Timeline for Taking a theorem as a definition and proving the original definition as a theorem
Current License: CC BY-SA 4.0
11 events
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| Aug 20, 2023 at 2:56 | history | edited | Buzz | CC BY-SA 4.0 |
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| Oct 1, 2021 at 15:43 | comment | added | BigbearZzz | @Yakk Intuitively what you said about circle and other geometric properties of triangle seems true, but that's not rigorous enough according to modern standard. The often-seen "geometric proof" of the derivatives of $\sin$ is usually either incomplete or circular (no pun intended), and the way to make it rigorous can be done via defining $\arcsin$ first by integration or by cheating and define $\sin$ via power series. | |
| Sep 29, 2021 at 6:09 | comment | added | liuyao | Sorry to digress, @LSpice (from the main point of this answer). I didn't mean that it should be introduced as the definition of sin to calculus students, just as an alternative, in the spirit of the original question. That may satisfy the purists, than the "line of x radian intersecting the circle" definition. | |
| Sep 28, 2021 at 21:26 | comment | added | Yakk | @LSpice Sine is just the ratio of sides of a particular right angle triangle whose far corner touches a circle. You don't need integration or differential equations to define it. Many of the properties can be difficult to get ahold of, but geometry can give you a lot far before you have to touch calculus. If you know that orbiting in a circle with constant thrust away from the center gives a circle, you can even get derivatives geometrically .. maybe, | |
| Sep 28, 2021 at 21:12 | comment | added | LSpice | @Yakk, re, I was borrowing @liuyao's terminology, in which proper definition was not defined (ha!), but the inverse of an integral was mentioned as a proper definition. I agree that $e$ and even $e^x$ can be defined using only limits, though I daresay it is a challenge to prove much about, e.g., its differentiability. How about sine and cosine (also without complex numbers)? | |
| Sep 28, 2021 at 20:30 | history | edited | Michael Hardy | CC BY-SA 4.0 |
`{\rm ...}` and `\operatorname{...}` do not always have the same effect, and the latter exists for good reasons.
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| Sep 28, 2021 at 20:06 | comment | added | Yakk | @lspice What do you mean "proper definition" and "these ingredients" (integration, differential equations?)? e can be uniquely defined without either of those. | |
| Sep 28, 2021 at 3:45 | comment | added | LSpice | @liuyao, re, "early transcendentals" usually, I think, means earlier than that: outside of Spivak, I think the transcendentals usually come even before integration, and certainly before any proof of an existence theorem for solutions to differential equations. Is there a proper definition of the transcendentals that does not require these ingredients? | |
| Sep 28, 2021 at 3:04 | comment | added | liuyao | Most calculus books we use today are labeled "early transcendentals", meaning that the transcendental functions (sin, cos, exp, ln) are introduced early, as opposed to defining them by power series. One might think they are not defined properly; but (to OP's point) they are just defined differently. E.g., sin is the inverse to arcsin, which is integral of an algebraic function (over complex domain); exp has 3 definitions. | |
| Sep 27, 2021 at 9:42 | history | made wiki | Post Made Community Wiki by Stefan Kohl♦ | ||
| Sep 27, 2021 at 1:52 | history | answered | Buzz | CC BY-SA 4.0 |