15, 2010, 12:24 amOperationCounterstrike says:
OK, I’d forgotten that equation, you’re right of course. My bad.
March 15, 2010, 12:35 amOperationCounterstrike says:
Full disclosure: I memorized 50 digits as a kid.
But I was into memorizing things. I memorized the entire HUNTING OF THE SNARK.
March 15, 2010, 12:36 amJohn Skookum says:
Yes! Thank you. You beat me to it.
Euler’s identity is as close as humans can get to knowing the mind of God. Here in one lovely equation are the five most important numbers in all of mathematics, and the three most fundamental operators: addition, multiplication, and exponentiation.
This simple little equation contains more fundamental truth than all the books ever written. It ranks among the top three or four achievements of the human mind, Maxwell’s equations and Einstein’s theories of relativity being the only other contenders I can think of. The combined works of Shakespeare, Michelangelo and Beethoven are filthy rags next to this majestic equation.
For those who do not understand what it means to raise a number to an imaginary power, you need to look at the last few chapters of an introductory college calculus book. It has to do with what happens when you do Taylor series expansions of trigonometric functions of imaginary numbers.
March 15, 2010, 12:38 amSteve says:
Thanks very much for those links, George!
March 15, 2010, 12:44 amAnthony says:
It’s possible that pi repeats some sequences (on a small enough scale, it certainly does) but ‘repeating’ means that a number continues repeating the same sequence of digits forever. Which would make pi a rational number, which has been proven false.
March 15, 2010, 2:11 amJohn Armstrong says:
Not to rain on the parade, but Euler’s formula is really sort of a trinket as far as mathematical statements go. There’s really nothing deep to it.
It’s very impressive to people with a superficial familiarity with mathematics, though. For some real meat, try Yoneda’s lemma.
March 15, 2010, 2:22 amJohn Armstrong says:
As for the irrationality of π (and thus the fact that its decimal expansion isn’t eventually repeating), it’s possible to show with nothing more than the formula for integration by parts. I don’t know whether Prof Volokh has installed any sort of LaTeX functionality here, and so I won’t terrify you by writing out raw TeX code. Basically, it’s something I could show to anyone in calc 2 and they should be able to follow, but I doubt they’d come up with it on their own.
March 15, 2010, 2:31 amPeter Twieg says:
Is it true that all irrational numbers continue infinitely without repeating? For example, what would the number .1037103710371037…. correspond to in rational terms?
I wouldn’t be surprised if there’s a simple proof showing that the set of repeating real numbers is larger than the set of rational numbers.
March 15, 2010, 3:08 amAnthony says:
1037/9999. All repeating numbers can be expressed as X/(10^N-1), where X is the repeating block and N is the length of the repeating block.
March 15, 2010, 3:11 amMike S. says:
Peter:
1037/9999.
You can show this as follows:
.10371037… = (.1037)*(1.000100010001….}
= 1037/10,000 * (1+ 1/10,000 + (1/10000)^2 …)
The latter term is a geometric series with multiplier 1/10,000 so its sum is 1/(1-1/10,000)=10,000/9999
A similar calculation can be done for any repeating decimal.
March 15, 2010, 3:19 amDoc Rampage says:
Although I admire your enthusiasm, I feel compelled to play the spoilsport and point out that if the identity were written as e^(pi*i) = -1 then it would lose three of those aspects that you are so enthusiastic about.
March 15, 2010, 3:28 amJim says:
Continuing infinitely is not an intrinsic property of irrational and transcendental numbers but is a property relative to the
base used to express a number.
In the base pi number system, pi = 1.
March 15, 2010, 4:09 amJohn Armstrong says:
Anthony and Mike do great work, and even indicate how to generalize the proof that any (eventually) repeating decimal represents a rational number.
Now: can you do the reverse? Any rational number’s decimal expansion is eventually repeating.
March 15, 2010, 4:12 amRicardo says:
On Euler’s equation, the reason why the equation is true is that e^ix = cos x + i*sin x for any real x. The reason for that in turn is that the Taylor series expansion of e^x looks an awful lot like the Taylor series expansions of sin x and cos x and when you plug an imaginary number into the e^x formula, you get an alternating series of terms that can be represented by the sin and cos functions.
So then the remarkable thing is not Euler’s equation or the way it is stated but rather the fact that the trigonometric functions have a kind of commonality with e^x. And that commonality derives from the fact that the derivative of sin x is cos x and the derivative of -cos x is sin x. This shows that sin and cos are mathematical cousins, so to speak, of e^x which is its own derivative. In both cases, you can take as many derivatives as you want of the original function and you will get something that still looks like the original function.
As to why that is the case for sin and cos, that’s where my own intuitive mathematics runs against a brick wall. But that commonality is the key to understanding how Euler’s equation works out so nicely.
March 15, 2010, 4:42 amOperationCounterstrike says:
The decision to define raising e to an imaginary power ix as (cos x) + i (sin x) (Euler’s formula) always struck me as an arbitrary decision, a convention, not necessarily natural or reflective of anything intrinsic.
Anyone wanna explain why I’m wrong? What INHERENTLY dictates that e^(ix) should equal (cos x) + (i sin x)?
March 15, 2010, 4:49 amOperationCounterstrike says:
Ricardo, thanx, missed your post before.
March 15, 2010, 4:51 amMike S. says:
Sure: but not without invoking Fermat’s little theorem to show that a number d, which will here represent the denominator of the fraction, relatively prime to 10 divides 10^n-1, where n is a number less than d (technically, the number of whole numbers less than d relatively prime to d) which is a simple result in group theory, but not one for which I can write a proof for a general audience with the amount of typing I fell like doing. Given that, we are done for rational numbers whose denominator is relatively prime to 10 (i.e. has no factors of 5 or 2), by the opposite of the method in my earlier post. And if the denominator has factors of 2 and/or 5, it can be written as a sum of a fraction whose denominator has only factors of 2 and 5 (and thus has a finite decimal expansion) and one whose denominator is relatively prime to 10. For example, 7/15 = 2/3-1/5. The sum of a terminating decimal and a repeating decimal is a repeating decimal, so we are done. (In my example 2/3 = .666… and 1/5 = .2 so 7/15 =.4666666…)
March 15, 2010, 5:59 amApperception says:
Yes, at this point in time, it’s so familiar it doesn’t seem profound to a jaded person. To a person with a more objective stance, of course, the profundity is easily recognized. But we don’t expect that of you. It’s ok.
March 15, 2010, 6:55 ammarkm says:
Ricardo: Plot out a couple of cycles of sin(x) and cos(x) and look at them. They are clearly the same function shifted by pi/2 (or 90 degrees). There are simple geometrical proofs of this, too, from the definitions of sin and cos on a right triangle.
Now, look at the slope of the functions (the 1st derivatives). The slope of sin(x) is steepest at the zero crossings, that is, at the peaks of cos(x). It is zero at the peaks of sin(x), which are the zeros of cos(x). It varies smoothly in between, just as cos(x) does. So it’s intuitively obvious that the first derivative of sin is shaped like cos, and the first derivative of cos is shaped like -sin.
It may take a good bit more work to establish the magnitude of the derivatives (that is, that d(sin(wx)/dx = w*cos(wx)).
OperationCounterstrike: The fourth derivative of cos(x) is thus cos(x). All the derivatives can be expressed as cos(x + n*pi/2). Similarly, all derivatives of e^ax are expressed using e^ax.
The relationship between the trig and exponential functions becomes much more interesting when you look at complex numbers as vectors on a plane and find the relationship between cartesian and polar coordinates:
z = x + iy
= |z|*(sin(theta) + i*cos(theta))
= |z|*e^(i*theta)
That is, e to an imaginary number is an angle. Multiplication by e^i*theta rotates the vector by theta. If theta = pi, the rotation is 180 degrees, equivalent to multiplying by -1.
Electrical engineers have been using these identities to simplify the calculations of AC circuits for over a century.
March 15, 2010, 7:30 amRicardo says:
Oh, the intuitive part I get. It’s why exactly d(sin(wx)/dx = w*cos(wx) I’ve never quite got. I do remember the rough outlines of why d(e^x)/dx = e^x.
March 15, 2010, 7:58 amDavid Chesler says:
I spent half a semester of physics-for-pre-meds not knowing that EEs call it j, not i. By which point it was too late to understand why. Which was part of why I dropped out of pre-med. And without three-semester physics I didn’t have the background for real engineering either, which is how I ended up a programmer, because the applied math requirements were the closest to what I’d already taken. (CS was then a specialty, just about to become a major.)
But I do know enough math to know that 3.14 is not at all pi, so why call March 14 Pi Day? (Unlike the nice palindromic or all-even-numbers or change-the-thousands-place days.)
March 15, 2010, 8:28 amTamerlane says:
Take the Taylor Series of cos(x) and isin(x). These series are absolutely convergent so they can be added together with a re-ordering of terms. By alternating the terms of the two series you get the Taylor series of exp(ix) which is also absolutely convergent. Therfore, cos(x) + i sin(x) is equal to exp(ix). This is rigorously proved in any first year analysis course at a decent university. The details of why absolutely convergent series can be equated this way and why the terms of absolutely convergent sequences can be re-ordered without changing the limit sum of the series is covered in such courses. The usual proofs involve some point set topology.
March 15, 2010, 9:13 amBemac says:
I would guess that in Europe, Pi Day falls on July 22.
And FWIW, 0.99999… = 1
March 15, 2010, 9:31 ammattski says:
Asking as a mathematical naif, what happened to 22/7?
March 15, 2010, 9:44 amuh_clem says:
Bemac: July 22nd is Pi Approximation Day. See http://bradley1969.blogspot.com/2009/07/happy-pi-approximation-day.html
David: Electrical engineers use the letter j for the simple reason that the letter i is already being used to represent electrical current.
March 15, 2010, 9:47 amAndyM says:
22/7 is a good approximation of pi; it is not exact.
If you’re trying to do calculations of the area of a circle by hand, 22/7 is probably “good enough” for what you’re doing; you don’t have infinitely long to do the multiplication anyway, so you have to use some kind of approximation…
March 15, 2010, 10:34 amBemac says:
Well, 3.14 is itself just an approximation. Any “Pi Day” will turn out to be “Pi Approximation Day.”
Had a 19th-century Indiana state senator had his way, pi would have be set by statute as 3. March might have become Pi Month in the Hoosier State. Think of the parades.
March 15, 2010, 10:48 amjustaguy says:
“Pi day” isn’t for five years — 3-14-15. To go even further at 9:26 GMT is Pi ?
March 15, 2010, 11:05 amuh_clem says:
But some approximations are better than others, which is why I serve my celebratory pie at precisely twenty-six and a half seconds after 1:59.
March 15, 2010, 11:08 amOrenWithAnE says:
Eugene Volokh is a man, meaning that he has two lungs but only one heart.
Eugene Volokh is a mammal, meaning that he has two lungs but only one heart; he is also a man [...]
March 15, 2010, 11:30 amJohn Armstrong says:
It’s true because of what other posters have been identifying as the connection between exponential and trigonometric functions, so if anything is profound it’s that connection.
As others have pointed out, the exponential with base e is the same as its own derivative. In fact, the exponential property — which allows us to translate multiplication and division problems into additions and subtractions — guarantees that an exponential function’s derivative will be a multiple of the function itself. There must be some base where that factor is 1, and that’s e.
On the other hand, the trigonometric functions are better seen as the coordinate functions of a point traveling around the unit circle at unit speed. Indeed, we parameterize the circle as (cos(t),sin(t)). Why does this matter? Because a particle moving around a circle is held on its path by a force pointing directly back towards the center. That is, the acceleration is a negative multiple of the position. And so the sine and cosine are both negative multiples of their own second derivatives (in fact, the factor is -1).
So we can use sin(x) and cos(x) as two solutions to the equation u” + u = 0, and general principles tell us that every other solution can be written as a linear combination of these two. But similarly we can use the exponential to construct a solution to this equation: e^(ix). And so there must be some equation relating the exponential to the trigonometric functions.
Not really that profound. Something like it had to be true.
March 15, 2010, 11:40 amAnthony says:
If you remember how to do long division, it’s fairly easy to show that any integer ratio must repeat. Long division is a simple iterative process:
Let N be the numerator, D be the denominator, and R be the remainder.
1) Set the value above zero to floor(N/D) and the R to N mod D.
2) Set the next decimal digit to floor(R*10/D) and R to (R*10 mod D)
3) Repeat step 3.
Now, given the nature of the mod operations, R must be an integer number in the range of 0 to (D-1); also, if R is ever 0, it will remain 0 from then on out. In addition, if R ever comes back to a value it has had before, it will at that point repeat. Since there are only (D-1) possible values that R can pass through (without winding up as a repeating zero), you must eventually repeat.
March 15, 2010, 1:53 pmMalvolio says:
It was my understanding there would be no math.
March 15, 2010, 2:11 pmJeff R. says:
The end of that sentence from the original post, by the way, should be “which means that it is not the root of any polynomial with rational coefficients.” Which goes some distance in explaining why the writers found it easier to elide the difference between irrational and polynomial…
(To express that in pure laymans terms, it means that you can take any rational numbers you want, and pi, and you can add them and multiply them in any order you want, any number of times, including multiplying pi by itself again and again and adding it to pi again and again, and no matter what you do, the final result will never be a rational number. Except in the cases where you cheat and every time pi is actually involved you end up multiplying it by zero, of course.)
March 15, 2010, 2:21 pmEugene Volokh says:
Jeff R.: Isn’t saying that x is a root of a polynomial with rational coefficients tantamount to saying that x is a root of a polynomial with integer coefficients?
March 15, 2010, 2:57 pmBABH says:
John Armstrong: “It is obvious that X.”
Commenters-without-math-doctorates: “Wait, why is that obvious?”
John Armstrong: [pauses in thought for a few minutes, then answers] “Yes, it is obvious.”
The plain fact that there is an equation relating the trigonometric functions to the exponential function is really cool. The special fact that the formula relating the two is so simple and elegant, and also involves the complex plane, is Deep Magic. It’s the sort of thing that makes one want to study more math.
March 15, 2010, 2:59 pmJeff R. says:
True. And I think that you may be able to get back to integers even if you start with non-transcendental real coefficients, even…
March 15, 2010, 3:03 pmJohn Armstrong says:
BABH: so it’s either obvious or profound?
March 15, 2010, 3:06 pmBABH says:
John,
I think of “obvious” as a technical term in math, meaning something like: “this result follows from other stuff you know. It may take a couple hours of work to derive it, but not hard work, so it’s obvious.”
In this case, then, we have a result that is both obvious and profound. You seemed to be denying that it was profound, simply because it is “obvious.” See, e.g., your statement at 2:22am:
March 15, 2010, 3:16 pmMaryanna says:
As a math atheist, I should be excused from this thread.
March 15, 2010, 3:40 pmRoger the Shrubber says:
Thank you. It’s amazing how many otherwise normal people will argue until they are blue in the face that 2.4999999….. isn’t 2.5, or (more commonly) that .9999….. isn’t 1.
March 15, 2010, 4:03 pmBABH says:
2+2=5, for very large values of 2.
March 15, 2010, 4:04 pmDuffy Pratt says:
“A is x and y, meaning (some essential property of x).”
It seems to me that this statement will always be true, though it might also sometimes be incomplete. Just because another statement might be more complete than the one given, does not imply that the more complete phrase is either necessary, or even better (depending on the circumstances). Of course I’ve always been a fan of the following syllogism, Socrates is a man; All men are mammals; therefore, Socrates is a man.
March 15, 2010, 4:16 pmMichael B says:
Recently read where an idiot savant has memorized pi out to literally tens of thousands of decimal digits.
March 15, 2010, 5:17 pmAnonsters says:
This may or may not be the reason I posted a quote from Nietzsche about evil Russians. ;)
Fantastic Radiohead song.
March 15, 2010, 6:38 pmFub says:
Not to mention incorporating the irrational and the imaginary.
[For the humor impaired, that was a joking jab at cataphatic theology. I don't doubt that there are mathematicians in foxholes.]
March 15, 2010, 6:53 pmJohn Armstrong says:
Maybe not foxholes, but they’re in snipers’ nests, calculating corrections for wind and distance.
March 15, 2010, 9:12 pmAnonsters says:
Nerds. ;)
March 15, 2010, 9:21 pmRicardo says:
Damn, that was pretty good. But the factor is only -1 when we express angles in radians, right? In that case, that’s the connection between e and pi. e is the base of the exponential function that guarantees its derivative is the original function. When you use the length of the semi-circle arc of unit radius (pi) to measure the angle of a semi-circle, then the second derivative of sin becomes -sin and cos becomes -cos.
March 15, 2010, 10:40 pmJohn Armstrong says:
Exactly, Ricardo. There are a number of parameters we could wiggle to get things to work out. Which exact exponential base do we use? What parameterization of the unit circle? What coefficients of sine and cosine? The point is that even if the famous formula weren’t true as written, some related formula would have to be.
March 15, 2010, 10:47 pmJaredS says:
Wait a minute.
The complex exponential function can be defined as the unique function f(z) such that f’(z)=f(z) and f(0)=1. If we stuck to degrees to define the trig functions, we might have f(i*x)=cos(x*180/pi)+i*sin(x*180/pi), but we would still have f(i*pi)=-1.
The relationship thus has nothing to do with how we measure angles (except inasmuch as it serves as yet another example of the superiority of radians).
March 16, 2010, 1:36 am