[HN Gopher] The symmetry that makes solving math equations easy
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The symmetry that makes solving math equations easy
Author : nsoonhui
Score : 42 points
Date : 2023-03-25 09:40 UTC (1 days ago)
(HTM) web link (www.quantamagazine.org)
(TXT) w3m dump (www.quantamagazine.org)
| photochemsyn wrote:
| 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
| [deleted]
| nightfly wrote:
| Cheating has never need hard to do. Before the internet there
| were friends/people that could be bribed
| moffkalast wrote:
| 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?
| anonymouskimmer wrote:
| 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.
| svantana wrote:
| 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.
| onos wrote:
| That's not right. For example, if you have access to any sort
| of radical you still can't solve the quintic.
| iamerroragent wrote:
| "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.
| geysersam wrote:
| 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
| scarecrw wrote:
| 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.
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