Information theory, which was developed by Claude Shannon starting in the late 1940s, deals with questions such as how quickly information can be sent over a noisy communications channel. Both the information carriers (e.g., photons) and the channel (e.g., optical fiber cable) are assumed to be classical systems, with well-defined, perfectly distinguishable states.
Now in a new paper, physicists Giulio Chiribella and Hlér Kristjánsson at the University of Oxford and the University of Hong Kong have proposed a second level of quantization, in which both the information carriers and the channels can be in quantum superposition. In this new paradigm of communication, the information carriers can travel through multiple channels simultaneously.
Mathematicians long wondered whether it’s possible to express the number 33 as the sum of three cubes — that is, whether the equation 33 = x³+ y³+ z³ has a solution. They knew that 29 could be written as 3³ + 1³ + 1³, for instance, whereas 32 is not expressible as the sum of three integers each raised to the third power. But the case of 33 went unsolved for 64 years.
Last month, I was at Foundations 2018 in Utrecht. It is one of the biggest conferences on the foundations of physics, bringing together physicists, philosophers, and historians of science. A talk I found particularly interesting was that of Alexander Blum, from the Max Planck Institute for the History of Science, entitled Heisenberg’s 1958 Weltformel & the roots of post-empirical physics. Let me briefly summarize Blum’s fascinating story.
In 1958, Werner Heisenberg put forward a new theory of matter that, according to his peers (and to every physicist today) could not possibly be correct, failing to reproduce most known microscopic phenomena. Yet he firmly believed in it, worked on it restlessly (at least for a while), and presented it to the public as a major breakthrough. How was such an embarrassment possible given that Heisenberg was one of the brightest physicists of the time? One could try to find…
He goes to class. There is an empty seat in front. He sits in the seat. He looks around. There are different people. He says “hi” to the girl next to him. She smiles. The teacher comes in. She closes the door. Everyone is silent. The first day of school begins.
Chapter 2 Water on the Floor
She is thirsty. She gets a glass of water. She begins to walk. She drops the glass. There is water on the floor. The puddle is big. She gets a mop. She wipes the water off. The floor is clean. She gets another glass of water. She drinks it. She is happy.
Chapter 3 Babysitting
Casey wants a new car. She needs money. She decides to babysit. She takes care of the child. She feeds him lunch. She reads him a story. The story is funny. The child laughs. Casey likes him. The child’s mom comes home. The child kisses Casey. Casey leaves. She will babysit him again.
Chapter 4 Twins
Jill and Jodi are twins. They look the same. But they act differently. Jill likes sports. She is good at basketball and golf. She is also loud. She talks all day. Jodi likes reading. She can read 300 pages a day. She is also quiet. She does not like to talk. Jill and Jodi still love each other.
Huge thanks to my friend Evan Miyazono, both for encouraging me to do this project, and for helping me understand many things along the way.
This is a simple primer for how the math of quantum mechanics, specifically in the context of polarized light, relates to the math of classical waves, specifically classical electromagnetic waves.
I will say, if you *do* want to go off and learn the math of quantum mechanics, you just can never have too much linear algebra, so check out the series I did at http://3b1b.co/essence-of-linear-algebra
Mistakes: As several astute commenters have pointed out, the force arrow is pointing the wrong way at 2:18. Thanks for the catch!
*Note on conventions: Throughout this video, I use a single-headed right arrow to represent the horizontal direction. The standard in quantum mechanics is actually to use double-headed arrows for describing polarization states, while single-headed arrows are typically reserved for the context of spin.
What’s the difference? Well, using a double-headed arrow to represent the horizontal direction emphasizes that in a quantum mechanical context, there’s no distinction between left and right. They each have the same measurable state: horizontal (e.g. they pass through horizontally oriented filters). Once you’re in QM, these kets are typically vectors in a more abstract space where vectors are not necessarily spatial directions but instead represent any kind of state.
Because of how I chose to motivate things with classical waves, where it makes sense for this arrow to represent a unit vector in the right direction, rather than the more abstract idea of a horizontal state vector, I chose to stick with the single-headed notation throughout, though this runs slightly against convention.
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In the late 1970s, Stroustrup applied the idea of “classes” to the C programming language to create a new language that allows for high level abstraction—but is efficient and close to the hardware.
What inspired you to create C++?
In the really old days, people had to write their code directly to work on the hardware. They wrote load and store instructions to get stuff in and out of memory and they played about with bits and bytes and stuff. You could do pretty good work with that, but it was very specialized. Then they figured out that you could build languages fit for humans for specific areas. Like they built FORTRAN for engineers and scientists and they built COBALT for businessmen. And then in the mid-’60s, a bunch of Norwegians, mostly Ole-Johan Dahl and Kristen Nygaard thought why can’t you get a language that sort of is fit for humans for all domains, not just linear algebra and business. And they built something called SIMULA. And that’s where they introduced the class as the thing you have in the program to represent a concept in your application world. So if you are a mathematician, a matrix will become a class, if you are a businessman, a personnel record might become a class, in telecommunications a dial buffer might become a class—you can represent just about anything as a class. And they went a little bit further and represented relationships between classes; any hierarchical relationship could be done as a bunch of classes. So you could say that a fire engine is a kind of a truck which is a kind of a car which is a kind of a vehicle and organize things like that. This became know as object-oriented programming or also in some variance of it as data abstraction. And my idea was very simple: to take the ideas from SIMULA for general abstraction for the benefit of sort of humans representing things… so humans could get it with low level stuff, which at that time was the best language for that was C, which was done at Bell Labs by Dennis Ritchie. And take those two ideas and bring them together so that you could do high-level abstraction, but efficiently enough and close enough to the hardware for really demanding computing tasks. And that is where I came in. And so C++ has classes like SIMULA but they run as fast as C code, so the combination becomes very useful.
What makes C++ such a widely used language?
If I have to characterize C++’s strength, it comes from the ability to have abstractions and have them so efficient that you can afford it in infrastructure. And you can access hardware directly as you often have to do with operating systems with real time control, little things like cell phones, and so the combination is something that is good for infrastructure in general. Another aspect that’s necessary for infrastructure is stability. When you build an infrastructure it could be sort of the lowest level of IBM mainframes talking to the hardware for the higher level of software, which is a place they use C++. Or a fuel injector for a large marine diesel engine or a browser, it has to be stable for a decade or so because you can’t afford to fiddle with the stuff all the time. You can’t afford to rewrite it, I mean taking one of those ships into harbor costs a lot of money. And so you need a language that’s not just good at what it’s doing, you have to be able to rely on it being available for decades on a variety of different hardware and to be used by programmers over a decade or two at least. C++ is not about three decades old. And if that’s not the case, you have to rewrite your code all the time. And that happens primarily with experimental languages and with proprietary commercial languages that change to finish—to meet fads. C++’s problem is the complexity part because we haven’t been able to clean it up. There’s still code written in the 80’s that are running and people don’t like their running codes to break. It could cost them millions or more.
This video is based on a proof from H. Vaughan, 1977.
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In the late 1940s, Richard Feynman invented a visual tool for simplifying particle calculations that forever changed theoretical physics.
As one of the most famous physicists of the 20th century, Richard Feynman was known for a lot. Early in his career, he contributed to the development of the first atomic bomb as a group leader of the Manhattan Project. Hans Bethe, the scientific leader of the project who won a Nobel Prize in Physics in 1967 (two years after Feynman did), has been quoted on what set his protégé apart: “There are two types of genius. Ordinary geniuses do great things, but they leave you room to believe that you could do the same if only you worked hard enough. Then there are magicians, and you can have no idea how they do it. Feynman was a magician.”
In his 1993 biography Genius, James Gleick called Feynman “brash,” “ebullient” and “the most brilliant, iconoclastic and influential physicist of modern times.” Feynman captured the popular imagination when he played the bongo drums and sang about orange juice. He was a fun-loving, charismatic practical joker who toured America on long road trips. His colleague Freeman Dyson described him as “half genius and half buffoon.” At times, his oxygen-sucking arrogance rubbed some the wrong way, and his performative sexism looks very different to modern eyes. Feynman will also be remembered for his teaching: The lectures he delivered to Caltech freshmen and sophomores in 1962 set the gold standard in physics instruction and, when later published as a three-volume set, sold millions of copies worldwide.
What most people outside of the physics community are likely to be least familiar with is the work that counts as Feynman’s crowning scientific achievement. With physicists in the late 1940s struggling to reformulate a relativistic quantum theory describing the interactions of electrically charged particles, Feynman conjured up some Nobel Prize-winning magic. He introduced a visual method to simplify the seemingly impossible calculations needed to describe basic particle interactions. As Gleick put it in Genius, “He took the half-made conceptions of waves and particles in the 1940s and shaped them into tools that ordinary physicists could use and understand.” Through the work of Feynman, Dyson, Julian Schwinger and Sin-Itiro Tomonaga, a new and improved theory of quantum electrodynamics was born.
Feynman’s lines and squiggles, which became known as Feynman diagrams, have since “revolutionized nearly every aspect of theoretical physics,” wrote the historian of science David Kaiser in 2005. “In the same way that computer-enabled computation might today be said to be enabling a genomic revolution, Feynman diagrams helped to transform the way physicists saw the world, and their place in it.”
To learn more about Feynman diagrams and how they’ve changed the way physicists work, watch our new In Theory video:
Decades later, as Natalie Wolchover reported in 2013, “it became apparent that Feynman’s apparatus was a Rube Goldberg machine.” Even the collision of two subatomic particles called gluons to produce four less energetic gluons, an event that happens billions of times a second during collisions at the Large Hadron Collider, she wrote, “involves 220 diagrams, which collectively contribute thousands of terms to the calculation of the scattering amplitude.” Now, a group of physicists and mathematicians is studying a geometric object called an “amplituhedron” that has the potential to further simplify calculations of particle interactions.
Meanwhile, other physicists hope that emerging connections between Feynman diagrams and number theory can help identify patterns in the values generated from more complicated diagrams. As Kevin Hartnett reported in 2016, understanding these patterns could make particle calculations much simpler, but like the amplituhedron approach, this is still a work in progress.
“Feynman diagrams remain a treasured asset in physics because they often provide good approximations to reality,” wrote the Nobel Prize-winning physicist Frank Wilczek three years ago. “They help us bring our powers of visual imagination to bear on worlds we can’t actually see.”
In the 20th century, math education was a cliched amalgam of authority and tradition, an impenetrable fortress to ideas and imagination that would “threaten” its history of compliance and conservatism.
While mathematics is a stake holder in truth, poetry and art speak to truths that are more accessible to all of us — especially since they speak to our hearts, first. But, even as something as wildly as endearing as the words of Robert Frost, somehow have had little currency in shaping math education.
But, that is because only mathematics is wild. Math education is not. It is in that gulf that we should see what needs to be done now in the 21st century.
We have to walk the walk. We want our students to be risk takers in the classroom. Well, so do we. We want our students to struggle. Well, so should we at times. We want our students to appreciate the failures of learning mathematics. Well, so should we.
Trying to make everything perfect — like a perfect landing in a gymnastics routine — is not only stressful, its inorganic intentions runs counter to experimenting, pushing boundaries…pushing ourselves. Guardrails are needed on highways. They shouldn’t be put up in our classrooms — especially with regards to what students can learn and when.
Math Education’s energy and direction is largely a function of the comfort zone of teachers. The general philosophy is that whatever is taught and presented must be, at first, in the comfort zone of the teacher — before it can be in the comfort zone of the students.
While there is some truth to that, there seems something very school about that — follow topics order, be aligned to a curriculum that you had no say in constructing, and get students to be tested. Don’t skid off the road.
Let Dan Finkel take some of the burden off your shoulders. In his highly watched TED Talk, the Third Principle(of Five)for Extraordinary Math Teaching is:
Your job — your desire — should be to want to find the answer. But, you shouldn’t have to a some walking math encyclopedia. Not only shouldn’t you, you also can’t. I possess some ridiculously low percentage of math knowledge in the universe — as do all of us — so, to implicitly communicate that is disingenuous. The truthful path, the easier path, is to not be worried about the answer. Most of the time you will know it, but if you don’t, get excited that moment with your students — it might not come again.
One of the most challenging ideas to any mathematician on the planet is the concept of Graham’s Number. To say it is a beast of a number is an understatement as large its very size. Truth be told, very few people on the planet really have the comfort and knowledge to explain the size of that number to themselves — let alone anyone else.
I for sure know I am not one of them. And yet, this very concept(and picture below is the book “Math Recess: Playful Learning in an Age of Disruption).
I tried to explain it, and even offered an upfront apology that my explanation, well-intentions and all, was going to fail. The idea of Graham’s Number was put in the book more as a metaphor for learning and teaching mathematics. We have to stretch the canvas of our collective imaginations and embrace all the wonderful attributes of failing, not being perfect, and just being human — before anything else.
When are kids ready for an idea like this? Honestly, not sure. But, I think it is unhealthy to dwell on it, if in the end, we become gate keepers of knowledge and ideas based on comfort zones and norms that have reached their expiration date.
How does one capture the curiosity of students in mathematics? Competency is not the road. Many have taken that road, fulfilled their obligations with courses completed and marks obtained, and never looked back at mathematics again.
That is not what I want for my own kids. They say you shouldn’t chase money in life. Chase your dreams and your heart and that will follow. Same with education. Don’t paper chase. Chase the fantastic ideas, heady, and edgy ideas of mathematics found everywhere in K to 12 mathematics — accessible to all — and you will garner those societal badges of knowledge acquisition.
Impossible ideas. Try them and share them. If they fail, well heck, try another one. Don’t spend your entire teaching career inside the guard rails — you just might have the grave misfortune of never failing.
Ironically, never failing will create the arid soil in which curiosity will fail.
And that is the final measurement for all of us — did we nurture curiosity or did we destroy it.
Few subjects bring up as much remembered pain and anxiety as math classes. The confusing symbols, the difficult procedures and the dreaded graphs and charts.
A few people are now even suggesting that learning math can be a traumatic experience, something survived, rather than learned.
The painful history many people have with math is a shame, because math is incredibly useful. Many of the best careers are coming out of STEM fields, and rely on understanding math. Understanding the news and world events, is increasingly becoming a lesson in statistics. Finally, math, properly understood, allows you to solve many of your own problems.
In this article, I’d like to explain how you can teach yourself any kind of math, whether it be statistics, algebra or algorithms.
Step One: Start with an Explanation
The first step to learning any math is to get a first-pass explanation of the topic.
There are many places where you can get this information. Some good resources that cover a huge swath of topics are:
KhanAcademy – Huge, free resources of videos on nearly every math topic
MIT OCW – These start at the university level, but they handle a lot of complicated math
Numberphile – Conversations with mathematicians about interesting math topics
Wherever you get your explanation, your first step is to watch it once, so that you feel you understand the basics of how it works.
What If I Don’t Understand the Explanation?
If you watch the explanation, but you didn’t understand it, there are two possible problems:
You’re lacking some prerequisites for understanding this piece of math. That means you need to back up and go through it again. If it feels like it “went too fast” or you don’t know what the teacher is doing, you may need to go back a few lessons and learn those better before proceeding.
You’re trying to cover too much without going onto practice. A good pattern is to watch a chunk of explanation and then try it yourself. If you only watch, but never practice, that’s a bit like watching videos on skiing and never hitting the slopes. Eventually the explanations will stop making sense because you’ll lack firsthand experience.
Try this: Watch an explanation once, in full, as your starting point.
Step Two: Do Practice Problems
Math isn’t something you watch and memorize, but something you do.
If you spend all your time watching videos, and then get to a set of problems, you may find it really hard to apply your math knowledge. This can lead to the feeling that you’re “bad at math” even though the problem is just that you’re using a lousy method to learn it.
You can fix this by getting to doing problems as soon as possible. A good problem should feel challenging, but not impossible. If you see the solution and don’t even understand how they got it, chances are you’re going too fast—go back and learn some of the basics before moving on.
What if I Don’t Have Problems to Solve?
If you lack provided problems, there’s a few things you can do:
Work through the problems done in the explanation, but without looking at the answer.
Create your own problems and try to solve them.
Try to prove concepts in the class. This is an advanced technique, but it is essential for truly grasping more complicated math.
Try this: After watching your explanation, do enough problems to feel comfortable that you understand the procedure.
Step Three: Know Why The Math Works
Having an intuitive grasp is very important for math in a way that it isn’t for other subjects. While having an intuition for vocabulary words in a foreign language can help, they still need to be memorized. However, memorizing math can be dangerous if it causes you to learn it without understanding.
The next step is to convince yourself that you know why the math works. My favorite technique for doing this is the Feynman Technique, which I demonstrate here:
The Feynman Technique takes some time, so you don’t need to apply it fully to every single facet of every math problem you face. Rather, apply it selectively on the most important concepts and those that seem confusing to you, despite sufficient practice.
Try this: Identify the core concepts in the math you’re learning, and use the Feynman Technique to convince yourself you understand them.
Step Four: Play with the Math
Practice is good, understanding is better, but playing with math is best.
Once you have solved some questions provided to you and convinced yourself you understand them, a natural extension of this is to try to play with the math you’ve been given. How do things change when you try changing the numbers or apply it to different problems?
For example, let’s say you just recently learned how to calculate compound interest. You can do the simple interest calculations on your own, and you understand why they work. How could you play with this math?
You could see what happens as the rate of compounding increases.
What would happen if interest were negative?
You could try to calculate your own savings if you invested them at different rates.
Try imagining how much of a mortgage you pay in interest, versus principal.
Excel is a good way to play around with math, since you can put the formulas in directly, without needing to do as much algebra or repeating calculations.
Try this: Take a topic of math you’ve learned recently and see how you could change the variables, apply it to different things and modify the formulas.
Step Five: Apply the Math Outside the Classroom
Ultimately, the goal for learning math should be to use it, not merely pass a test. To do that, however, you need to break your understanding free of the textbook examples and apply it to real world situations.
This is more difficult than just solving a problem. When you solve a problem, you will start to memorize the pattern of the solution. This often allows you to solve problems without really understanding the principles behind how they work.
Applying math to real life, in contrast, requires recognizing the situation, translating it into math and then solving the problem you’ve created. This is strictly harder than solving problems, so if you want to be able to actually use what you learn, you need to practice this.
Try this: Take a topic you’ve learned recently in math and try to find a real-life situation where you could calculate it, using your own numbers or estimates if those aren’t available.
This All Sounds Like Too Much Work!
Doing all these five steps on every single thing you learn in math is going to be time consuming. That’s fine, you don’t need to do this for every little thing you need to learn.
Instead, think of this like a progress bar. Every math concept you learn can go from steps one through five, deepening your knowledge and increasing the usefulness of the math each time. Some concepts will be important enough that you’ll want to apply them thoroughly. Others will be infrequent enough that just watching the explanation is all the time you can spare.
In particular, you should try to focus on the most important concepts for each idea. Math tends to be deep, so often in a full semester class, there may only be a handful of really big ideas, with all the other ideas being simply different manifestations of that basic concept.
Most first-year calculus courses, for instance, all center around the concept of a derivative, with everything being taught merely being different extensions and applications of that core idea. If you really understand what a derivative is and how it works, those other pieces will be much easier to learn.
A Comprehensive Course in Analysis is number one in my analysis list of should reads! if you know it properly then this post has nothing to say to you. but if you are one of those people in mathematics that have not yet heard , then I suggest you to follow this link: A Comprehensive Course in Analysis. And more info: About the Author and the set