The symmetry that makes solving math equations easy
quantamagazine.org
The symmetry that makes solving math equations easy
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Re: The symmetry that makes solving math equations easy
#2Nice 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:
Re: The symmetry that makes solving math equations easy
#3I 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.
Re: The symmetry that makes solving math equations easy
#4Nice 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 wan…
Cheating has never need hard to do. Before the internet there were friends/people that could be bribed
Re: The symmetry that makes solving math equations easy
#5I 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.
Re: The symmetry that makes solving math equations easy
#6I 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.
Re: The symmetry that makes solving math equations easy
#7Nice 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 wan…
[deleted]
Re: The symmetry that makes solving math equations easy
#8I 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.
"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.
Re: The symmetry that makes solving math equations easy
#9Nice 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 wan…
Cheating has never need hard to do. Before the internet there were friends/people that could be bribed
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?
Re: The symmetry that makes solving math equations easy
#10I 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.