Scruffy's math program for kids

[Sherlock Holmes]

You're not worth arguing with. You're a complete asshole and an idiot if you think math or this thread has anything to do with Donald Trump.

No I'm not because you don't stand a chance against me.

Fuck you, monkey boy. Go jerk off with your computer programs. YOU are why we need Scruffy's math program.

Yawn.

It's pretty damn obvious you have no idea what you're talking about. Even your fucktard AI says "generalizing" Pythagoras means increasing dimensionality, and EVEN THAT is wrong. Your fucktard AI bot is corrupting the minds of youth.

So no you admit there IS a generalization telling us what it "means" for the past few days you've been denying there was any such thing. Yes the theorem has been generalized in different domains I never said otherwise.

Now shut up and go away, you miserable little troll. You are the poster boy for Scruffy's math program. It's too late for you but it's not too late for the kids. STOP MOLESTING THE CHILDREN. Get the fuck out of their way and let them learn. Stop lying to them, you dumb fucking leftard skank.

Good lord, you are useless utterly inept and incompetent and in need of psychiatric help.

 

[Sherlock Holmes]

You want to see the California math curriculum?

Here it is.

Video: Overview of Mathematics Framework - Common Core State Standards (CA Dept of Education)

Video overview of the Mathematics Framework.

www.cde.ca.gov

Yes, I'll show your clueless ass how to teach mathematics. Give me 6 months with your dumbest student and I'll give you back a mathematician.

Go ahead, look at the California curriculum in detail. Compare it with the Chinese plan I posted earlier.

The numbers don't lie. California public school graduates come out STUPID, their test scores have been dropping every year for the past 20 years. Whatever they're being taught, it is FAILING.

Freud would have had so much fun with you.

 

[Sherlock Holmes]

See?

You don't care about the students, you care about Trump.

I never once mentioned Trump in this thread, except to deny his relevance.

This is about MATH, not Trump. Why the fuck are you bringing up Trump?

Perhaps because Trump needs help with arithmetic, after all look at the trouble he got into with his accounting

You try that in my kids' class and I'll have you forcibly ejected and fired.

That kind of stupid shit is EXACTLY why your students are failing

I dearly wish I was a student in your class, I'd have the class howling as I tied you up in knots.

 

[Sherlock Holmes]

See? Some moron thinks he can educate a PhD by looking up some dumbass AI on Google.

I regularly eat PhD's for breakfast, don't forget I respect or destroy arguments based on their consistency and validity not on the certifications of the person making those arguments, so appealing to your own authority doesn't help you here.

 

[scruffy]

Scruffy says:

You will learn the material or you won't graduate.

There are no participation trophies in Scruffy's math class.

You can use any and all tools available to you, but you WILL learn the material.

Bay Area innovator uses AI to tackle California students’ declining math skills

Peter Relan, chairman of GotIt! Education, calls MathGPT, “the AI that instructors trust and students love.”

www.mercurynews.com

 

[scruffy]

Elon Musk "gets it".

Because he has kids

Elon Musk's next venture set to launch this year: a private preschool

Elon Musk's private preschool in Texas is accepting applications for up to 21 students with subsidized tuition in the first year.

www.businessinsider.com

I have kids too.

And pretty soon they'll have kids.

We want our kids to be smart and capable, not DUMBED DOWN.

Only 1/3 of America's children are performing at grade level.

The first goal has to be getting that number up to 2/3.

We need some changes to make that happen.

 

[Sherlock Holmes]

Elon Musk "gets it".

Because he has kids

One of whom despises him.

Elon Musk's next venture set to launch this year: a private preschool

Elon Musk's private preschool in Texas is accepting applications for up to 21 students with subsidized tuition in the first year.

www.businessinsider.com

I have kids too.

God help them.

And pretty soon they'll have kids.

We want our kids to be smart and capable, not DUMBED DOWN.

Right, so being a trumpanzee scared of drag queens and vaccines and science and facts is not dumbing down in your little world.

Only 1/3 of America's children are performing at grade level.

That's because there's a teacher shortage and there's a teacher shortage because the pay is crap, you pay peanuts you get monkeys.

The first goal has to be getting that number up to 2/3.

Take a look at China, they can tell you where you're going wrong.

We need some changes to make that happen.

Wow, that's insightful, one needs to make changes in order to change things, what a clever boy.

 

[justoffal]

Traditionally, math begins with addition, subtraction, and the multiplication tables in the 3rd grade or so. Then, there is an introduction to geometry, algebra, and calculus in high school. Finally, there is usually a course in differential equations in college, and maybe some complex analysis if you're in a STEM field, and somewhere along the way there is also probability and statistics.

Scruffy says, this is not enough. To be competent in the modern world, a college graduate should understand differential geometry and tensor calculus (same thing, kind of), topology, and abstract algebra including but not limited to the theory of groups, rings, modules, and categories. Waiting for graduate school to understand these things is a guarantee of difficulty. These topics should be well studied prerequisites for any kind of graduate work.

Scruffy proposes to condense the course work at the high school level, as the beginning of a new mathematics program. BEFORE we start learning about triangles, we need an exposure to surfaces. Otherwise, there is no motivation for triangles, trigonometry, or conic sections. 7th grade is the perfect time for a "survey of mathematics" course. Why are we learning abstract algebra? What good is it? Why is calculus important? What is topology, and what is differential geometry?

In 6th grade, is when we want to cover basic algebra, combinatorics, and probability.

The 8th grade algebra course should include systems of linear equations, matrix math, and change of coordinates in addition to polynomials.

The 9th grade geometry course should include manifolds and an introduction to dot product, cross product, and tensor product.

The 10th grade calculus course should include multi variable calculus and de-emphasize the memorization of integrals and derivatives. It should specifically include an introduction to differential equations. Students should understand Stokes' and Green's theorems by the end of the course.

All this will prepare the student for the typical 11th grade introduction to physics, which should include Hamiltonian and LaGrangian mechanics (introductory dynamics, the LaPlacian, what is a wave equation, and so on). Math in 11th grade should be fun, not rigorous. It should include topology, abstract algebra, and probability as an introduction to quantum science. It should also specifically include an introduction to non-Euclidean geometries. It should provide motivation for the continuing study of math in college.

This is an example of a math curriculum that provides motivation for continuing study:

Erlangen program - Wikipedia

en.wikipedia.org

Then, by the time we get to college, we ALREADY know what an orbit is, before we start taking first year physics and chemistry. We know what polar coordinates are BEFORE we have to solve problems involving angular momentum, and thus we can easily and intuitively understand concepts like quantum spin, and we can already start to manipulate them using spinors and quantum computing gates.

First year math in college should be hard core differential equations. (Since we already learned linear algebra and multi variable calculus in high school). It should also include an introduction to stochastic equations (like Brownian motion, which fits right in with first year chemistry and physics).

Second year math should be differential geometry and topology, with a focus on tensor calculus. By the end of the second year in college, students should be able to seamlessly transition between coordinates systems, both geometrically and algebraically. The course should include an introduction to measure theory and invariances, so we understand measure preserving and conformal transformations, and it should also include the mechanics of distances and angles on curved surfaces.

After all this, third year college math should focus on abstract algebra and algebraic topology, starting with immersions and including fiber bundles, sheaves, foliations, simplexes with graph theory, and so on.

Finally, senior level.math in college should once again be fun and motivational. Dimensionality is a good topic, fractional dimensions, dusts, space filling curves, fractal geometries, and so on. Electives might include a course in quantum field theory, for instance the creation and annihilation operators tie right in with dimensionality and also pertain to neural networks and machine learning. Advanced probability theory could be another elective - the Ito, Langevin, and Malliavin calculuses, how to solve non-equilibrium equations in open systems, information geometry, stochastic dynamics and stability, and entropy.

Anyone who graduates from college should be conversational with mathematics. It doesn't mean you have to pull equations out of your butt, it just means you have to be familiar with the principles. You don't have to know what a Lorentz boost is unless you're a physicist, but you should know enough about coordinates systems so someone could explain it to you in 5 minutes.

This business of graduating from college without knowing how to balance a checkbook has to stop. Scruffy says: if you want to graduate, you WILL learn math. If you can't do it, go to trade school instead. (Where you'll learn it anyway lol).

Great idea wrong approach.

You start in kindergarten. Like the Chinese do.
You cannot inundate a young mind with mathematical complexity or you will poison them.
From kindergarten on 80% of learning experience should be math. There are a lot of different ways math can be taught. It can be taught in conjunction with spelling, reading, art and even music.

If you start at this age by the time they are ready to leave primary School you will have a completely different crop of math students ready for the higher complexities with much less of a challenge.

Children in the first grade should be introduced to Cartesian graphics... They should be encouraged to use them for everything from shopping lists to video game scores. We are really missing an opportunity here because the woke agenda has siphoned off valuable learning time.

 

[scruffy]

Great idea wrong approach.

You start in kindergarten. Like the Chinese do.
You cannot inundate a young mind with mathematical complexity or you will poison them.
From kindergarten on 80% of learning experience should be math. There are a lot of different ways math can be taught. It can be taught in conjunction with spelling, reading, art and even music.

If you start at this age by the time they are ready to leave primary School you will have a completely different crop of math students ready for the higher complexities with much less of a challenge.

Children in the first grade should be introduced to Cartesian graphics... They should be encouraged to use them for everything from shopping lists to video game scores. We are really missing an opportunity here because the woke agenda has siphoned off valuable learning time.

Okay. Now fast forward to what I said earlier. Every college graduate needs to be conversant with quantized angular momentum. Because chemistry, because relativity, and because quantum computing.

Kids should understand vectors thoroughly before entering college. In college they should start learning about bivectors, which is the natural way of representing angular momentum. Bivectors use the wedge product from Grassmann algebras to define orientation. Otherwise, you have to use cumbersome tricks like the right hand rule to figure out if you're spinning left or spinning right.

An algebra with both a wedge product and a dot product is called a Clifford algebra. It is a geometric algebra. It's used in quantum computing. These are the types of things kids should be learning in college algebra. NOT "systems of linear equations", which is high school stuff.

Wedge products are also oriented tangent planes in differential geometry. When you're dealing with angular momentum the tensor product is of limited utility. The wedge product is a different kind of exterior product that assigns orientation to your surfaces. It provides an easy way to understand flows on curved surfaces, thereby leading naturally into the topology of fiber bundles and foliations. No STEM student should graduate college without understanding these things.

 

[Sherlock Holmes]

Okay. Now fast forward to what I said earlier. Every college graduate needs to be conversant with quantized angular momentum. Because chemistry, because relativity, and because quantum computing.

Kids should understand vectors thoroughly before entering college. In college they should start learning about bivectors, which is the natural way of representing angular momentum. Bivectors use the wedge product from Grassmann algebras to define orientation. Otherwise, you have to use cumbersome tricks like the right hand rule to figure out if you're spinning left or spinning right.

An algebra with both a wedge product and a dot product is called a Clifford algebra. It is a geometric algebra. It's used in quantum computing. These are the types of things kids should be learning in college algebra. NOT "systems of linear equations", which is high school stuff.

Wedge products are also oriented tangent planes in differential geometry. When you're dealing with angular momentum the tensor product is of limited utility. The wedge product is a different kind of exterior product that assigns orientation to your surfaces. It provides an easy way to understand flows on curved surfaces, thereby leading naturally into the topology of fiber bundles and foliations. No STEM student should graduate college without understanding these things.

Shut up dick head, you routinely post bullshit and when I demonstrate that and provide my sources to back up what I say, you stamp your feet and throw a tantrum. You're unhinged, now fuck off.

 

[scruffy]

Quantum states are technically "spinors", which are almost like tensors of rank 1/2.

They are the "square roots" of vectors.

To derive the spinor construction, you first break up the 3-by vector into two 2-by's. (These are the "Pauli vectors").

Once you have the 2-by's, you can manipulate them using unitary SU2 matrices, which are nothing more than functions that act on the state vectors by matrix multiplication.

If you take the (FREE) quantum computing course from IBM, this is the first thing you'll learn. You will immediately see how to generate an entangled Bell state using a pair of quantum gates, which are matrices. (In this case, a Hadamard gate and a CNOT gate).

What IBM doesn't tell you is, there's an angle doubling relationship between the state space and the physical space.

To understand this, you can look at the polarization of light, and also the Stern-Gerlach experiment (1922) involving the induction of a magnetic dipole in electrically neutral atoms using the angular momentum of the electrons.

The importance of the SG experiment is that it doesn't create a full spectrum superposition - instead, it creates a QUANTIZED superposition. You get spin up, and spin down, and that's it.

The associated Grassmann algebra shows us why this happens. It's actually very easy to understand this way. Otherwise it's confusing because you can do the SG experiment with silver atoms in which NONE of the electrons have any net angular momentum.

The dipole behavior of silver is due to the SPIN (internal) angular momentum of the 47th electron in the 5S orbital. The spin is specifically NOT from the electron orbiting the nucleus like a spinning ball. (The Bohr model taught in freshman chemistry is WRONG and only serves to confuse. To see this all you have to do is look at the 5S orbital, which has a shape and it's not a spinning ball).

So, instead of states being described by position and momentum, we have a state vector |¥>, representing the wavefunction. To get the magnitude of the wavefunction we can't use the ordinary norm, because wavefunctions are complex numbers. So we have to use the complex conjugate to calculate the norm (which ensures the result is real). The complex conjugate is represented as <¥|, so the norm just becomes <¥|¥>. (Some people put a little dagger on the complex conjugate, but in this case it's implicit - and familiarity with implicit daggers and implicit tensor summations is a big part of the learning curve for these types of algebras - but at the end of the day they're still linear algebras and you have all the same concepts of scalar fields like temperature and vector fields like electric potential).

To represent a superposition of spin up and spin down, you just use a linear combination, like this:

¥ = a |0> + b |1>

Where a and b are coefficients that represent the likelihood of measuring one state or the other (in other words, they are probabilities - or more accurately the square roots of probabilities, because our quantum state is the square root of a physical vector). As always the probabilities have to add up to 1.

The key to understanding the square root behavior of spinors is to realize that in physical space, the difference between spin up and spin down is 180 degrees, whereas in the state space it's only 90 degrees. In the state space, the spin up and spin down states are orthogonal, which means we can use them as the basis for linear combinations. Thus a "ket" (like |¥>) can be written in terms of its components as a 1x2 column vector, which reduces spinor math to ordinary linear algebra. Therefore "quantum superposition" becomes just a fancy way of saying linear combination.

The square root property can be clearly seen when the state space is viewed as a circle in the complex plane. Here, physical angles are square rooted (ie cut in half in terms of the complex angle), and probability amplitudes are square rooted (a probability of 50% becomes a coefficient of 1/√2, and you can check that this causes the sum of probabilities to add up to 1).

To rotate one state vector into another, all we have to do is multiply by a unitary matrix (one whose conjugate transpose is the same as its inverse, which means the length of our vector doesn't change). To eliminate the redundancy caused by.phase, we choose the specific matrix whose determinant is 1, and this group of operations is called SU2 ("special" unitary 2x2 matrices, which are 2 dimensional because our quantized basis is two dimensional, we only have spin up and spin down and those states are orthogonal).

This is all of quantum mechanics in a nutshell, everything else is just practice with matrix multiplication.

The Grassmann algebra for this construction treats the quantum states as anticommuting variables, in other words a*b = -b*a. That's all you need to know to understand quantum field theory and condensed matter physics.

See? That wasn't so hard, right? High school kids could do this stuff. The only part of it we don't usually get in high school is the complex numbers. But what the hell, it's just a circle, it's 2π like any other circle, and ultimately this should be taught at the exact same time as elementary trigonometry.

Pythagoras leads naturally to sines and cosines, and the basic concepts of angles, norms, and rotation operators (which are nothing more than square matrices). PHASE is the only concept missing from this analogy, and it's trivially easy to introduce it in the context of this discussion. PHASE is the reason we want our unitary matrices to have determinant 1, so if a kid can understand that, he's ready for college.

 

[ShahdagMountains]

Elon Musk "gets it".

Because he has kids

Elon Musk's next venture set to launch this year: a private preschool

Elon Musk's private preschool in Texas is accepting applications for up to 21 students with subsidized tuition in the first year.

www.businessinsider.com

I have kids too.

And pretty soon they'll have kids.

We want our kids to be smart and capable, not DUMBED DOWN.

Only 1/3 of America's children are performing at grade level.

The first goal has to be getting that number up to 2/3.

We need some changes to make that happen.

Has Sherlock got kids?

 

[ShahdagMountains]

Shut up dick head, you routinely post bullshit and when I demonstrate that and provide my sources to back up what I say, you stamp your feet and throw a tantrum. You're unhinged, now fuck off.

Such language. From an “urbane and elegant” individual such as yourself.

 

[scruffy]

btw, abstract algebra is only 300 years old.

The Babylonians knew how to solve polynomials in 1700 BC, but to get from there to rings took another 3000 years.

Emmy Noether is the mother of rings. Her father Max Noether dabbled in the concepts of moduli and quotient groups, but it was Emmy (Einstein's math tutor) who put the rings on the map, laying the foundation for what is today simply called "algebra".

Poincare is still the MAN for the relationship between algebra and geometry. This guy:

 

[Sherlock Holmes]

btw, abstract algebra is only 300 years old.

The Babylonians knew how to solve polynomials in 1700 BC, but to get from there to rings took another 3000 years.

Emmy Noether is the mother of rings. Her father Max Noether dabbled in the concepts of moduli and quotient groups, but it was Emmy (Einstein's math tutor) who put the rings on the map, laying the foundation for what is today simply called "algebra".

Poincare is still the MAN for the relationship between algebra and geometry. This guy:

View attachment 1063325

To which question is this a response?

 

[scruffy]

In college, we want STEM students to become familiar with optimization.

This topic is too complex and too sophisticated for high school, but it makes use of all the tools learned in high school, namely algebra, calculus, and geometry.

High school students learn Newton's physics, F = mA. Newton's physics is based on derivatives, acceleration being the second derivative of position. Therefore, calculus. Newton and Leibniz are birds of a feather when it comes to calculus. The "d" of derivative calculus is Leibniz's "infinitesimal increment", and an integral is simply a sum over all such increments.

In college, kids learn that there is a dual relationship between points and lines. For a function f(x), every "point" of the function (meaning f(x) for every position x), has a line associated with it - the line is its derivative at x, which is sometimes called the "tangent" of the function at x. The tangent line has an equation y = mx + b, and "points" of the function can be recovered by solving the tangent equations for every x.

In college, kids are asked to generalize this duality, when they learn that Newton's force laws are equivalent to Hamiltonian and LaGrangian mechanics. Except that the latter are energy based, instead of force based. And this is where it becomes essential to have a solid foundation in mathematics. Because even though energy is easier than force (because force requires coordinates, whereas energy is just a scalar that's independent of the coordinate system), the energy formulation requires us to solve the principle of least action, which is an optimization problem.

Hamilton and Lagrange are famous for their least action mechanics, but the principle of least action actually goes back to a guy named Maupertuis (pronounced mo-per-twee) around the turn of the 18th century, about 50 years before Lagrange and 100 years before Hamilton. Maupertuis was working on Fermat's corpuscular theory of light. His work was the first to relate wave mechanics to particles.

Maupertuis's principle - Wikipedia

en.wikipedia.org

Today, the relationship between particles and waves is at the core of every form of physics. Not only is it foundational to Hamiltonian and LaGrangian mechanics, but if you're into quantum computing it's at the heart of Schrodinger's equation, and if you're studying machine learning you'll need to apply Legendre transforms on convex hulls. These are all forms of "energy optimization", and the math requires an understanding of Noetherian symmetries which means rings and abstract algebra, as well as tensors and differential geometry. Such topics are impossible without a firm grasp of basic vector calculus, which includes systems of linear equations.

Therefore, we want kids to have a firm grasp of mathematics before they graduate high school. Because if you graduate college with a STEM degree, you're expected to know how to solve optimization problems. Newton's F = mA basically answers the question "what happens next?". The derivatives are time dependent. But Maupertuis is the guy shooting a free throw into the basket, he's calculating "what are the allowable trajectories" regardless of how long the ball stays in the air.

Turns out, ants can solve these problems too, just not as precisely. Biological systems are inherently built to provide these real world answers, because the purpose of the brain is to optimize behavior in real time. Brains can't afford to ask "what happens next", it's too slow and too costly to keep asking that question. Brains need to see all possible paths in real time, the same way Feynman diagrams represent all possible paths between a beginning state and an end state.

And BRAINS work on linear algebra, they do lots and lots of matrix manipulation with very few actual derivatives. They use the Maupertuis trick, the duality between points and lines, whenever they need to process derivatives. The duality between points and lines is developed in projective geometry, which means the Riemann formulation, which has everything to do with complex numbers, which in turn leads immediately to the Clifford algebras that describe quaternions and octonions and other hypercomplex number systems. (Hamilton was the first to study quaternions, he found it "necessary" in the formulation of Hamiltonian mechanics). And in turn, Clifford algebras are quantizations of the exterior algebra. So now you begin to understand the significance. The first four Clifford algebras are the real numbers, the complex numbers, the quaternions, and the split biquaternions. Clifford algebras define the bundle of differential forms on a smooth manifold, for example on a Riemann surface the tangent spaces have a natural quadratic form induced by the metric. This is how you handle a spin field, where every point in space has a quantized angular momentum associated with it. The Clifford duality will give you the spin bundle associated with every tangent plane.

If you understand these things, STEM will be a breeze for you. If you don't, it'll be very difficult and you'll end up having to memorize hundreds of equations. All of physics can be derived easily and quickly from the Clifford geometry. Everything from Maxwell's equations to Schrodinger's equation to quantum spin states and even the sophisticated optimizations involved in machine learning. If you understand the Clifford math you'll be conversational with ALL of this stuff, and with a little practice and a few examples you'll be able to handle any real world challenges.

This is what we need from our educational system. Particle in a box is yesterday. Bohr's model of the atom is downright WRONG, it shouldn't be taught at all. Kids need to learn how and why "all possible paths" arises naturally from wave mechanics, and why quantization is just a natural consequence of the underlying algebra.

 

[scruffy]

Fun with math!

This is one of the missing pieces, that kids should learn EARLY - but don't learn today till they get to college.

Here's some elementary geometry.

We talked about the relationship between points and lines in the previous post. That's calculus, but there's an easier way to understand it which is projective geometry.

Every kid likes movies, right?

So, put a movie screen at the point x=1, and extend the screen along the y axis.

Now tell the kids, "your eye is at the origin (the point 0,0). What do you see?"

What you see, is called a stereoscopic (or stereographic) projection.

And, what's so special about that? Why does it matter? The answer is: lines become points. Draw a line, y = mx + b. Every point in that line, projects to the SAME point on your movie screen.

(Of course, since you're a mathematician, you're going to show the kids a Riemann sphere at the same time, and show them the similarities).

So this is fun with points and lines. But here's where the real education starts.

What about circles? Can you change a line into a circle?

Yes, you can. Begin by taking a small piece of the line - a line "segment" (otherwise known as an interval). Now bend it, either up or down, so the ends join. Voila - a circle.

You can easily see that the longer the line segment, the bigger the circle. Now kids - what is the relationship between the length of the line segment and the size of the circle? That's right! The length of the line segment is the same as the circumference of the circle - which means the radius of the circle is the interval length divided by 2π.

Now who can tell me, how long the interval has to be, if we want a circle of radius 1? That's right! The answer is 2π.

So now we're going to have some more fun. Let's draw some shapes inside the circle. How about... a triangle? How about... a square? What's next? Right! A pentagon, a hexagon, in fact we can draw a polygon with any number of sides. Let's call the number of sides N, so we have an N-sided polygon. When N=3, we have a triangle. When N=4, we have a square. And so on.

So let's take a triangle, which has 3 sides. Now who can tell me, where the top of the triangle should be? What is the top of the triangle? What's it called? That's right, it's called a vertex. A triangle has 3 sides, and 3 vertexes. A square has 4 sides, and 4 vertexes. How many vertexes does an N-sided polygon have? Which vertex is the top? Does it matter?

You can see, that we can rotate our polygon, so the vertex can be anywhere we want. It can be on top, or on the right, or on the bottom. So let's learn some math. Let's put a vertex on the right, so it lines up with the x axis. Now what can we say about all the other vertexes?

Let's imagine we're going to go around the circle, in a counter-clockwise direction. What do we see when we do this? Well, we're going to visit every vertex of our polygon, and eventually we're going to get back to where we started. Can we do the same thing with our original line segment? No, we can't. Why not? That's right, because the ends aren't joined. Who can tell me the mathematical word for joining the ends of the line segment?

< crickets >

The word is "compact". It means you can get anywhere from anywhere, by moving in one direction. If we cut a hole in the circle, are we still compact? No, we're not, because we can't travel around it any more. Is our original line segment compact? No, it isn't, because if we travel in one direction we fall off the end. But the circle is compact, because we can keep going around and around and we always stay on the circle.

Now kids - if we have a circle of radius 1, with an N-sided polygon inside it, and we walk around the circle in a counter-clockwise direction, and every step we take brings us to the next vertex, how many steps do we have to take before we get back to where we started? That's right! N steps. Now, what happens if instead of walking around, we simply rotate the circle and the polygon? How far do we have to rotate the circle to get to the next vertex? That's right! 2π / N. If we do that N times, we get back to the beginning, just like when we were walking around.

So let's say we have a magic wand that can rotate the circle for us, so we don't have to walk around. We can be lazy, and just use our magic wand to move the circle. Every time we wave our magic wand, the circle moves by one vertex. How many times do we have to wave the wand to get back to the original vertex? That's right! N times. Now let's say the wand is really magic, and if we wave it up the circle moves counter-clockwise but if we wave it down it moves clockwise. So if we wave it up and then down, the circle moves one way and then the other. So if we wave the wand up once and then down once, what happens? That's right, we get back to where we started. So waving the wand up and then down, is the same as waving it up N times in a row.

If we have the wand up twice and then down twice, what happens? That's right, we get back to where we started. How about if we wave it up 3 times and then down 3 times? Same thing, right? We get back to where we started. How about if we wave it up 100 times and down 100 times? Same thing, we always get back to where we started, no matter how many vertexes there are. But if we have N vertexes and we wave the wand up and down 100 times, is that the same as waving the wand up N times? No, it isn't, is it. Unless N happens to equal 100. So who can tell me what the math word is for the special property of vertexes?

< crickets >

The word is roots. If we have a circle of radius 1 and we draw an N sided polygon inside it, we say that there are N "roots" of 1. Roots is an easier word than vertexes, right? So if we wave our magic wand up one time, we are visiting a positive root, and if we wave it down one time, we're visiting a negative root. And because we have a compact circle, the positive roots eventually become negative, and the negative roots eventually become positive.

Congratulations children, you're very smart. You've just learned topology, group theory, and complex numbers in one easy lesson.

 

[scruffy]

So then, in college, you learn the fancy word "isomorphic". It means "looks the same". Has the same shape, same behavior.

You learn that the roots of unity form a group, and you learn that the group is isomorphic to rotations of vectors in the complex plane. Eventually by the time you study physics and chemistry, you learn that the Pauli matrices are isomorphic to the quaternions.

And more importantly, you learn about the difference between the Pauli spinors that describe quantum states, and the Weyl spinors that describe general relativity. You learn what a "double cover" is, and why Weyl spinors have left handed and right handed (chiral) versions but Pauli spinors don't.

All these things trace right back to the Clifford algebras. Which in turn trace right back to understanding of roots.

First a child learns what a square root is, he learns there are two of them, a positive version and a negative version (a "double" cover, as it were). Then he learns about the imaginary number i, by solving equations that require the square roots of negative numbers. Then he learns about the complex plane, and Euler's formula (which is easily understood algebraically in terms of its Taylor expansion. Finally, he learns that the commutativity rules for systems of multiple imaginary numbers are just permutation groups involving a minus sign.

Therefore when someone (like a prospective employer) asks him to solve a problem in special relativity, he knows how to derive the Minkowski metric without having g to look it up. Because he knows the Clifford algebra that applies to 1 time and 3 space dimensions. He can change coordinates for a Lorentz boost because he knows how to transform the Weyl spinors, and this way he can tell his employer what spin "looks like" at very high velocities. If you put a qubit into a beam of light (like a gigahertz laser), what does that look like? How come you start seeing g symmetries that you didn't see before, and how come you start seeing unusual states of matter and energy that you didn't see before?

This understanding starts when you're FIVE YEARS OLD. If you miss it, you're not going to be able to keep up. Teachers need to learn to start planting the seed in kindergarten, and water it every year. Square roots are 5th grade, right after long division. Complex numbers and imaginary roots are 6th grade. Groups and projective geometry are 9th grade, right after systems of equations and matrix math. Calculus is 10th grade, and instead of memorizing integrals we start teaching differential geometry. Then instead of slacking off in 12th grade we introduce complex analysis - be abuse by that time we should already have enough physics to handle impedances and transmission lines. That way, we hit the ground running when we enter college.

First year college physics and chemistry we learn all about rotation - polarization of light, angular momentum, change of coordinates in rotating reference frames, rotation operators, spinors, and rotatíonal invariances. Second year we learn the methods of differential equations and tensor math, and we learn all about curvature and the Levi-Civita connection. Third year we do topology - bundles, sheafs, simplices, graphs, covers, and embeddings. Fourth year we do optimization, variational methods, perturbation, stability, and stochastic dynamics. By the time we graduate college we know all about control systems, we can control anything whether discrete, continuous, or fractional - and whether quantized or not.

It all starts in kindergarten. Children need to be motivated to learn these things. Teachers need to get their students interested, it's a big part of the job. Start with "who wants to learn how to hotwire a car", and by the end of the semester your kids will know all about ignition systems and automotive electronics.

 

[ReinyDays]

College graduates need English 101 more than they need Math 521 ... if you're not communicating, why bother? ...

Accounting is the highest form of mathematics ... money is the only thing worth counting ... the purest form of American communications ...

See? ... I didn't need twenty-five paragraphs, just two sentences ... brevity, the sign of an educated person ...

 

[Moonglow]

my kids were doing trig by ninth grade.