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Once I renamed imaginary numbers to 2D numbers internally it made more sense. Do you know any other examples in math where fixing terrible naming makes the con
by kowdermeister 8y ago
Once I renamed imaginary numbers to 2D numbers internally it made more sense.
Do you know any other examples in math where fixing terrible naming makes the concept easier to digest?
- another-cuppa 8y agoIt's not a terrible name, it's just that we've forgotten the real meaning of "complex" and use it as a synonym for "complicated".
- mrob 8y agokowdermeister wrote about imaginary numbers, not complex numbers. "Imaginary" is a terrible name, because all numbers are imaginary. Even integers are imaginary, in the sense that the percentage of integers with actual physical representations is 0%, rounded to however many decimal places you like. "Real" is a terrible name for numbers for the same reason. "Complex" was originally justifiable but it should be renamed to reflect changes in the English language.
- tangentspace 8y agoWhat would happen if we started renaming mathematical objects to reflect changes in the English language? English will continue to evolve, and the vast body of mathematical literature would have to be constantly rewritten. Wouldn't that cause much, much greater confusion due to mathematical nomenclature being a moving target rather than remaining stable?
- mrob 8y agoMathematical nomenclature is already a moving target. And the symbols don't have to change, only the English reading of them, e.g. ∫ is from the latin "summa" (sum), but you don't need to know that to read it as "integral"
- tangentspace 8y agoI can't think of any examples of mathematical objects being renamed, do you have any? There may be some amount of drift over time, but you can go to the research library of any university math department and find books from pre-WWII that are still totally readable because although style has shifted somewhat, the basic nomenclature hasn't.
- deleted 8y ago[deleted]
- sorokod 8y agoThere is "random variable" a.k.a function.
- ginnungagap 8y agorandom variable as a shortening of "measurable function from a probability space" doesn't seem too bad to me
- sorokod 8y agoDon't see how this works as the extended version doesn't mention random or variable.
- ginnungagap 8y agoMy point is that is isn't just a function, but a measurable one (with weak restriction on the measure space it has as domain) so we can't call it just "function" as you were suggesting. The actual name might be debatable, but a shortened name rather than the full definition definitely makes sense here. Random variable seems like a good choice of name to me, thinking about the intuition we're formalizing with it.
- tangentspace 8y agoBut you propose to simply relabel as function, which doesn't work in general because random variable corresponds to a specific type of function. You could compromise by calling it a probability function, but then you start to collide with other uses of that word. I agree random variable is awkward, though. I always avoided stats courses because it's full of so much jargon that collides with nomenclature used by mathematicians.
- sorokod 8y agoYes, it's not just a function - there are strings attached but we deserve a better name - probability function is x100 better. "random variable" is exceptionally bad in that it leads the mind in irrelevant directions.
- Analemma_ 8y agoI don't know about math, but at least in physics, "quantum teleportation" is a terrible name; it made way more sense to me when somebody called it "quantum telegraphy" instead.
- vinchuco 8y ago> Do you know any other examples https://youtu.be/lFIYKmos3-s https://youtu.be/lFIYKmos3-s Where "complex numbers" is Chinese, "imaginary numbers" is German, and "2D numbers" is English in this analogy.
- empath75 8y agoThey're not simply 2d numbers, though. ℝ^2 is also a 2d number, but has different multiplication rules, for example.
- tangentspace 8y agoIntuitively elements of R2 are not numbers, they're vectors, or points in a 2 dimensional vector space. You can also treat complex numbers as vectors in the 2 dimensional complex plane.
- westoncb 8y agoI think there's definitely something to your suggestion here. For instance, there is no result to sqrt(-1) on the real line, but it becomes possible in the complex plane because you have an additional degree of freedom. One qualifier I'd add though is that they're 'asymmetric' 2D numbers. The real vs imaginary axes have different behaviors. If I multiply by the positive unitary value on the imaginary axis (i.e. 'i'), I get CCW rotation by 90 degrees; with the negative imaginary unit I get CW rotation; positive real unit, no change occurs; negative real unit, rotation by 180 degrees. Actually that's something I wonder about—could there be a 'symmetric' version of that? Maybe something where the real axis behaves more like the imaginary axis, but maybe some signs are flipped or something. I guess an important aspect of how using the complex plane is beneficial is the fact that it combines these disparate elements though—we can start with something real, do certain transformations to it in our more broadly capable complex plane, then bring it back into the real line.
- Jtsummers 8y agoReduce these to the integral powers of i: i^0, i^1, i^2, i^3. The symmetries occur two apart (0, 2; 1, 3). Higher integral powers are equivalent to the 4 listed, i^n = i^(n mod 4). Really, each power corresponds to an additional 90 degree rotation. 0th = 0 degree rotation (just scaling), 1st = 90 degree rotation, 2nd = 180 degree rotation, 3rd = 270 degree rotation (equivalently -90 degree). So that seeming CW rotation is also 3 CCW rotations.
- westoncb 8y agoSure, I guess that's a consequence of general rotational symmetry. The main thing I was pointing out were the different geometric behaviors of multiplication by real vs imaginary units—not so much the difference between negative and positive units on either axis.
- Jtsummers 8y agoSeparate out the components, rotation and reflection and scaling. -1 is reflection. i is rotation by 90 degrees CCW. -i is both rotation by 90 degrees and reflection. Positive reals are scaling. So -ai is really rotation by 90 degrees, reflection, and then scaling by a.
- mrob 8y ago"Intrinsic curvature" should be called something else, because people frequently drop the "intrinsic". This implies the existence of a higher dimensional space which usually doesn't exist. "Curvinture" would be a better name.
- westoncb 8y agoHmm. I only see 'intrinsic curvature' used when there is a potential confusion about whether it's intrinsic or extrinsic (otherwise it's just 'curvature'). And if that is the case, so that there is an embedding in a higher dimensional space, then intrinsic/extrinsic curvature are excellent terms (imo). They were some of the first math terms I came across where I really appreciated the selection of terminology.
- ajkjk 8y agoDefinitely: "determinant" should be called "volume".
- johnfn 8y agoVolume of what?
- deleted 8y ago[deleted]
- deleted 8y ago[deleted]
- tangentspace 8y agoOf the parallelepiped generated by the columns of the matrix. (https://textbooks.math.gatech.edu/ila/determinants-volumes.html https://textbooks.math.gatech.edu/ila/determinants-volumes.h...) The result of the determinant might be negative, which doesn't make sense for a volume, so you need to take the absolute value to interpret it that way. Another way to think of it is the change in volume caused by the linear transformation determined by the matrix (http://algebra.math.ust.hk/determinant/01_geometry/lecture3.shtml http://algebra.math.ust.hk/determinant/01_geometry/lecture3....)
- mkl 8y agoExcept they can be negative, and relate to area for 2D, not volume, and would require defining volume in dimensions higher than 3, which is not necessarily obvious or intuitive for beginners. "Signed scaling factor" is more general, and some courses (e.g. ones I teach on) use that to introduce the idea.
- ajkjk 8y agoI think that 'N-volume' including area, volume, and higher dimensional analogs is pretty understandable to a student. The fact that it's signed is more subtle, but still is not a big problem. The benefits to intuition of calling it 'volume' far outweigh the negatives. Anyway, 'scaling factor' still leaves the signs mysterious.
- deleted 8y ago[deleted]
- SilasX 8y agoThis might be one: Instead of rational numbers, call them "[integer-]ratio numbers" -- they're the numbers that can be expressed as a ratio of integers. When you call them rational numbers, it sounds like you're saying they're the only numbers that are "smart" or that "make sense" -- the connection to ratios is obscured.
- dTal 8y agoBut... they aren't just 2D numbers. That would be vectors. Yes, they behave like vectors under addition, but they also have a ton of other behaviour that '2D numbers' doesn't help with at all, like what it means to raise a number to a complex power.
- kim031 8y agoA great explanation about imaginary numbers: https://www.quora.com/How-do-you-explain-the-imaginary-number-i-to-a-layman/answer/Nathan-Pflueger?srid=ujOO&share=1 https://www.quora.com/How-do-you-explain-the-imaginary-numbe...