Timeline for Examples of common false beliefs in mathematics
Current License: CC BY-SA 2.5
18 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 1, 2021 at 21:42 | comment | added | Rainb | I mean, if you're being pedantic, if x is on degrees inside the sine, it's also in degrees outside the sine, which cancels out, so it's 1. | |
| Feb 9, 2021 at 10:08 | comment | added | José Hdz. Stgo. | Well, that is no the only misconception regarding $\lim_{x \to 0} \frac{\sin x}{x}$: maa.org/sites/default/files/pdf/mathdl/CMJ/Richman160-162.pdf | |
| Apr 16, 2019 at 20:47 | comment | added | user137767 | when I first saw this, I thought that we are somehow considering $x$ as a homogeneous variable in non-zero degree and then extended the definition of sine to graded variables. Given that the power series of sine is manifestly not homogeneous, this was confusing. | |
| Apr 10, 2011 at 20:41 | comment | added | user11235 | @Laurent Moret-Bailly: The definition of $\frac{\sin x}x$ in degrees is the number you get when you type it into your calculator while forgetting to push deg/rad/grad first. | |
| Mar 11, 2011 at 17:47 | comment | added | user9072 | @JBL: Well, there are also some universities outside the US ;) This is not standard, yet not unusual though becoming rarer, in certain parts of Europe: In HS one learns about trig. func. in a geom. way; about diff./int. without a formal notion of limit, mainly rat. funct; in any case that limit wouldn't show up explictly. (Maybe 'invisibly' if derivative of trig. functions are mentioned.) Then, at univ. at the very start you take (real) analysis: constr. of the reals, basic top. notions(!), continuity,...,series of functions as application powerseries, and as appl exp and trig. func. | |
| Mar 11, 2011 at 16:26 | comment | added | JBL | @unknown, students in the US learn trigonometric functions some time in high school and learn limits in a first calculus course, typically in the last year of high school or first year of college. The words "open" and "dense" are not defined in any of these courses, except maybe that some students know what an open interval is. | |
| Mar 11, 2011 at 16:01 | comment | added | user9072 | @JBL, what Sivaram says is taken directly from the question, an example of what is asked for. Granted, this is slightly more advanced. Yet, the second example given 'open dense sets in R' is (in certain uni-curricula) something that comes up earlier than sin (at the level of rigor needed to talk about limits). @Laurent Moret-Bailly, yes and no: define sind(x)= sin(pi x /180), to ask what the limit of sind(x)/x is is not meaningless. And, on varios calculators pressing 'sin' gives this 'sind' (or at least they have that option). | |
| Mar 6, 2011 at 16:13 | comment | added | Laurent Moret-Bailly | To my knowledge, there is only one sine function, and it is a map from $\mathbb{R}$ to $\mathbb{R}$ (or from $\mathbb{C}$ to $\mathbb{C}$, if you insist). What does it mean for a real number to be "in degrees"? And if you redefine the sine function as a map from $A$ to $\mathbb{R}$ where $A$ is the space of "angles" (whatever this means), then $\frac{\sin\,(x)}{x}$ is meaningless. | |
| Mar 6, 2011 at 15:08 | comment | added | JBL | Sivaram, I guess you are complaining that some other answer wasn't downvoted (but I'm not sure because I'm certainly not going to go back and look through all past answers to check; but anyway how would you know it hadn't been down-voted and then later up-voted, as this one)? But the second example is obviously less elementary -- the first one is taught to every high school student or college freshman who takes calculus. The phrase "sin(x) is bounded in the complex domain" is incomprehensible to anyone not in a first complex analysis course. | |
| Mar 6, 2011 at 8:04 | comment | added | user11000 | @JBL: Probably... I however feel that it is equally elementary to think that $\sin(z)$ is bounded in the complex domain. | |
| Mar 6, 2011 at 6:54 | comment | added | Eric Naslund | @Sivaram: I think it received down votes because noone likes degrees. 360 is quite an arbitrary choice. | |
| Mar 6, 2011 at 4:40 | comment | added | JBL | Probably, it received down-votes because the question says, "The properties I'd most like from examples are that they are from reasonably advanced mathematics (so I'm less interested in very elementary false statements...)." Certainly, I'm inclined to down-vote on that basis. | |
| Feb 27, 2011 at 14:45 | comment | added | Thierry Zell | +1. The limit when $x$ is in degrees is an exercise in many calculus textbooks (or equivalently, the derivative of $\sin (x degrees)$. Yet, it seems people are slow to pick up on it. Your point was made by Deane Yang in this answer: mathoverflow.net/questions/40082/… (and no one found anything wrong with it then...) | |
| Feb 24, 2011 at 22:54 | comment | added | darij grinberg | I can't believe one actually can believe this... | |
| Feb 24, 2011 at 17:47 | comment | added | user9072 | +1 from me. I will try to keep this potential source of confusion in mind until I teach on this the next time. | |
| Feb 23, 2011 at 15:54 | comment | added | user11000 | @downvoters: Kindly provide a reason for the down votes. | |
| Feb 23, 2011 at 15:08 | comment | added | Yaakov Baruch | maybe not one of the best answers here, but why the down votes? | |
| Feb 23, 2011 at 7:49 | history | answered | user11000 | CC BY-SA 2.5 |