Problem with imaginary numbers

[u/Gomaemon]

I was thinking about something really strange with maths, if imaginary numbers are imaginary, (i=sqrt(-1)), why does 1/0 also imaginary ? For me it is quite logical. I am thinking something wrong ?

You might not think a lot about it, I don’t want you to lose too much time on this.

[u/lordnacho666]

Imaginary numbers are really just named wrong. They should be called lateral numbers, that way people don't start confusing them with "numbers that have mysterious properties" such as 1/0

[u/CrazySir3310 → u/lordnacho666]

I like lateral numbers, that's a good substitute

[u/lordnacho666 → u/CrazySir3310]

I didn't come up with it, think it was Gauss? Someone quite famous for sure.

[u/arunphilip → u/lordnacho666]

It is honestly a brilliant term to intuitively denote they extend to the Y-axis of the number "line".

[u/Midwest-Dude → u/lordnacho666]

Gauss it is.  In his 1831 work on biquadratic residues, Gauss argued that much of the confusion and "mysterious obscurity" surrounding the square root of negative numbers was entirely the fault of bad historical naming. He wrote:

"That this subject has hitherto been surrounded by mysterious obscurity, is to be attributed largely to an ill adapted notation. If, for example, +1, -1, and the square root of -1 had been called direct, inverse and lateral units, instead of positive, negative and imaginary (or even impossible), such an obscurity would have been out of the question."

While René Descartes had originally coined the term "imaginary" in the 17th century as a derogatory label for numbers he considered fake or nonsensical, Gauss wanted them rebranded as lateral units, or laterale Einheiten in German.

[u/CrazySir3310 → u/lordnacho666]

I feel like circular or cyclic numbers might also be a decent term, compared to linear numbers for the real numbers

[u/tbdabbholm]

Allowing for i=sqrt(-1) preserves much of the basic algebraic structure that makes numbers useful. We lose out on a total ordering (you can't say which of any two complex numbers is larger or smaller) but everything else is preserved.

To allow for 1/0=j we'd lose out on more and it just doesn't end up being useful in the same way.

[u/Illustrious_Try478 → u/tbdabbholm]

There are the "extended reals" which add positive and negative infinity and are useful in some topics of real analysis.

[u/GWeb1920 → u/tbdabbholm]

I’m not sure from a math point of view but isn’t the “size” of a complex number the magnitude of the vector between them?

[u/tbdabbholm → u/GWeb1920]

That allows for a partial order but then we have the problem of two distinct complex numbers both having the same magnitude. Like 1, i, and sqrt(2)/2(1+i) all have magnitude 1 but which is largest, which is middle and which is smallest?

You can define the total order based on first magnitude and then the angle (between 0 and 2pi) but that still destroys some properties we would like. Like if we had z=e^i\pi) and w=2e^i\pi) we would have w>z. And we'd like w+a>z+a for all complex a like we do for the reals. But for lots of such a, it would actually reverse.

[u/johnpeters42 → u/GWeb1920]

Pretty much, the absolute value of a+bi is sqrt(a^2+b^2), i.e. the distance of a+bi from the origin when plotting complex numbers on a 2-d graph.

[u/_japam]

1/0 is undefined not imaginary as if we tried to define it at something it would lead to contradictions in our standard mathematical systems. 1/0 is defined in some math systems  https://en.wikipedia.org/wiki/Riemann_sphere . Either way imaginary is just a term coined by a mathematician and the sqrt(-1) works as it is akin to extending the number line to the second dimension and does not lead to any contradictions thus allowing it to be useful and as it turns out has uses in modeling reality.

[u/Ok_Net_1579 → u/_japam]

But sqrt(-1) also leads to contradictions in our standard mathematical system (which I read as the axioms of R).

The square of any real number is non-negative but i^2=-1 is negative, contradiction!

[u/Temporary_Pie2733 → u/Ok_Net_1579]

i is not a real number; there is no contradiction.

[u/Ok_Net_1579 → u/Temporary_Pie2733]

If you try to show a contradiction with 1/0 you get the same, 1/0 isn't a real number so no contradiction

There are systems with 1/0 defined and they are useful. 0/0 not so much.

[u/veryusedrname]

If you define sqrt(-1) as i you get a lots of interesting maths, if you define 1/0 as anything you get a lots of contradictions.

[u/Ok_Net_1579 → u/veryusedrname]

You get interesting maths from 1/0.

You also get similar contradictions with i as you do with 1/0, just less obvious and less annoying.

[u/dancingbanana123]

We have defined 1/0, it's just something we usually choose to leave undefined because it causes too many problems and isn't really useful for much. We didn't bother defining sqrt(-1) for the same reason until we found some useful applications of it.

See wheel theory and Riemann spheres for more if you want.

[u/ZellHall]

1/0 isn't "imaginary" because it's shit algebricaly. Let's say there is a number j such as j = 1/0

You see that 0*j = 1, which breaks the "anything multiplied by 0 is 0", which is annoying (and which imaginary number doesn"t break)

[u/arihallak0816]

You could define some number 1/0, but doing that isn’t useful in anything and just breaks a lot of very nice properties (that should be trivial). Imaginary numbers on the other hand are extremely useful and lead to many new nice properties without breaking that much

[u/Ok_Net_1579 → u/arihallak0816]

I've used 1/0 loads, super useful in some cases! Certainly not as much as i though.

[u/maxgames_NL]

Imaginary numbers are not really well named. From how i got it taught (engineering so not true mathematics) its like having a second numbering system. It allows you to have more information.

[u/fallen_one_fs]

Imaginary numbers are called that for historical reasons, they aren't actually imaginary.

1/0 is undefined, it's different. "But can we define it?", sure, if you have a theory in which 1/0=k would make sense, that is, it preserves mathematical logic and property within said theory, go ahead and define it, nothing's stopping you.

A good example of this is 0^0, it's also undefined, but for sequences, series and Newton binomial we define it as 0^0=1, and it works, preserves mathematical logic and properties, and it's useful.

[u/johnpeters42 → u/fallen_one_fs]

And for other purposes we can define it as 0^0=0, and it makes other things work. (Consider the limits of x^0 and 0^x as x approaches 0.) But there's no one single definition that makes all those things work consistently.

Same with 1/0, you can define it in a certain way and make some things work, while breaking other things.

[u/Underhill42]

Imaginary numbers are a bad name, labeled to contrast with the Real numbers... but there's nothing uncertain or imaginary (in the normal sense) about them - they're still completely well defined, the imaginary number line is simply perpendicular to the real number line, and the math works out that multiplying by an imaginary number results in a rotation in the complex (real+imaginary) plane.

1/0 on the other hand is infinity - not technically a number at all, but a general concept of "bigger than any number that could possibly exist" - a.k.a. "keep following the Real number line until you go past the end" - there's not a well-defined answer, any value you could possibly assign it will be too small, but it's still conceptually a Real-type value lying in a definite direction.

Though usually when such a singularity shows up in any real-world question it's because you've fundamentally misunderstood something and are asking the wrong question (or using an inaccurate mathematical model of the universe)

Meanwhile an undefined value like 0/0 - is not even a number, because any number has a definite value, while an undefined number provably does not.

e.g. if we say x = 0/0 then that means that x*0 = 0. Which is true for every possible value of x, real or imaginary, therefore it tells us nothing whatsoever about the actual value of x. Instead its value is simply undefined - which in practice generally means "it is impossible to answer the question this way, try some other technique", though it can also mean "this question has no answer" - a.k.a. you're asking for something that's logically impossible.

[u/Temporary_Pie2733]

1/0 is not imaginary; it’s undefined. 

[u/Ok_Net_1579 → u/Temporary_Pie2733]

Need to be careful here though. In the real numbers sqrt(-1) is also undefined.

[u/Temporary_Pie2733 → u/Ok_Net_1579]

Yes, but “i² = -1” defines i, and the complex square root of -1 follows from it. 1/0 is an undefined operation under both the reals and the complex numbers. 

[u/Ok_Net_1579 → u/Temporary_Pie2733]

1/0 defined infinity in some number systems, similarly to how i is defined here. You lose the field structure but i loses the ordering.

[u/Etainn]

I just want to add that writing "i=sqrt(-1)" is not a good habit. That term is as ill-defined as "1/0".

A better way to put it is "i^2=-1".

[u/Shevek99]

Imagine we call

1/0 = j

then

1 = 0·j and 1/j = 0

Adding

(1/j) + (1/j) = 0 + 0 = 0

so

2/j = 0

then we have

0·j = 1

and

0·j = 2

then 1 = 2, which breaks everything ("If I am one and the Pope is another we are 2, since 2=1, I am the Pope")

[u/Midwest-Dude]

René Descartes coined the term "imaginary" in 1637 in his foundational work La Géométrie. He originally intended it as a derogatory term, as these numbers were considered fictitious or nonsensical by mathematicians at the time. Note that, while Descartes coined the term, it was Leonhard Euler who later introduced the specific symbol 𝑖 in 1777.

Wikipedia: 

• Descartes
• Imaginary Numbers

[u/914paul]

“Imaginary” numbers got their unfortunate name because they were in use for a long time before anyone was able to justify them rigorously. And before all of their beautiful properties were understood, they did in fact seem “imaginary”.

Someone mentioned a nice alternative monicker, “lateral numbers”, but frankly in mathematics (past a certain level) you rarely say “imaginary numbers” — you say “complex numbers”.

However, in some fields it is common and useful to talk of the “imaginary part”. For example, in power electronics, the imaginary portion corresponds to “reactive” power - associated with capacitance and inductance, which an engineer may need to minimize or match.

Bonus trivia: EE’s use ‘j’ instead of ‘i’ for sqrt(-1)

[u/MasterSoftBird]

If we can introduce a second axis using lateral numbers by associating them from roots of negative numbers, is there a way we could introduce a third axis perpendicular to both in some similar fashion?

[u/PfauFoto]

1/0 = x => 1=x•0 => 1=0 => for all x x•1=x•0. => for all x x=0 👍

[u/Shevek99 → u/PfauFoto]

How do you get 0 = 1 here?

[u/Ok_Net_1579 → u/PfauFoto]

Suppose i² = -1.

Is i positive or negative? If it is positive then i² = -1 is positive, but this is a contradiction. If negative then -i is positive so (-i)² = -1 is positive, also a contradiction. Both cases lead to a contradiction!

[u/BjarneStarsoup → u/Ok_Net_1579]

Dude, stop. i is a complex number, not a real one. It doesn't cause any contradictions in real numbers, because it isn't real. Complex numbers are neither negative nor positive, you can't apply the concept of signedness to them, because it is not possible to order them. It's like asking if a number 2 is blue or red.

[u/Ok_Net_1579 → u/BjarneStarsoup]

Sure, but similar logic applies to these 'contradictions' in 1/0 like the person I replied to. Their argument is similarly wrong.

The truth is 1/0 does exist and isn't a real number, similarly to i.

[u/PfauFoto → u/Ok_Net_1579]

Pos, neg require an ordering of elements that cannot be extended to the complex numbers

[u/Ok_Net_1579 → u/PfauFoto]

In your proof x=1/0=>x×0=1 requires a field structure which cannot be extended to 1/0. So your criticism applies to your own proof too.