Timeline for teach that $\frac10$ not defined properly
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Feb 11, 2018 at 16:55 | comment | added | Kevin | The other option is to use the projective reals instead of the extended reals, because then you have $\infty = - \infty$ and $y=\frac{1}{x}$ is continuous. | |
| Jan 15, 2018 at 19:31 | comment | added | user797 | (also, I find the $x/0$ form a particularly sensitive issue, since one of the common errors students make computing limits is ignoring the sign on the denominator when it converges to zero) | |
| Jan 15, 2018 at 19:25 | comment | added | user797 | @DanielR.Collins: Yes, something has to be lost, but losing continuity is a poor poison to pick, since it sours one of the major applications of the number system: they simplify the computation of limits by continuously extending the operations involved. I'm not sure what you were getting at with your last sentence, but note that $\infty / 1 = \infty$ and there are a number of other values we need to leave undefined as well. | |
| Jan 15, 2018 at 18:06 | comment | added | Daniel R. Collins | @Hurkyl: Extended reals aren't a field, so any way you cut it something gets lost (pick your poison, basically). The alternative is to not have division be onto the whole set under discussion. | |
| Jan 15, 2018 at 16:08 | comment | added | user797 | @DanielR.Collins: That's an unusual convention, since it would mean division is not continuous on its domain; usually $1/0$ is left undefined, for the reason of this answer. Although one can conceive of a "positive zero", so that they can write $x/0^+ = \infty$. | |
| Jan 13, 2018 at 23:35 | comment | added | Daniel R. Collins | One caveat would be this opens up a counterargument of inspecting $1/|x|$, say. Second caveat is when I see extended reals defined (i.e., $\infty$ as a usable number), it actually is defined that $x/0 = sgn(x) \cdot \infty$ (assuming $x \ne 0$; Ray, Real Analysis, Appendix A). | |
| Jan 13, 2018 at 20:01 | history | answered | Sue VanHattum♦ | CC BY-SA 3.0 |