Why aren't we introducing the main ideas of Calculus *much* earlier, in grade school, so students aren't bombarded with a million new ideas at once?
When we teach calculus, we *at once* introduce (at least) 5 new concepts (limits, tangent lines/slopes, derivatives, area, integrals), all intertwined together. There's no reason for this! Most of these ideas can be introduced *much* earlier, one at a time. Here's an attempt at.. pic.twitter.com/mWed3rS9sn
Hey everybody! Gather round; I just discovered this incredible ancient way for multiplying! It's so simple; here's what you do: Say you want 3x4. So you make 3 horizontal lines, and 4 vertical lines, and count the intersections! I can't believe nobody ever showed me this!(cont'd) pic.twitter.com/iCdgjvpDo7
No time for full thread on this, but this is my preferred method for long division (for kids... maybe adults too). Guess whatever you want, doesn't have to be the right order of magnitude, or even the right sign. (Of course -144=-12x12; the idea is to keep using single digit...) https://t.co/8zpeZMtwV7pic.twitter.com/AhOnqheJnU
— Alex Kontorovich (@AlexKontorovich) July 6, 2020
Thread on why the "evidence" for the Collatz Conjecture may not be so strong:
Unpopular opinion: The 3x+1 Conjecture might be False!
Here's why I think this may be the case. (thread)
My first paper, with Y. Sinai in 2002, https://t.co/Jv9rKAh29T proves that 3x+1 paths are a geometric Brownian motion (in a precise asymptotic sense), with drift log 3/4 < 0
I couldn't help but start thinking about this again (sigh; you delete enough emails with a subject line, it seeps into your subconscious) and had a new idea I hadn't explored before (& saw nothing in a quick search of literature). Here it is.
Thread on the origins of trigonometry (the Sun and Half-Moon):
Ah both the sun and the (almost) half-moon in the sky! The birthplace of trigonometry! To be a half-moon, the sun-moon-earth triangle has a right angle at the moon. That means we can calculate the relative distances of sun & moon to earth! Point one hand to the sun and (cont?d) pic.twitter.com/qzOpPZkpOJ
Thread on new conjectures about Fibonacci numbers (based on joint work above with Jeff Lagarias):
1/Spoke today about a fun new conjecture (joint with Lagarias) about Fibonacci numbers, explainable to grade-schoolers: There's a constant we call beta ? 0.373365, so that the total number of prime factors of F_{2n}, divided by log n, has lower limit beta https://t.co/GWBzOn3iy9pic.twitter.com/NZnYoLEx9O
Thread on decidability of Goldbach and Riemann hypothesis (see sub threads for more discussion...)
Always great to hear @stevenstrogatz on @Radiolab, here talking about Godel incompleteness. Just one quibble: if Goldbach is undecidable, then it?s true! Same holds with any finitely-falsifiable statement, and could be an insane way to prove the Riemann hypothesis! (cont?d) https://t.co/WkK9aJAnWk
Langlands program in a tweet: I give you a recipe for coefficients, e.g.:
a_{2n} = (-1)^n/n!, a_{odd}=0.
Prove that the resulting power series
F(x)=1-x^2/2+x^4/4!-x^6/6!+...
has ?non-obvious? symmetry F(x+2pi)=F(x).
?Program?: Discover automorphy F(x)=cos(x).
Now generalize
— Alex Kontorovich (@AlexKontorovich) July 3, 2019
Why is the Riemann hypothesis hard? (Spoiler: if we actually knew the answer, we might have a better chance of proving it...):
Why is the Riemann Hypothesis hard? Just one reason (of very many): it's not an analytic question. Here are the values of zeta on the 1/2-line (where at least 40% of the zeros are, all should be) and the 4/5-line, where none should be. The latter gets arbitrarily close to 0, i.o. pic.twitter.com/qSHhwwCY6X
"Factor City" where the building blocks are... primes!
Welcome to Factor City! Here building blocks are primes, and the city shows the prime factorizations of 0 to 399, organized in 20x20 streets/avenues. Blue=2, Red=3, Green=5, Orange=7, Yellow=11, Purple=all other primes. (There is no building 0, and building 1 is black.) Cont'd pic.twitter.com/tVvMsfVAsG
Proof: Let x^x^x^x... = 2. Then x^(x^x^...)=2, so x^(2)=2, so x=?2. Let y^y^y^... = 4. Then y^(y^y^...)=4, so y^4=4, so y=4^(1/4)=?2. So x = y, and x^x^x... = y^y^y... QED.
The local-global conjecture for Apollonian packings: "grade" the packing by height=curvature; every large enough, admissible height has some circles in it:
Very cool, thanks! Somehow it reminded me of this pic I made a long time ago. It's an Apollonian packing, except the circles are "lifted" up to height = bend. So the local-global conjecture says that, at all "admissible" (large enough) heights, there actually is a circle there... pic.twitter.com/0xgtAtaKFn
Actually I was just reminded of how I learned the general identities from Conway, using this insane thing called "Bernouilli integration" (apparently that's how Jacob did it), see attached. Is there a pretty proof that sums of 4th powers has sums of squares as a factor?... pic.twitter.com/mons2IgTfu
People seem to be surprised by how simple the notion of Pi is:
With Pi Day just around the corner, let?s remember what Pi is all about.
After washing your hands thoroughly, cut the crust off a pizza pie and lay it across four others. You?ll see that the crust spans a little more than 3 pies. That?s Pi ? 3.14.
This is just awful news. My introduction to "real" mathematics came in the fall of my freshman year, when I took Linear Algebra from Conway. We have 100 Conway stories from that one course alone. How do you learn vector spaces? Compute dimension of the space of 3x3 magic squares! https://t.co/8EnbSzGpZq
I think this video is an apt representation of exactly what Conway was dreading would happen, hence his hatred for the Game of Life. I myself have discussed only his recreational contributions on here. So let's correct that.
Cardano, Tartaglia, and the Cubic Equation (also, why imaginary numbers were accepted *before* negative numbers!...):
Yeah, this is one of the craziest stories in the history of math, and deserves to be better known!
In the mid-1500s, Cardano (building on del Ferro and Tartaglia, not entirely amicably...) solved the cubic equation. The solution goes something like this. https://t.co/STpNbCP3F1pic.twitter.com/107xn2kx1X
You have a (discrete) function, in green, and can control every input. The differences are computed in red (derivative). Game: make the red into a straight line going up, and then change it to go down. (Solution: parabolas)
The tricky thing about prime numbers is that they?re defined by what they?re not. A composite number is a
— Alex Kontorovich (@AlexKontorovich) May 11, 2021
What is conditional expectation?:
The concept of conditional expectation is (surprisingly?) difficult for students, until they do the following exercise:
Let X and Y be iid 6-sided dice rolls, and let Z=X+Y be the sum. What is E(X|Z)? Def: E(X|Z)=sum x * Prob(X=x|Z). Not so enlightening. But space of values ... https://t.co/z8ogdmvnIG
Subtlety in some elementary geometry constructions and the parallel postulate:
My favorite thing in elementary Geometry that I learned waay too late is Hartshorne's (very natural) notion of "efficiency", that is, executing a construction using the fewest number of ruler or compass moves (unlike Euclid, who uses previously established constructions freely)..
If you really mean "eyeball," you could just take a bunch of v's (say the 10x10 grid), and draw arrows connecting v to the image A.v under your linear operator, A. You'll immediately see whether there are two, one, or no eigenvectors... pic.twitter.com/fa0RNIT2cY
Oh no! We placed an order for an infinite number of bottles and bottle caps, but the caps company labels them 1, 2, 3,... while the bottle company labels them 0, 1, 2,... what will we do? ??
— Alex Kontorovich (@AlexKontorovich) May 16, 2022
What might math and math publishing look like in 20-50 years?
What will math research look like in 20 years? I have no idea, and am pretty sure I'm not alone. Will Interactive Theorem Provers (eg, Lean) be the next phase in math evolution? Maybe, maybe not. (How long did it take for everyone to learn TeX?) Announcing https://t.co/O1bRjx65zg
— Alex Kontorovich (@AlexKontorovich) May 15, 2020
On the evolution of Proof and Rigor in Mathematics. And on PhD education.
Wow, fascinating paper on arxiv by @XenaProject and others: https://t.co/NdKMAEChNH on the nature of mathematical rigor. ?Proof? is the heart and soul of our project; if we don?t know what it means, then we don?t know anything! But what exactly is accepted *by the community*...
Perspective art and projective geometry (for kids). How easy it is to miss a beautiful and simple idea for millennia!
It's actually mind-boggling how artists first figured this out (not mathematicians!) in the early 1400s. Here's one from 1426 that's very well drawn and *almost* understands points at infinity. And by 1435, they've figured out enough to draw things like the second image. How... https://t.co/patWArGrndpic.twitter.com/z3P6IZe2jF