Sarah

“afab chests aren’t inherently sexu—” please say breasts please just say breasts please please please stop throwing the term “afab” around say tatas say titties say big bahoona bazingoroos if you must

not to be that guy but if you’re talking about breasts, say breasts. if you’re talking about pregnancy, say pregnancy. you don’t have to bend over backwards coming up with wacky shit like “uterus-having pregnancy-carrier” or “afab presenting nipples” to be more inclusive. half the time, it’s not inclusive anyway. so just be specific. a person with breasts can simply be “a person with breasts.” it’s easy, it’s fun, it’s free

I don't really think I agree with this sentiment. In my opinion, student intuitions are often right or at least reasonable. Numbers have historically meant many things but the modern notion of number, as something 'measurable' and concretely realizable by a decimal expansion seems so engrained, and this heavily integrates into a known geometry so utterly profound (Cartesian coordinates). This basis of 'real number' as 'measurement' is its utility in pretty much every other field (physics, engineering...) and at its core, this is what 'measurable reality' is for many. Complex numbers, by contrast, don't have such an intrinsic definition. Any definition just makes it seem less like a number and more like a convenient shorthand to wrap up a structure based in two numbers and so, too students, seems to be 'skewing and cheating on what it means to be a measurable number'. This is why the case with negatives as invented doesn't seem to be on par with complex numbers and its worth recognizing this 'obviously evident' intuition of the student. It feels just as much a number as 'squaring' matrices. Ask someone what's the square root of 2 and one can give a precise answer using other numbers, ask someone what's the square root of -1 and it seems someone is just throwing out a symbol i and calling it the square root of -1. I'm no historian of science or math, but from what I've seen, such sentiments carry across the sciences too where, because of complex numbers are not 'measurable' and are always considered a 'convenience' and not a 'true number'. Like, out of the few instances I know, it was used for complex impedance, Schrodinger's equation but originally when it was written, it used entirely real variables and I think it was mentioned imaginary number as a convenience. The quaternions used to formulate electromagnetism until the vectorial form won out. The irony is that for mathematicians, it felt that, perhaps because of the niceties with algebra and analysis, everything was seen as like the fundamental numbers and everything defined using them (eg, this is how 'manifolds' first became a thing: through riemann surfaces). Such happened too in the study of quadratic forms as Weyl pointed out in something I can't find rn. In relativity, complex numbers were used because it allowed a 'Pythagorean' form for space-time distance, but realizing this to not be a physical aspect, came to realize that actually they should investigate a novel geometry and a different 2 signature bilinear form and pseudoriemannian spaces and I believe these kinds of explorations lead to representation theory. So I feel I have more an issue with the reverse, where many seem to act overzealous as treating complex numbers as like an implicitly provable aspect of math (I've seen the argument 'its proven n degree polynomials must have n roots and these are the complex roots' and this common kind of lame portrayal). Indeed, it'd be just as wrong to say there *is* a square root of -1 since such a thing is structurally contingent and, unlike for how mathematicians portrayed it, complex numbers are not the more 'complete' numbers. Although, most school books do explicitly mention at the very least no 'real solutions' to x^2=-1 and not 'no solution'. This issue is one that cuts deep into the very nature of math and what it means to be real or even useful in math. I'm not even sure a good pedagocial answer has been given for complex numbers. The algebra used to define it doesn't give a clear reason why the analysis its known for works the way it does. I think probably the most successful way to teach them is through the notion of polar coordinate representations and it like a number system of 2D geometry and only introduce the 'cartesian representation' later. At the very least, you can depict its most widespread application in the physics and engineering rather than treating it as a game.

I think its very important to remember that math isn't just the abstract system playground mathematicians explore in. This view of math is actually a very high level type perspective that is developed only from very gradual math experience that its easy to forget that it does not at all translate to others. So while yes given how you view math its easy to see how its taught as not portraying it right, and that's probably true to an extent from what I've seen, it's also worth considering the hindsight bias that has from your level of experience, and how you need to consider how students themselves understand it. I think it is quite evident that there are reasons complex numbers are treated differently to the reals and one shouldn't ignore that at the very least, and the student questions on these should be really considered: this isn't about physical intuition but context as a whole. When I say 'implicitly provable aspect', I don't mean in terms of proof systems but as some evidently justified aspect, physical or not. Proofs and formalizations are built on top of math to recontextualize it: mathematicians were able to work with naturals and real numbers without a formalization. And some of these demands (like axiom of infinity) is only a thing in fitting a specific formalization, namely conceptualizing natural number arithmetic as an entire singular structure taking the totality of naturals into one set itself. This is in itself not needed for naturals or speaking of their 'potentially infinite' nature. So yes while it true numbers don't have a definite meaning and can be defined by what they do, there aren't things pertaining to existence both in logic, physical reality, and 'something else' that makes something quantify like a number and have a more tangible existence for most people. It's a valid view when we think of other worldviews outside of math itself. So, while in some part teaching is at play, its also worth considering the concept itself and not treating it 'all as equal'. Because try as I might, there is hardly any totally 'direct' digestible examples of complex numbers I can think of. It doesn't feel its as 'essential' and implicitly a 'number' of a number system as something as clearly quantifiable as real numbers. The closest example to showcase complex numbers I can think is phase and waves: and that requires at least trig, so precalculus. All of this requires at least the crucial abstraction learned throughout algebra and geometry, and that can't be rushed through for most. I'd argue though such a thing isn't even understood until calculus, and where you might actually see actual use. In earlier years, numbers are learned through gradually concrete means because that's all there is to go off of. Rationals aren't some operation playground but roughly explained by its representation as proportion. One can learn about decimals now. This is all contextualized on a number line, which is a continuous geometric object: with these together, one is naturally lead to the concept of real numbers and the demonstrated truth of irrationals. Then, these build the coordinate grid, where functions, linear equations need to be explained. Complex numbers can be maybe introduced here but I'm not sure how to bridge the geometric image of the rotation to the actual complex arithmetic itself. I mean thinking it now, and from how I've seen it taught at tutor places and just, gosh how critically missed basic concepts in algebra are, students rushed to algebra before they can do arithmetic, etc., like a massive overhaul of the system can be done. But, I'm still wary over your perception of complex numbers as something so analogous to teach as say the reals.

I don't really think I agree with this sentiment. In my opinion, student intuitions are often right or at least reasonable. Numbers have historically meant many things but the modern notion of number, as something 'measurable' and concretely realizable by a decimal expansion seems so engrained, and this heavily integrates into a known geometry so utterly profound (Cartesian coordinates). This basis of 'real number' as 'measurement' is its utility in pretty much every other field (physics, engineering...) and at its core, this is what 'measurable reality' is for many. Complex numbers, by contrast, don't have such an intrinsic definition. Any definition just makes it seem less like a number and more like a convenient shorthand to wrap up a structure based in two numbers and so, too students, seems to be 'skewing and cheating on what it means to be a measurable number'. This is why the case with negatives as invented doesn't seem to be on par with complex numbers and its worth recognizing this 'obviously evident' intuition of the student. It feels just as much a number as 'squaring' matrices. Ask someone what's the square root of 2 and one can give a precise answer using other numbers, ask someone what's the square root of -1 and it seems someone is just throwing out a symbol i and calling it the square root of -1. I'm no historian of science or math, but from what I've seen, such sentiments carry across the sciences too where, because of complex numbers are not 'measurable' and are always considered a 'convenience' and not a 'true number'. Like, out of the few instances I know, it was used for complex impedance, Schrodinger's equation but originally when it was written, it used entirely real variables and I think it was mentioned imaginary number as a convenience. The quaternions used to formulate electromagnetism until the vectorial form won out. The irony is that for mathematicians, it felt that, perhaps because of the niceties with algebra and analysis, everything was seen as like the fundamental numbers and everything defined using them (eg, this is how 'manifolds' first became a thing: through riemann surfaces). Such happened too in the study of quadratic forms as Weyl pointed out in something I can't find rn. In relativity, complex numbers were used because it allowed a 'Pythagorean' form for space-time distance, but realizing this to not be a physical aspect, came to realize that actually they should investigate a novel geometry and a different 2 signature bilinear form and pseudoriemannian spaces and I believe these kinds of explorations lead to representation theory. So I feel I have more an issue with the reverse, where many seem to act overzealous as treating complex numbers as like an implicitly provable aspect of math (I've seen the argument 'its proven n degree polynomials must have n roots and these are the complex roots' and this common kind of lame portrayal). Indeed, it'd be just as wrong to say there *is* a square root of -1 since such a thing is structurally contingent and, unlike for how mathematicians portrayed it, complex numbers are not the more 'complete' numbers. Although, most school books do explicitly mention at the very least no 'real solutions' to x^2=-1 and not 'no solution'. This issue is one that cuts deep into the very nature of math and what it means to be real or even useful in math. I'm not even sure a good pedagocial answer has been given for complex numbers. The algebra used to define it doesn't give a clear reason why the analysis its known for works the way it does. I think probably the most successful way to teach them is through the notion of polar coordinate representations and it like a number system of 2D geometry and only introduce the 'cartesian representation' later. At the very least, you can depict its most widespread application in the physics and engineering rather than treating it as a game.

Its generally my opinion that in discourse of media descriptions become fit to different feelings as a description of why this is good nor but which alone is not necessary justification of it. It's not really about relatability I think as much the extent it feels justifiable and plausible given their mindset. Since the goal of any story is to essentially build a world the author deceives you to be experiencing even when you both know its fiction. Characters whose sense of reasoning, purpose, or action that feels its in break with a person but of simply a character doing things because plot is the real issue.