https://pluralistic.net/2026/03/19/jargon-watch/

[khleedril@cyberplace.social]

@Phosphenes @johnpaulflintoff @pluralistic The reason you couldn't make it work is because there is only one way to multiply two negatives, and that's how we do it.

Happy to be corrected...

[hopfgeist@digitalcourage.social]

@khleedril @Phosphenes @johnpaulflintoff @pluralistic it's a bit more complicated than that. The definitions of multiplication and addition are somewhat arbitrary, but the ones that we use are consistent and turned out to be extremely useful to describe physical phenomena and make accurate predictions. Other definitions are possible, and even necessary when dealing with objects other than simple numbers.

[phosphenes@mastodon.social]

@hopfgeist @khleedril @johnpaulflintoff @pluralistic

Imaginary numbers are an open admission that our math is not entirely consistent. If you can get rid of imaginary numbers, you have resolved an inconsistency.

[hopfgeist@digitalcourage.social]

@Phosphenes @khleedril @johnpaulflintoff
Imaginary numbers are completely consistent, which just means that there are no internal contradictions. Consistency is not the same as "common sense" and it does not mean that all concepts that can be mathematically expressed must have an intuitive interpretation in the tangible world.

[dearlove@mathstodon.xyz]

@Phosphenes @johnpaulflintoff @pluralistic Number systems have properties. In this case multiplication together with addition (which is where negation comes from) form a ring, which has a certain set of those properties. For multiplication of negative numbers to be negative we have to give up at least one of those properties, the really important one of distribution, the one that links addition and multiplication. I'm not sure if what you've got left is very useful in that case (no interesting properties, not a useful model of reality).

[phosphenes@mastodon.social]

@hopfgeist @khleedril @johnpaulflintoff

The square root of a negative number provably does not exist. Imaginary numbers both do and do not exist.

[phosphenes@mastodon.social]

@dearlove @johnpaulflintoff @pluralistic

Agree, it's the uselessness of a fully symmetrical system that makes it less interesting.

A system with a self-contradiction is kind of like a shirt with more buttons than holes. You can start buttoning at the top and find the discontinuity at the bottom, or start at the bottom and find it at the top.

The glitch is like a bubble you can push around, so we choose to push it to where it is the most useful.

[khleedril@cyberplace.social]

@Phosphenes @hopfgeist @johnpaulflintoff In the realm of natural numbers, neither -1 nor 0.5 exist. In the realm of real numbers, the square root of -1 does not exist. In the realm of complex numbers, all these things do exist.

[khleedril@cyberplace.social]

@Phosphenes @hopfgeist @johnpaulflintoff @pluralistic They are an admission that the number system is incomplete--until you get to complex numbers when you arrive at a formal algebra, in which all algebraic problems have solutions.

[patrys@mastodon.online]

@Phosphenes @johnpaulflintoff @pluralistic

It’s a fun exercise because it forced me to think how to show this without resorting to flashy terms like “algebraic rings”.

In your algebra, some of the following axioms have to be false:

-x = 0 - x
x * y = y * x
x * (a - b) = x*a - x*b

Because if all are true:

-x * -y = (0 - x) * (0 - y) = (0 - x) * 0 - (0 - x) * y = 0 - (0 - x) * y = 0 - y * (0 - x) = 0 - (y * 0 - y * x) = 0 - 0 + y * x = y * x = x * y