No need to rationalize the denominator. Maybe in 1920, but not in 2020.
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Replying to @divbyzero @abusch38
Agree. It's a decent exercise in arithmetic/algebraic manipulation when learning the rules, but mostly a waste of student time and good will at the university level. I want them to understand why things are true. Why rationalize? Is there a reason? 1/sqrt(3)=sqrt(3)/3, so why?
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Before the age of calculators it was much easier to divide sqrt(2) by 2 than it was to divide one by sqrt(2). But now, who cares?
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I was doing high school maths before (pocket) calculators were available and, as far as I can remember, at that time I only ever saw sin(45°) as 1/√2, not √2/2.
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For real? I've never been in a class (as a student) or in a school (as a teacher) that was okay with that. Craziness.
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I believe that. Students forget all kinds of math as they progress, but "don't leave a square root in the denominator" is rock solid in all their brains—like it is "don't divide by zero." I'm exaggerating, but not too much. They get very uncomfortable when I tell them it is OK
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Replying to @divbyzero @abusch38 and
to leave an expression as 1/sqrt(2), and most of them ignore my invitation. That tells me that it is firmly implanted in the high school curriculum. Speaking only for myself, I think that should change.
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Replying to @divbyzero @abusch38 and
Any system of mathematics education that inculcates a widespread and stubborn disposition to mistake a pure convention for a fundamental law is fatally flawed. We might dismiss as accidental this single instance of perversity, were there not so many others, equally flagrant.
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Replying to @MathPrinceps @divbyzero and
Exactly. PEMDAS. Always put the function in standard form. It takes time to unspool these beliefs and get to the actual mathematics. It seems to be a part of the strength of active learning/IBL. Students get back some of the power that they lose to (unneeded?) simplifications.
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Replying to @ef_math @divbyzero and
Alas, the essence of the problem is that the teacher corps, lacking real mathematical understanding, often cannot distinguish between conventions and axioms. Which is in turn a consequence of the irresponsible, self-serving indifference of mathematicians, who merely shrug at it.
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Mathematical teaching ought to have at least a passing resemblance to mathematical practice, with which only mathematicians are sufficiently acquainted to convey. But mathematicians (vastly) prefer to do mathematics, and to view pedagogy as a fraught and futile backwater.
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Replying to @MathPrinceps @ef_math and
Most mathematicians are indifferent even to the mathematical misery of their own children in school, or, if not actually indifferent, insufficiently motivated to try to do anything material about it. A little discreet hand-wringing is usually enough to assuage their consciences.
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Replying to @MathPrinceps @ef_math and
I think it has more to do with curriculum than secondary teachers not knowing. Their stuff. If the textbook manufacturers put everything in simplest form, the teacher is not going to fight that fight.
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