Abstract
How wonderful that we have met with a paradox. Now we have some hope of making progress.
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Notes
- 1.
Except 2.
- 2.
This path is called “the Dragon Curve”.
- 3.
Unfortunately so. It would be really nice to know if you’re the first or not.
- 4.
“Math experience” doesn’t mean studying double-super-calculus in your third PhD. Even if you practice arithmetic, there are provably innumerable things to find.
- 5.
Eight sides.
- 6.
A “bight” is a segment in the middle of a rope.
- 7.
Mathematicians excel at coming up with worst-case examples to force other mathematicians to be mildly paranoid and over-cautious in their theorems. Physicists, by and large, don’t suffer from this affliction. I say “every imaginable signal”, because if you’re imagining an example right now (without knowing one of the weird counter-examples), then you’re almost certainly imagining a signal that can be broken down into simple waves.
- 8.
Middle C = 261.6 Hz.
- 9.
Something like “Where’s Waldo?” would be an exception.
- 10.
“What people won’t notice” is a guiding principle behind data compression in digital media. By taking advantage of auditory and visual illusions and weaknesses in human perception, we can dramatically reduce the amount of data required to make sound and pictures that are practically perfect... as far as humans can tell.
- 11.
One such example can be found in Section 2.5 .
- 12.
There are a couple of subtle requirements on f(x) in order to ensure that it has a Fourier Transform. In particular, f(x) must be “integrable”, which means that \(\int f(x)\,dx\) is equal to some finite number. That forces f(x) to go to zero as x goes to ±∞, which is why \(f(x)e^{-2\pi i n x}|{ }_{-\infty }^{\infty }=0\).
- 13.
Or, more likely, a bar.
- 14.
Such as every individual number being just as likely as every collection of numbers.
- 15.
Technically, the gravitational interaction of more than three object is chaotic (this is called the “three-body problem”), it’s just that the Sun’s gravity is so much more important to each planet than all the other planets, that everything is more-or-less in a two-body system with the Sun. The time scales on which the chaotic nature of our solar system is important is on the order of billions of years.
- 16.
Which is why there’s a lot of Sun and a little of Jupiter and practically nothing of anything else.
- 17.
Hats off to the brave climatologists and meteorologists who wade through it all the same.
- 18.
This is a love born of e’s utility in calculus, but if you don’t know calculus, then it really does seem totally arbitrary.
- 19.
Specifically, \(x_n\to \frac {r-1}{r}\).
- 20.
Here M and m are the different masses involved, R is the distance between them, and G is the gravitational constant that describes how strong gravity is in our universe.
- 21.
There are solutions, such as three equal-massed planets orbiting in a figure-eight “juggling pattern”, but they are generally unstable and physically unrealistic.
- 22.
- 23.
That’s not to say that we can’t say anything. For example, a set of three objects may be “gravitationally bound”, meaning that there isn’t enough kinetic energy to fling any one of the bodies away. In that case we could rule out the system falling apart, but still be unable to say exactly where each object will be in a couple thousand years.
- 24.
A finite number of people will travel less than any given distance, while an infinite number will travel farther.
- 25.
This is subtle, but you cannot apply this same argument to rational numbers because the decimal expansion of rational numbers is far more restricted than those of real numbers. In particular, rational numbers have a “repetend”, a series of digits that it (eventually) repeats forever. For example, \(\frac {3}{11}=0.2727272727\ldots \) Section 4.6 goes into more detail.
- 26.
π = 3.1415926535897932384626433832795028841971693993751…
- 27.
“Today” = 2017.
- 28.
π is defined to be the ratio of the circumference to the diameter of any given circle.
- 29.
It’s a big, complicated universe after all.
- 30.
A “geometric series” is a never-ending sum of the powers of a number. As luck would have it, this is an easy thing to calculate. \(r+r^2+r^3+\ldots =\frac {r}{1-r}\) as long as |r| < 1.
- 31.
This is “the equivalence of the contrapositive”: “P implies Q” is logically equivalent to “not Q implies not P”. For example, “if it is raining, then it is wet” is logically equivalent to “if it is not wet, then it is not raining”. In this case, “if a number has a repeating decimal, then it is rational” is equivalent to “if a number is irrational, then it does not have a repeating decimal”. This sort of logical judo is a mathematician’s bread and butter.
- 32.
E.g., \(\frac {18}{12}=\frac {3}{2}\).
- 33.
The binomial expansion theorem is: \((c+d)^n=\sum _{j=0}^n \frac {n!}{j!(n-j)!}c^jd^{n-j}\).
- 34.
The first, second, and third derivatives are normally written as f ′(x), f ′′(x), and f ′′′(x). For the sake of clarity we’ll write these as f (1)(x), f (2)(x), and f (3)(x) and just so that nobody has to count k hash marks the kth derivative is f (k)(x).
- 35.
\({k \choose n}=\frac {k!}{(k-n)!n!}\) is the number of ways to choose n objects out of a set of k objects. For example, there are 3 ways to choose one object out of three: \({3\choose 1}=\frac {3!}{(3-1)!1!}=\frac {1\cdot 2\cdot 3}{1\cdot 2\cdot 1}=\frac {6}{2}=3\). Since it’s literally just counting permutations, the choose function is never anything other than an integer.
- 36.
The odd-numbered derivatives have a different sign, but the fact that they’re integers isn’t changed by that.
- 37.
The product rule is \(\left [c(x)d(x)\right ]^\prime = c^\prime (x)d(x)+c(x)d^\prime (x)\).
- 38.
And bravo to you.
- 39.
Incidentally, this definition is equivalent to declaring the small angle approximation: \(\sin {}(x)\approx x\).
- 40.
This is why one is given a special status. For example, 12 can be factored into 12 = 2 ⋅ 2 ⋅ 3 or 12 = 2 ⋅ 2 ⋅ 3 ⋅ 1 or 12 = 2 ⋅ 2 ⋅ 3 ⋅ 1 ⋅ 1 and so on. The prime factors (2, 2, and 3) are unique, but you can toss in as many ones as you like.
- 41.
I’ll wait.
- 42.
Fermat’s Last Theorem says that if a, b, c, and n are counting numbers, then a n + b n = c n has solutions in a, b, and c only when n = 1 or n = 2.
- 43.
This is not the standard notation for the modulus. The established notation is to actually write “X mod N”, but I find that arduous. The defining characteristic of mathematicians is sloth, so occasionally writing some of an extra word is basically torture. Many alternatives have been proposed.
- 44.
Far more common than the Carmichael numbers are “Fermat pseudoprimes”, so named because they sometimes pass Fermat’s Little Theorem but are not prime. For example, \(\left [2^{340}\right ]_{341}=1\) says that 341 might be prime and \(\left [3^{340}\right ]_{341}=56\) says 341 definitely isn’t. Here 2 is called a “Fermat liar” and 3 is called a “Fermat witness”. Fermat pseudoprimes can be caught by trying out a few different “bases” until a Fermat witness is found. It almost never takes more than two tests. Carmichael numbers are Fermat pseudoprimes where every base is a Fermat liar. At least, every base that shares no factors in common with the Carmichael number is a Fermat liar. But by the time you’ve accidentally picked a base with a factor in common with the N you’re testing, you’re already done: N isn’t prime.
- 45.
This level of pedantic paranoia is not without precedent. For example, the Pólya Conjecture says that more than half of the numbers less than any given N have an odd number of prime factors. Empirically, this would appear to be true, since the first counterexample doesn’t show up until N = 906, 150, 257. Just because something works the first few hundred million times isn’t a good enough reason to say that it always works.
- 46.
“Primes is in P” means that the problem of determining whether or not a number is prime can be solved in “polynomial-log time”. The log of N is roughly proportional to the number of digits in N, so you can describe the running time of AKS as \(O\left ([\text{number of digits in }N]^{21/2}\right )\).
- 47.
“To date” = January 2016.
- 48.
“Why are we still using these things?”
- 49.
When analyzing crypto systems, different “attacks” are considered. For example, a “man in the middle attack” involves hijacking a communication channel in order to pose as one or more of the parties involved. A “rubber hose attack” is an actual term of art meaning “grab someone who knows the password and beat it out of them”.
- 50.
Ron Rivest, Adi Shamir, and Leonard Adleman invented RSA in April of 1977 at MIT. Recognizing its power and the potential for its abuse, they quietly mailed the algorithm to Martin Gardner. Gardner published it in his column “Mathematical Games” in the August 1977 issue of Scientific American. By the time American intelligence agencies knew what was happening, it was way to late to classify RSA.
- 51.
Including: the theoretical and philosophical backbone of modern computer theory (“Turing Machines”), one of the most widely accepted goal posts for artificial intelligence (“the Turing Test”), and even an early excursion into mathematical biology (“The Chemical Basis of Morphogenesis”) wherein he described how a wide variety of surprisingly complex patterns, such as a leopard’s spots or the leopard’s skin itself, can form spontaneously under the right chemical conditions.
- 52.
Presumably so that they can dress up as a fellow prisoner to help intrepid (but gullible) heroes escape, only for those heroes to find themselves in a trap of a more devious nature!
- 53.
To “encode” a word into Pig Latin you put all the consonants before the first vowel at the end and add an “-ay”. If the word begins with a vowel you leave it and add “-way”. This is done relentlessly until someone in earshot can’t stand it any more.
- 54.
Or you can read through the Gravy below and think of it as exactly what it is.
- 55.
Technically, a “pseudo-random” number. In fact, this is one method for producing lots of random-enough numbers without using actual randomness.
- 56.
This isn’t the standard notation for modular arithmetic. That would be: 9 + 5 mod 12 ≡ 14 mod 12 ≡ 2 mod 12. The standard notation is awkward enough that a lot of mathematicians just make up their own. Case in point.
- 57.
A common question that comes up at this point is “what if you pick an x that isn’t coprime to m?” There are two answers. First, the primes used to create an encryption modulus are so ludicrously large that it is extremely unlikely in the time since RSA was invented that any message has ever accidentally been a multiple of one of the prime factors. If you want something to worry about, worry about lightning striking your computer just as you hit “send”. Second, it doesn’t matter. \(\left [x^{j\varphi (m)+1}\right ]_m=x\) is always true, for any x, when the prime decomposition of m has no square or higher powers (e.g., m = 30 = 2 ⋅ 3 ⋅ 5 but not m = 12 = 22 ⋅ 3), which is exactly the case for the moduli used in encryption, m = pq. So, for example, \(\left [2^{\varphi (10)}\right ]_{10}=\left [2^4\right ]_{10}=\left [16\right ]_{10}=\left [6\right ]_{10}=6\ne 1\), but \(\left [2^{\varphi (10)+1}\right ]_{10}=\left [2^5\right ]_{10}=\left [32\right ]_{10}=\left [2\right ]_{10}=2\).
- 58.
Pick a candidate number, p, and any random number a such that 1 < a < p − 1. If [a p−1]p≠1, then p is not prime. But if [a p−1]p = 1, then p is very likely to be prime (especially if it’s large).
- 59.
Starting with A and B you subtract smaller and smaller combinations of A and B from each other until you find the smallest possible combination. This is the greatest common divisor and if it’s 1, then A and B are coprime.
- 60.
Alternatively you can use \(\ell = \left [k^{\varphi (\varphi (m))-1}\right ]_{\varphi (m)}\). However, in order to calculate φ(n), you need to know the prime factors of n. The factors of m are known, because m = pq, but the factors of φ(m) may not be known so φ(φ(m)) may not be easily calculable.
- 61.
There is a huge body of work dedicated to these special cases. Suffice it to say: there are lots of weird special cases where breaking the encryption can be done, but they don’t show up accidentally very often and those that are known are usually avoided when generating new encryption keys.
- 62.
A “linear combination” of A and B is any sum of the form xA + yB.
- 63.
Pronounced “Oiler”, as in “one who oils”.
- 64.
For example, “x 5 − 2x 4 + 3x 3 + πx 2 − 57x + 1” is a “5th degree polynomial”.
- 65.
Complex numbers include an i. For example, 3 + 2i or − 5i.
- 66.
a(bc) = (ab)c.
- 67.
a(b + c) = ab + ac.
- 68.
Fourier transforms (Section 4.2) provide another good option. FTs change derivatives into multiplication by a variable, so if you write the FT as \(\hat {f}(k) = FT[f(x)]\) and the inverse FT as \(f(x) = FT^{-1}\left [\hat {f}(k)\right ]\), then \((2\pi ik)^n\hat {f}(k)=FT\left [f^{(n)}(x)\right ]\) and \(f^{(n)}(x)=FT^{-1}\left [(2\pi ik)^n\hat {f}(k)\right ]\).
- 69.
When a mathematician says a function is “smooth”, they mean that it is differentiable; usually many times or infinitely differentiable. After integrating, the resulting function is always at least once differentiable.
- 70.
The natural numbers are: 0, 1, 2, 3, ...
- 71.
With the exception of the poles at each negative integer, where Γ(N) jumps to infinity.
- 72.
The “poles” are (usually) locations where there’s a division by zero and the function jumps to infinity.
- 73.
“Analytic continuation” is a process where you look at a little region of an analytic function, figure out how that region of “soap film” is behaving, and use that to figure out what the rest of the function is. Often this is easier than it sounds: the “analytic continuation” of f(x) = x (where x is a real number) is f(z) = z (where z is any complex number).
- 74.
This will be done below.
- 75.
When you take the root of a negative number, there are suddenly difficulties in defining the “principal solution”. For example, \(4^{\frac {1}{2}}=\sqrt {4}=2,-2\) with the principal solution typically chosen to be the positive one: 2. On the other hand, \((-4)^{\frac {1}{2}}=\sqrt {-4}=2i,-2i\); these are both just as “far” from the positive numbers, so which should be the principal solution?
- 76.
People who know calculus.
- 77.
In particular, being analytic which basically means that it “smoothly connects the dots”.
- 78.
Zermelo-Fraenkel set theory.
- 79.
“≡” means “is defined as”.
- 80.
Nobody actually does this, but it is important when you’re thinking about defining how things should work. Rational exponents, \(\frac {a}{b}\), can be defined as “the bth root of the ath power”, such as 91.5 = 27. But to define irrational exponents you need a construction for numbers like the one discussed. For example, the limit \(2^\pi \equiv \left \{2^3,2^{3.1},2^{3.14},2^{3.141},2^{3.1415},\ldots \right \}\) tells you anything you’re likely to want to know about 2π.
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Cottrell, S.S. (2018). Not Things. In: Do Colors Exist?. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-64361-8_4
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