The symmetry that makes solving math equations easy(quantamagazine.org)
quantamagazine.org
The symmetry that makes solving math equations easy
https://www.quantamagazine.org/the-symmetry-that-makes-solving-math-equations-easy-20230324/
49 comments
It's fortunate that we so seldom need to solve cubic or higher order polynomial equations. Quadratic equations come up in physics all the time, because it's very common to have a constant acceleration. But when does one ever come across a constant jerk?
As others pointed out, this is just a fallacy. Physics (as well as Math, CS, and all other disciplines) is full of non-linear, chaotic problems that are way too hard to solve, so academia doesn't like to teach them to new students and even experts aren't delighted to deal with them, because it's frustrating that we can't solve them (in a "beautiful" and "elegant" way). My own hot take is that Physics is stuck on a theory of everything for the last 100 years, because we are stuck on some fundamental limitations what Math is able to model.
>My own hot take is that Physics is stuck on a theory of everything for the last 100 years, because we are stuck on some fundamental limitations what Math is able to model.
Your previous assumption reads more like math being able to model it, but scientists preferring elegant, aesthetic and simplistic solutions.
Looking at my PhD physics neighbor in student housing a few years back, it doesn't seem true though. He was juggling multi-page equations for his quantum field theory thesis.
In control engineering it's more true though. I remember one of my professors say that many people only use linear controls because there they can "prove" the stability of the system. In contrast to the nonlinear, i.e. everything else, control theory where such proofs were harder to get by.
However, he continued, proving the stability of a linear controller interacting with a linearized system could never "prove" the system's stability as a whole.
Your previous assumption reads more like math being able to model it, but scientists preferring elegant, aesthetic and simplistic solutions.
Looking at my PhD physics neighbor in student housing a few years back, it doesn't seem true though. He was juggling multi-page equations for his quantum field theory thesis.
In control engineering it's more true though. I remember one of my professors say that many people only use linear controls because there they can "prove" the stability of the system. In contrast to the nonlinear, i.e. everything else, control theory where such proofs were harder to get by.
However, he continued, proving the stability of a linear controller interacting with a linearized system could never "prove" the system's stability as a whole.
From experience linear controls is probably more popular because there's a larger set of tools in the toolbox, so to speak, and the nature of the systems is that this is (usually) close enough for modelling nonlinear systems which are close enough to linear (in fact you may introduce nonlinear terms in your controller to try to remove nonlinearities in your system). It's not just having a 'proof' of stability, it's having methods to analyse the system and optimise the controller, which are much more difficult to apply in general to non-linear systems. It's much like how engineers will generally just stick to classical models of physics despite them being 'wrong', because in practice they are close enough and also much easier to deal with.
> because we are stuck on some fundamental limitations what Math is able to model.
This seems more a limitation of what (human) mathematicians are able to model, rather than something fundamental in math itself. Perhaps in a few centuries we will have invented mathematical tools that allow the mind to grasp nonlinear systems better, similar to how the invention of better construction tooling has enabled the construction of ever larger buildings.
This seems more a limitation of what (human) mathematicians are able to model, rather than something fundamental in math itself. Perhaps in a few centuries we will have invented mathematical tools that allow the mind to grasp nonlinear systems better, similar to how the invention of better construction tooling has enabled the construction of ever larger buildings.
On the other hand, physics is filled with problems that are far worse behaved than even large polynomials. 3+ massive bodies interacting under Newtonian gravity. 2+ massive bodies interacting in General relativity [0], anything in quantum field theory, almost all differential equations you run into outside of a textbook on differential equations.
[0] To be far, I'm not actually positive that the 2 body problem in GR is actually computationally harder than massive polynomials, but there is no exact solution to it; and understanding the approximation is definitely harder.
[0] To be far, I'm not actually positive that the 2 body problem in GR is actually computationally harder than massive polynomials, but there is no exact solution to it; and understanding the approximation is definitely harder.
I've worked with several.
I did audibly chuckle at this one. Good job, doesn't happen to me often online!
I think I waited 25 years for the opportunity to unleash this particular dad joke.
For when one would come across a constant jerk - how about the usage case of putting a stopped vehicle into motion? You gradually push the accelerator or throttle from zero to maximum at a constant pace - that's increasing the acceleration linearly (ignoring friction/drag), thus constant jerk, which the occupants will perceive as a smooth start. This probably doesn't often get solved numerically, but it's something you do instinctively do.
>It's fortunate that we so seldom need to solve cubic or higher order polynomial equations.
We do, it's just that we choose different formulations for problems which require such solutions, or straight up don't solve them.
Think about the unsolvable n-body problem from classical mechanics. An equivalent of that problem exists in most scientific disciplines. We just choose not to think of those problems from the polynomial solving angle.
We do, it's just that we choose different formulations for problems which require such solutions, or straight up don't solve them.
Think about the unsolvable n-body problem from classical mechanics. An equivalent of that problem exists in most scientific disciplines. We just choose not to think of those problems from the polynomial solving angle.
I always wondered why that is for a long time (in basic physics that at least). It wasn't until grad school that I understood that position and velocity are enough to uniquely determine a trajectory on a manifold.
IMO the bigger part is that numerical root finding works really well in practice. There's plenty of times you need to solve for roots of equations, but getting within 1e-6 is often good enough.
A rocket could be an example, assuming constant thrust and linearly decreasing mass, the decreasing mass would result in constant jerk.
>when does one ever come across a constant jerk?
Read the news lately?
Read the news lately?
Doesn't every startup come bundled with one?
There’s an interesting detail that the article omits, which is that in that long expression, it’s possible to end up with complex numbers under the cube root sign even if the roots of the polynomial are all real! The formula was discovered before mathematicians accepted the idea of imaginary numbers which made the formula suspect in its early history.
I have the opportunity to introduce (or re-introduce) quadratics to students fairly regularly, and I'm eager to incorporate this understanding! I've often highlighted the symmetry of the quadratic formula, though usually we get there via completing the square, rather than this translation approach.
I desperately wish students got more practice with function transformations. It's a powerful tool that far too many students leave high school without understanding.
I desperately wish students got more practice with function transformations. It's a powerful tool that far too many students leave high school without understanding.
I feel like this article is a bit backwards - solving quadratic equations is only easy if you have access to the square root function, which by definition is a solver of quadratic expressions. Without it, one needs to resort to iterative root-finding, which works for polynomials of any order.
That’s not right. For example, if you have access to any sort of radical you still can’t solve the quintic.
But if you have access to an extended set of operations (ultraradicals), in particular an operation that solves a parameterized quintic, you can solve all quintics.
https://en.m.wikipedia.org/wiki/Bring_radical
https://en.m.wikipedia.org/wiki/Bring_radical
I believe that doesn't extend beyond degree 6 polynomials though. Isn't that Hilbert's 13th problem?
"Without it, one needs to resort to iterative root-finding, which works for polynomials of any order."
I believe that's what they mean for quintics. It's been a while for me apologies if I'm miss remembering here.
I believe that's what they mean for quintics. It's been a while for me apologies if I'm miss remembering here.
Just use Newton's method!
I can't tell if this is a joke. Newton's method will give you an approximate answer.
You can write the answer as a recurrence in symbolic form.
Or better yet, gpt
Computing (or computing with) a square root is much easier than solving a quadratic equation!
- A square root’s result is always nonnegative (or in the complex case, has nonnegative real part) and unique, whereas the quadratic equation has two solutions.
- You can manipulate the root function symbolically much better than the roots of an equation. sqrt(2)^2 is 2, but figuring out that `solution(x^2-2=0)^2` reduces to the same number is much less obvious, and even more so for more complicated roots.
- A square root’s result is always nonnegative (or in the complex case, has nonnegative real part) and unique, whereas the quadratic equation has two solutions.
- You can manipulate the root function symbolically much better than the roots of an equation. sqrt(2)^2 is 2, but figuring out that `solution(x^2-2=0)^2` reduces to the same number is much less obvious, and even more so for more complicated roots.
>A square root’s result is always nonnegative (or in the complex case, has nonnegative real part)
I don’t know what definition you’re using, but if we take the square root of Y to be a number X such that X*X = Y, then the above statement isn’t true.
I don’t know what definition you’re using, but if we take the square root of Y to be a number X such that X*X = Y, then the above statement isn’t true.
The principal square root (ie, the one with positive part) is commonly referred to as the square root, especially since the radical symbol is explicitly defined to produce the principal root.
It just seems odd, because the reason why the quadratic equation has/can have two solutions, is due directly to the square root having two solutions.
Very nicely written article! Easily approachable for someone in high school.
I just had one problem in reading it - the somewhat inconsistent use of fonts - often in the same line was not pleasant. The numeric zero ('0') in particular looks like character 'oh' ('o').
I just had one problem in reading it - the somewhat inconsistent use of fonts - often in the same line was not pleasant. The numeric zero ('0') in particular looks like character 'oh' ('o').
If you're interested in exploring the beauty of symmetry further, and want to dive deeper into the world of math, I highly recommend checking out the book "LOVE and MATH: The Heart of Hidden Reality" by Edward Frenkel.
Here's a long quote from "LOVE and MATH" that gives a taste of the book:
"The story of Galois is one of the most romantic and fascinating stories about mathematicians ever told. A child prodigy, he made groundbreaking discoveries very young. And then he died in a duel at the age of twenty. There are different views on what was the reason for the duel, which happened on May 31, 1832: some say there was a woman involved, and some say it was because of his political activities. Certainly, Galois was uncompromising in expressing his political views, and he managed to upset many people during his short life.
It was literally on the eve of his death that, writing frantically in a candlelit room in the middle of the night, he completed his manuscript outlining his ideas about symmetries of numbers. It was in essence his love letter to humanity in which he shared with us the dazzling discoveries he had made. Indeed, the symmetry groups Galois discovered, which now carry his name, are the wonders of our world, like the Egyptian pyramids or the Hanging Gardens of Babylon. The difference is that we don’t have to travel to another continent or through time to find them. They are right at our fingertips, wherever we are. And it’s not just their beauty that is captivating; so is their high potency for real-world applications.
...
What Galois had done was bring the idea of symmetry, intuitively familiar to us in geometry, to the forefront of number theory. What’s more, he showed symmetry’s amazing power.
...
The general formula for the solutions of the quadratic equations was already known to the Persian mathematician Al-Khwarizmi in the ninth century (the word “algebra” originated from the word “al-jabr,” which appears in the title of his book). Formulas for solutions of the cubic and quartic (degree 4) equations were discovered in the first half of the sixteenth century. Naturally, the next target was a quintic equation (of degree 5). Prior to Galois, many mathematicians had been desperately trying to find a formula for its solutions for almost 300 years, to no avail. But Galois realized that they had been asking the wrong question. Instead, he said, we should focus on the group of symmetries of the number field obtained by adjoining the solutions of this equation to the rational numbers – this is what we now call the Galois group.
The question of describing the Galois group turns out to be much more tractable than the question of writing an explicit formula for the solutions. One can say something meaningful about this group even without knowing what the solutions are. And from this one can then infer important information about the solutions. In fact, Galois was able to show that a formula for solutions in terms of radicals (that is, square roots, cubic roots, and so on) exists if and only if the corresponding Galois group has a particularly simple structure: is what mathematicians now call a solvable group. For quadratic, cubic, and quartic equations, the Galois groups are always solvable. That’s why solutions of these equations may be written in terms of the radicals. But Galois showed that the group of symmetries of a typical quintic equation (or an equation of a higher degree) is not solvable. This immediately implies that there is no formula for solutions of these equations in terms of radicals.
...
Galois’ work is a great example of the power of a mathematical insight. Galois did not solve the problem of finding a formula for solutions of polynomial equations in the sense in which it was understood. He hacked the problem! He reformulated it, bent and warped it, looked at it in a totally different light. And his brilliant insight has forever changed the way people think about numbers and equations.
Here's a long quote from "LOVE and MATH" that gives a taste of the book:
"The story of Galois is one of the most romantic and fascinating stories about mathematicians ever told. A child prodigy, he made groundbreaking discoveries very young. And then he died in a duel at the age of twenty. There are different views on what was the reason for the duel, which happened on May 31, 1832: some say there was a woman involved, and some say it was because of his political activities. Certainly, Galois was uncompromising in expressing his political views, and he managed to upset many people during his short life.
It was literally on the eve of his death that, writing frantically in a candlelit room in the middle of the night, he completed his manuscript outlining his ideas about symmetries of numbers. It was in essence his love letter to humanity in which he shared with us the dazzling discoveries he had made. Indeed, the symmetry groups Galois discovered, which now carry his name, are the wonders of our world, like the Egyptian pyramids or the Hanging Gardens of Babylon. The difference is that we don’t have to travel to another continent or through time to find them. They are right at our fingertips, wherever we are. And it’s not just their beauty that is captivating; so is their high potency for real-world applications.
...
What Galois had done was bring the idea of symmetry, intuitively familiar to us in geometry, to the forefront of number theory. What’s more, he showed symmetry’s amazing power.
...
The general formula for the solutions of the quadratic equations was already known to the Persian mathematician Al-Khwarizmi in the ninth century (the word “algebra” originated from the word “al-jabr,” which appears in the title of his book). Formulas for solutions of the cubic and quartic (degree 4) equations were discovered in the first half of the sixteenth century. Naturally, the next target was a quintic equation (of degree 5). Prior to Galois, many mathematicians had been desperately trying to find a formula for its solutions for almost 300 years, to no avail. But Galois realized that they had been asking the wrong question. Instead, he said, we should focus on the group of symmetries of the number field obtained by adjoining the solutions of this equation to the rational numbers – this is what we now call the Galois group.
The question of describing the Galois group turns out to be much more tractable than the question of writing an explicit formula for the solutions. One can say something meaningful about this group even without knowing what the solutions are. And from this one can then infer important information about the solutions. In fact, Galois was able to show that a formula for solutions in terms of radicals (that is, square roots, cubic roots, and so on) exists if and only if the corresponding Galois group has a particularly simple structure: is what mathematicians now call a solvable group. For quadratic, cubic, and quartic equations, the Galois groups are always solvable. That’s why solutions of these equations may be written in terms of the radicals. But Galois showed that the group of symmetries of a typical quintic equation (or an equation of a higher degree) is not solvable. This immediately implies that there is no formula for solutions of these equations in terms of radicals.
...
Galois’ work is a great example of the power of a mathematical insight. Galois did not solve the problem of finding a formula for solutions of polynomial equations in the sense in which it was understood. He hacked the problem! He reformulated it, bent and warped it, looked at it in a totally different light. And his brilliant insight has forever changed the way people think about numbers and equations.
Nice write-up! Incidentally I ran all the exercises through ChatGPT which seems to have completely crushed them, with work shown. Forget about assigning grades via math homework, it only took a few minutes (though I should go through and check it by hand I think). [edit looking at the answers in more detail it seems to go down some strange rabbit holes that probably were not what the author intended]
Also, if one wanted to extend this article a little, ask it about the role quadratic equations with complex roots played Gauss's derivation of the fundamental theorem of algebra. Here's the core of the output:
> "Suppose we have a quadratic equation with complex roots. The roots of the equation are given by the formula:
x = (-b ± sqrt(b^2 - 4ac))/2a
> "If b^2 - 4ac is negative, then the roots of the equation are complex numbers. Gauss recognized that the complex roots of a quadratic equation come in conjugate pairs. That is, if one root is a + bi, then the other root is a - bi, where a and b are real numbers."
> "Gauss used this fact to show that any polynomial equation with complex coefficients can be factored into linear factors with complex roots. He did this by taking pairs of complex conjugate roots and combining them into quadratic factors. Then he repeated this process until all the roots had been combined into linear factors."
Numberphile has a video on this and the rest of Gauss's proof:
https://youtu.be/shEk8sz1oOw
Also, if one wanted to extend this article a little, ask it about the role quadratic equations with complex roots played Gauss's derivation of the fundamental theorem of algebra. Here's the core of the output:
> "Suppose we have a quadratic equation with complex roots. The roots of the equation are given by the formula:
x = (-b ± sqrt(b^2 - 4ac))/2a
> "If b^2 - 4ac is negative, then the roots of the equation are complex numbers. Gauss recognized that the complex roots of a quadratic equation come in conjugate pairs. That is, if one root is a + bi, then the other root is a - bi, where a and b are real numbers."
> "Gauss used this fact to show that any polynomial equation with complex coefficients can be factored into linear factors with complex roots. He did this by taking pairs of complex conjugate roots and combining them into quadratic factors. Then he repeated this process until all the roots had been combined into linear factors."
Numberphile has a video on this and the rest of Gauss's proof:
https://youtu.be/shEk8sz1oOw
Cheating has never need hard to do. Before the internet there were friends/people that could be bribed
Getting help on homework isn't really cheating. I had one professor who sort of teased the class when we admitted we were mostly working on homework alone. He felt we wers suspposed to be working in groups.
I think the only reason homework is graded at all is because if it wasn't, teachers assume the students wouldn't do it. The tests are there to test you. The homework is just there for practice. But if you just let someone else do your homework, you're unlikely to pass the test. Conversely, if you can score well on the tests without doing the homework, you shouldn't have to do it, it's just a waste of time.
Still that's kind of like the "you won't always have a calculator with you" argument. Going to other humans takes time, persuasion and can be expensive. This is a glorified calculator that's free and one click away.
Is it really cheating when it's just using another tool in the box? People should learn to do more with everything at their disposal, not arbitrarily limit themselves. Should I not use a 3D printer because I ought to sculpt by hand? Must I not use a regular printer because I should write and draw everything with a pencil?
Is it really cheating when it's just using another tool in the box? People should learn to do more with everything at their disposal, not arbitrarily limit themselves. Should I not use a 3D printer because I ought to sculpt by hand? Must I not use a regular printer because I should write and draw everything with a pencil?
We teach kids in the hopes they will be more able to verify truth or falsity when they can't do the hard work themselves.
This is why black boxes are deprecated in schooling, until the point comes when we believe the students understand enough to be able to verify truth or falsity, at which point we let them use black boxes so that they can learn more. But still, when they are using those black boxes to learn more, we're keeping them from using a black box of the "more". Again, in the hopes that they'll understand the "more" enough to be able to verify the truth or falsity of the results of that "more". Once they can do this, they've hopefully graduated, and are free to use a black box of the "more".
Use black boxes from the get-go and no one realizes that soylent green is people. Or that the Morlocks feed on the Eloi.
This is why black boxes are deprecated in schooling, until the point comes when we believe the students understand enough to be able to verify truth or falsity, at which point we let them use black boxes so that they can learn more. But still, when they are using those black boxes to learn more, we're keeping them from using a black box of the "more". Again, in the hopes that they'll understand the "more" enough to be able to verify the truth or falsity of the results of that "more". Once they can do this, they've hopefully graduated, and are free to use a black box of the "more".
Use black boxes from the get-go and no one realizes that soylent green is people. Or that the Morlocks feed on the Eloi.
Still if the end goal is to be able to verify the result, learning the entire process is usually not required. Almost everything we use is a black box and while it's good to know the basics to have a good understanding, it's also not a bad idea to strike a balance when dedicating limited learning time between required background and things that are practically useful.
This ultimately becomes vocational specialization.
Solving exercises have never been a practical problem. The solution is already known by the one asking the question, the only point is training and skill assessment.
The point of solving these exercises is for you to train yourself in manipulating functions, maybe so that you can later do it in more complex and novel situations where neither ChatGPT nor WolframAlpha will find the solution for you (or worse, give you a wrong answer). It doesn't mean they are useless tools, actual scientists and engineers use them, but because they actually solved the exercises when they learned to do maths, they are able to recognize when they fail and adjust their queries to have they do what they need.
And while the "you won't always have a calculator with you" is mostly wrong nowadays. And it also was before smartphones since pocket calculators were extremely cheap and small so you could always have one on you if you wanted to, and they were actually more reliable than smartphones. What is important is not to put your entire trust into the calculator. You have to recognize that if you calculator tells you that $42 + 10% is $63, you messed up somewhere. It is the same with ChatGPT, it can help you, but if you don't know your subject, you won't be able to differentiate between helpful answers and hallucinations, making it worse than useless.
The point of solving these exercises is for you to train yourself in manipulating functions, maybe so that you can later do it in more complex and novel situations where neither ChatGPT nor WolframAlpha will find the solution for you (or worse, give you a wrong answer). It doesn't mean they are useless tools, actual scientists and engineers use them, but because they actually solved the exercises when they learned to do maths, they are able to recognize when they fail and adjust their queries to have they do what they need.
And while the "you won't always have a calculator with you" is mostly wrong nowadays. And it also was before smartphones since pocket calculators were extremely cheap and small so you could always have one on you if you wanted to, and they were actually more reliable than smartphones. What is important is not to put your entire trust into the calculator. You have to recognize that if you calculator tells you that $42 + 10% is $63, you messed up somewhere. It is the same with ChatGPT, it can help you, but if you don't know your subject, you won't be able to differentiate between helpful answers and hallucinations, making it worse than useless.
Does this work for more advanced maths? It's a shame the amount you can spend per book for self-study, without access to the exercise answers needed to confirm your comprehension.
No.
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