# Square root of a negative number? **URL:** \ **Category:** Factual Questions\ **Created:** [October 29, 2001, 12:54am UTC](https://boards.straightdope.com/t/square-root-of-a-negative-number/90196 "2001-10-29T00:54:07Z")\ **Posts on this page:** 19\ **Page:** 2
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [October 29, 2001, 6:53am UTC](https://boards.straightdope.com/t/square-root-of-a-negative-number/90196/21 "2001-10-29T06:53:18Z")
We recently got our hands on the symbol font, activated using the [**sym] and [**/sym] tags, so now we can do this: _e[sup]i[sym]p[/sym][/sup]_ + 1 = 0 Anyway, the key to imaginary numbers is to realize that different sorts of numbers have different uses. If I’m a shepherd counting my flock, I just need positive integers: I have 100 sheep. If I find myself with 99 and a half, then that means that the wolves got to one of them, and I only have 99 live sheep. Now, maybe, I settle down and start growing grain. I’m not about to count all the seeds, so I measure them in bushels: Now, 99.5 bushels is worse than 100 bushels, but it’s still better than 99. Fractions are now useful. Well, civilization advances some more, and I need to go to a banker to get a loan to plant my first crop. The banker approves me for a loan of 100 sheckles… Which means that I’m now in debt. Negative numbers are suddenly useful. Civilization advances some more, and I decide that since I was a failure as a farmer, I’ll take up physics or engineering. And all of a sudden, I’ve got imaginary impedences or wave amplitudes or spacetime intervals. An imaginary number of sheep is still meaningless, of course, but then, so is a negative or fractional number of sheep. ---
**Author:** ![Crafter\_Man](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/crafter_man/32/458_2.png) [@Crafter\_Man](https://boards.straightdope.com/u/Crafter_Man)\ **Post date:** [October 29, 2001, 3:06pm UTC](https://boards.straightdope.com/t/square-root-of-a-negative-number/90196/22 "2001-10-29T15:06:01Z")
> [@](#): > > _Originally posted by ElvisL1ves \* > \*\* > So why do EE’s care, especially?_\* 'Cause we don’t want to flunk out of school… ---
**Author:** ![GulDan](https://avatars.discourse-cdn.com/v4/letter/g/22d042/32.png) [@GulDan](https://boards.straightdope.com/u/GulDan)\ **Post date:** [October 29, 2001, 5:26pm UTC](https://boards.straightdope.com/t/square-root-of-a-negative-number/90196/23 "2001-10-29T17:26:04Z")
I think the real question is, if complex numbers are represented in two dimensions with real being the x-axis and imaginary being the y-axis. What could the z-axis represent? ---
**Author:** ![Enola\_Straight](https://avatars.discourse-cdn.com/v4/letter/e/dec6dc/32.png) [@Enola\_Straight](https://boards.straightdope.com/u/Enola_Straight)\ **Post date:** [October 29, 2001, 5:39pm UTC](https://boards.straightdope.com/t/square-root-of-a-negative-number/90196/24 "2001-10-29T17:39:54Z")
Unless you have another variable, Z is undefined and therefore the z-axis is irrelevant. ---
**Author:** ![Enola\_Straight](https://avatars.discourse-cdn.com/v4/letter/e/dec6dc/32.png) [@Enola\_Straight](https://boards.straightdope.com/u/Enola_Straight)\ **Post date:** [October 29, 2001, 5:49pm UTC](https://boards.straightdope.com/t/square-root-of-a-negative-number/90196/25 "2001-10-29T17:49:40Z")
This discussion reminds me of a term I learned on the boards…QUARTIONS. Instead of an x,y,z,t (for time) axes, use four imaginary dimensions…i,j,k,l. To this admittedly ignoant pseudo mathematician, the quartion seems to be a convenient method of rotating a solid on each axis, plotted over time. ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [October 29, 2001, 7:00pm UTC](https://boards.straightdope.com/t/square-root-of-a-negative-number/90196/26 "2001-10-29T19:00:16Z")
> [@](#): > > \*Originally posted by Enola Straight \* > \*\*This discussion reminds me of a term I learned on the boards…QUARTIONS. > > Instead of an x,y,z,t (for time) axes, use four imaginary dimensions…i,j,k,l. > > To this admittedly ignoant pseudo mathematician, the quartion seems to be a convenient method of rotating a solid > on each axis, plotted over time. \*\* FWIW, the correct spelling is “quaternions”. They are useful for rotations, but they’re also interesting in and of themselves. And when you’re representing something with n axes, it’s very common to ignore any other axes, or set all those coordinates to zero. ---
**Author:** ![GulDan](https://avatars.discourse-cdn.com/v4/letter/g/22d042/32.png) [@GulDan](https://boards.straightdope.com/u/GulDan)\ **Post date:** [October 29, 2001, 9:03pm UTC](https://boards.straightdope.com/t/square-root-of-a-negative-number/90196/27 "2001-10-29T21:03:30Z")
I realize that the z-axis doesn’t need to represent something. But, if the y-axis represents a set of numbers that cannot be represented by the x-axis, can a third set of numbers be found that cannot be represented by either the x or y axes be found? I.E. Does there exist (or can there be found) a set of numbers that is not a subset of the complex numbers? ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [October 29, 2001, 10:04pm UTC](https://boards.straightdope.com/t/square-root-of-a-negative-number/90196/28 "2001-10-29T22:04:19Z")
> [@](#): > > \*Originally posted by GulDan \* > \*\*I realize that the z-axis doesn’t need to represent something. But, if the y-axis represents a set of numbers that cannot be represented by the x-axis, can a third set of numbers be found that cannot be represented by either the x or y axes be found? I.E. Does there exist (or can there be found) a set of numbers that is not a subset of the complex numbers? \*\* Yeah, sure. There’s lots of 'em. Problem is, you lose some nice feature of arithmetic if you attempt to move beyond the complex numbers–the most common problem is that multiplication becomes non-commutative (i.e., a \* b isn’t necessarily equal to b \* a). There is a slight technicality here, but I’m not going to explain it unless someone asks about it. ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [October 29, 2001, 10:33pm UTC](https://boards.straightdope.com/t/square-root-of-a-negative-number/90196/29 "2001-10-29T22:33:57Z")
There’s also not as much incentive to use other number systems, since the complex numbers (unlike the reals) are complete. That means that any equation you can write with complex numbers, you can solve with complex numbers. In the reals, equations like ln(x) = -1, sqrt(x) = -1, or sin(x) = 2 don’t have solutions, but in the complex numbers, they all do. No matter what complex numbers you stick in, you’ll always be able to get the equations to work. ---
**Author:** ![The\_Ryan](https://avatars.discourse-cdn.com/v4/letter/t/7feea3/32.png) [@The\_Ryan](https://boards.straightdope.com/u/The_Ryan)\ **Post date:** [October 29, 2001, 10:48pm UTC](https://boards.straightdope.com/t/square-root-of-a-negative-number/90196/30 "2001-10-29T22:48:30Z")
GulDan: of course there’s a set of numbers like that. Every set that you think of _can_ be used as a set of numbers. The question is whether it is _useful_ to do so. > [@](#): > > \*Originally posted by Chronos \* > There’s also not as much incentive to use other number systems, since the complex numbers (unlike the reals) are complete. Actually, the reals are complete (every Cauchy sequence is convergent). What they aren’t is _closed under algebraic operations_. > [@](#): > > That means that any equation you can write with complex numbers, you can solve with complex numbers. In the reals, equations like ln(x) = -1, sqrt(x) = -1, or sin(x) = 2 don’t have solutions, but in the complex numbers, they all do. What’s ln 0? ---
**Author:** ![GulDan](https://avatars.discourse-cdn.com/v4/letter/g/22d042/32.png) [@GulDan](https://boards.straightdope.com/u/GulDan)\ **Post date:** [October 30, 2001, 12:47am UTC](https://boards.straightdope.com/t/square-root-of-a-negative-number/90196/31 "2001-10-30T00:47:48Z")
If the complex numbers is truly closed under all algerbraic manipulation, does it bother anyone else that we have finally hit the end of the road numerically? For many years negative and complex numbers weren’t considered (for that matter, so were irrational numbers) as existing in any sense of the word. Perhaps there is an underlying assumption, much like Euclids 5th postulate, that we are working under, preventing us from seeing the next catagory of numbers. And yes, I would like information on why mult. would necessarily be no longer commutative. ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [October 30, 2001, 12:54am UTC](https://boards.straightdope.com/t/square-root-of-a-negative-number/90196/32 "2001-10-30T00:54:04Z")
> [@](#): > > \*Originally posted by GulDan \* > \*\*If the complex numbers is truly closed under all algerbraic manipulation, does it bother anyone else that we have finally hit the end of the road numerically? For many years negative and complex numbers weren’t considered (for that matter, so were irrational numbers) as existing in any sense of the word. Perhaps there is an underlying assumption, much like Euclids 5th postulate, that we are working under, preventing us from seeing the next catagory of numbers. And yes, I would like information on why mult. would necessarily be no longer commutative. \*\* But we haven’t hit the end of the road in terms of number systems. Get yourself a good book on abstract algebra and you’ll see all the stuff that’s left to do. FWIW, the proof that any algebraic extension of C (that’s the field of complex numbers) loses some properties of arithmetic is fairly advanced, so don’t hold your breath on that one. And if some kind mathematician wanted to give an exact statement of that theorem, I’d appreciate it. On the other hand, there are a couple other directions with new numbers. Do a google search on any of the following: transfinite cardinals, transfinite ordinals, surreal numbers, hyperreal numbers, supernatural numbers. Just be prepared to invest some time–some of these are very difficult to get your mind around. ---
**Author:** ![Cabbage](https://avatars.discourse-cdn.com/v4/letter/c/f07891/32.png) [@Cabbage](https://boards.straightdope.com/u/Cabbage)\ **Post date:** [October 30, 2001, 3:46am UTC](https://boards.straightdope.com/t/square-root-of-a-negative-number/90196/33 "2001-10-30T03:46:17Z")
**ultrafilter** : Not sure if this is what you’re thinking of, but I would word the theorem as “There is no algebraic extension field of the complex numbers”. This is an immediate consequence of the complex numbers being algebraically closed. On the other hand, there are transcendental extension fields of the complex numbers; one example would be the set of rational functions with complex coefficients. ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [October 30, 2001, 3:58am UTC](https://boards.straightdope.com/t/square-root-of-a-negative-number/90196/34 "2001-10-30T03:58:13Z")
> [@](#): > > \*Originally posted by Cabbage \* > \*\ ***ultrafilter** : > > Not sure if this is what you’re thinking of, but I would word the theorem as “There is no algebraic extension field of the complex numbers”. > > This is an immediate consequence of the complex numbers being algebraically closed. > > On the other hand, there are transcendental extension fields of the complex numbers; one example would be the set of rational functions with complex coefficients. \*\* That might be it. I’d need to do more reading on extension fields, though. ---
**Author:** ![Boris\_B](https://avatars.discourse-cdn.com/v4/letter/b/fbc32d/32.png) [@Boris\_B](https://boards.straightdope.com/u/Boris_B)\ **Post date:** [October 30, 2001, 4:18am UTC](https://boards.straightdope.com/t/square-root-of-a-negative-number/90196/35 "2001-10-30T04:18:07Z")
Well, I don’t think we should get hung up on the semantics. A lot of numbers have names that aren’t particularly complimentary; number can be unnatural, negative, irrational, or imaginary. Fractions can be improper. Let’s just not try to apply some social standards of politeness here. Imaginary numbers are _not_ real. They may be relevant, useful, logical, and accepted, but they cannot be real. The set of real numbers is defined, quite specifically, as excluding all imaginary numbers. Let’s just no go thru too many definitions of the word “real” in a single discussion. ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [October 30, 2001, 4:40am UTC](https://boards.straightdope.com/t/square-root-of-a-negative-number/90196/36 "2001-10-30T04:40:36Z")
> [@](#): > > \*Originally posted by Boris B \* > **They may be relevant, useful, logical, and accepted, but they cannot be real. The set of real numbers is defined, quite specifically, as excluding all imaginary numbers.** Imaginary numbers can be constructed, just like real numbers. How are they any less real? ---
**Author:** ![Boris\_B](https://avatars.discourse-cdn.com/v4/letter/b/fbc32d/32.png) [@Boris\_B](https://boards.straightdope.com/u/Boris_B)\ **Post date:** [November 4, 2001, 11:07pm UTC](https://boards.straightdope.com/t/square-root-of-a-negative-number/90196/37 "2001-11-04T23:07:34Z")
> [@](#): > > \*Originally posted by ultrafilter \* > \*\* > > > [@](#): > > > > \*Originally posted by Boris B \* > > **They may be relevant, useful, logical, and accepted, but they cannot be real. The set of real numbers is defined, quite specifically, as excluding all imaginary numbers.** > > Imaginary numbers can be constructed, just like real numbers. How are they any less real? \*\* By definition they are not Real. Again, I’d like to keep to one definition per word. If you’d like to apply synonyms of real in its _non-mathematical_ senses (like genuine, true, or actual) to the Set of Imaginary Numbers, that is fine by me. I really have nothing against these numbers. They do, however, lie entirely outside the Set of Real Numbers, and I’d like to keep to the mathematical definition of the word “real” at least on this thread. ---
**Author:** ![Boris\_B](https://avatars.discourse-cdn.com/v4/letter/b/fbc32d/32.png) [@Boris\_B](https://boards.straightdope.com/u/Boris_B)\ **Post date:** [November 4, 2001, 11:09pm UTC](https://boards.straightdope.com/t/square-root-of-a-negative-number/90196/38 "2001-11-04T23:09:48Z")
Although my response proves that I have gotten hung up on the semantics, despite my intentions. ☹ ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [November 4, 2001, 11:15pm UTC](https://boards.straightdope.com/t/square-root-of-a-negative-number/90196/39 "2001-11-04T23:15:52Z")
> [@](#): > > \*Originally posted by Boris B \* > \*\* > > > [@](#): > > > > \*Originally posted by ultrafilter \* > > \*\* > > > > > [@](#): > > > > > > \*Originally posted by Boris B \* > > > **They may be relevant, useful, logical, and accepted, but they cannot be real. The set of real numbers is defined, quite specifically, as excluding all imaginary numbers.** > > > > Imaginary numbers can be constructed, just like real numbers. How are they any less real? \*\* > > By definition they are not Real. Again, I’d like to keep to one definition per word. If you’d like to apply synonyms of real in its _non-mathematical_ senses (like genuine, true, or actual) to the Set of Imaginary Numbers, that is fine by me. I really have nothing against these numbers. They do, however, lie entirely outside the Set of Real Numbers, and I’d like to keep to the mathematical definition of the word “real” at least on this thread. \*\* OK, I see what you’re saying, and yes, complex numbers aren’t real numbers. FWIW, the reals aren’t defined to exclude imaginaries, but it’s pretty easy to show that they do (i.e., there is no real number whose square is -1). I was thinking in a completely different sense. [Previous page](https://boards.straightdope.com/t/square-root-of-a-negative-number/90196.md?page=1)