It's the other way around: real numbers are imaginary. All numbers are imaginary. Numbers are very useful concepts, and can be used in various ways to describe parts of reality. These descriptions can stop making sense if inappropriate kinds of numbers are used (e.g. -5 makes perfect sense as a displacement, but not as a count, and 5i makes perfect sense as part of a description of simple harmonic motion, but not as a displacement).
The use of the word "imaginary" to describe parts of complex numbers is quite unfortunate [1], because it misleads people into thinking there's something fake or weird about them, leading to confusion and entire articles trying to convince people they're valid. They're a totally normal, expected, and widely used feature of number systems as we understand them today.
The name they've got comes from how surprising and foreign they seemed to the people who first encountered them. Irrational numbers were similarly surprising and foreign to the people who first encountered them. Today we're perfectly comfortable working with irrational numbers like π, but the original reaction lives on in other meanings of "rational" and "irrational": based reasoning, and not based on reasoning, respectively.
"Imaginary" seems like a tougher hurdle, and while the article says "imaginary numbers" many times, mathematicians and physicists don't really. We talk about complex numbers, and describe complex numbers with 0 real part as "pure imaginary". "Imaginary" and "real" are just labels for the axes of the complex plane, and neither is literal in this context.
[1] As is the word "complex", which most commonly now means "complicated", but is used here with another meaning: made up of multiple parts.
Before I studied math, I always slightly resented imaginary numbers as being "math wankery" and just defined because mathematicians had a compulsion to generalize and define new nonsense because they could, and not because it made any sense to.
On the way to changing my mind, I learned that the Fundamental Theorem of Algebra only works for complex numbers (and not "real" numbers), the beauty and simplicity of rotations in the complex plane, but maybe most convincing to me was a history lesson about quaternions.
Quaternions are an extension of the complex numbers, but they're not typically taught in higher math education these days, which contradicted my resentment that mathematicians were just obsessed with getting more and more abstract and general for the sake of it. Of course, they were in vogue in the 19th century (Maxwell's equations were originally written down using them), but mathematicians soon realized they just weren't as useful or as "nice" philosophically as complex numbers and just about anything you can accomplish with quaternions were better accomplished with vectors of complex numbers.
That story played a big part in persuading me that there really is something special about complex numbers -- that maybe they're the most "natural" or "real" numbers of them all.
Also, note the origin story of the complex numbers:
Contrary to legend, they weren't discovered out of a desire by mathematicians to have roots to all quadratics such as x^2 + 1 = 0. It's perfectly sensible for an equation like that to just lack a solution: this just means the standard parabola never drops below zero. Analogously, if we calculate a rocket's payload mass to Low Earth Orbit and the answer is negative, we don't feel a need to find some deep meaning behind negative mass: we just say the rocket can't get to orbit at all. Simple.
It's cubics for which complex numbers were introduced. Cubics (with real coefficients), unlike quadratics, always have real roots, since one arm goes to +∞ and the other to -∞, so it has to cross the x axis somewhere in between. But when the cubic formula was finally discovered, it had this strange property that you frequently had to take square roots of negative numbers, then add those weird square roots to "regular" numbers, and if you just shut up and calculated, the weird parts would always cancel out and you'd get a "regular" number that solved the original equation. That is, you had to pass through the complex numbers in order to find the real solutions.
While Maxwell's equations certainly can be written with quaternions, they were not originally written that way. Maxwell originally just "wrote them out", meaning component-by-component. He had 20 equations! That's why H is sometimes used for magnetic fields: the electric fields were E, F, and G.
Nowadays we usually write 4 vector equations, or 2 in the language of differential forms.
If you do any electronics or signal processing (digital or analog) at all[1], you stop believing that complex numbers are "math wankery" immediately and embrace them as the only thing that's, uh, real.
You mention the Fundamental Theorem of Algebra, I would also add analytic signals. Those are complex by nature, and yet they make so much more sense than real signals ("real" in both senses: non-imaginary and "real world"). In fact, it turns out, real/real world signals are better represented as the sum of two analytic signals.
It's so weird how real numbers, the numbers we consider to be the normal ones, are the special case in a universe that seems to favor complex numbers for the most fundamental things.
[1] Of course also physics in general, but physics can be theoretical, while engineering is almost always rooted in practicality.
I didn't even know what "Quaternions" were until I started to get into 3D graphics and found that they're often used instead of degrees or radians to avoid camera "gimbal lock".
One of my favorite places where imaginary numbers pop up is in solving recurrence relations[0]. For instance, the Fibonacci numbers are an example of a recurrence relation and you can obtain the closed form solution by solving the characteristic polynomial. However, you can also get a closed form solution involving imaginary numbers when the roots of the polynomial are imaginary, even though the sequence only contains real numbers! From that you can express the result completely with real numbers if you use the Euler transformation of complex numbers.
Unfortunately this article repeats the same misconceptions about imaginary/complex numbers and their role in physics as most popular articles do.
Imaginary numbers were invented too early, that is why the usual explanations following the historical development are so mysterious. The notion of Sqrt(-1) does not help to explain what they actually are.
In fact imaginary "numbers" are not numbers. They cannot be ordered by size.
Imaginary "numbers" are also not vectors in the geometrical sense. This is because you can multiply two geometrical vectors and get a scalar as result (scalar product). But if you multiply two imaginary "numbers" you get another imaginary "number", so this is different to a geometrical vector. In fact imaginary "numbers" are transformations or operators, because they multiply like operators/matrices and they act on geometrical vectors like rotation matrices. Group theory makes all that very clear: Imaginary "numbers" are just the elements of the group SO(2), the group of rotation operations in 2D (multiplied with a scaling factor).
You can add a non-zero number which squares to zero to your number system [edit: but this doesn't let you divide by zero]. This results in the "dual numbers" and is practically useful for "automatic differentiation", where we represent quantities in the form x + x'ε, and have the rule that ε² = 0.
Quantities which square to zero are also implicit in spacetime. A "lightlike" vector which has equal displacements in space and time between two spacetime “events” (e.g. the displacement between two points along the path of a photon) has a squared length of 0, compared to "timelike" vectors with negative squared length and "spacelike" vectors with positive squared length. (conventions about signs vary from source to source)
In other contexts it makes sense to define 1/0 to be the quantity ∞. There are two relevant models here, with different practical applications. One model where we add a single number ∞ which is also equal to –∞, and makes the number line "wrap around" into a circle. Another model where we add two separate numbers +∞ and –∞ at the two ends of the number line.
No, but there are algebraic structures that allow for this, like the Riemann sphere. The proper way to talk about this is the concept of "zero dividers". For Z the only zero divider is zero, 0/0=0.
The closest thing to what you describe is the the dual number (that together with the imaginary and hyperbolic numbers make the geometric numbers), which has zero dividers, and is defined as k^2=0 (where k is not in R).
This is a very interesting number and it can help clean up a lot of problems when using complex numbers.
No, or at least not in a field. From the field axioms you can directly proof that 0x=(a-a)x = ax - ax =0, and therefore 0 doesn't have a well defined inverse. You can look at other algebraic structure, but those behave less like numbers.
Yeah. Great mind teaser. I can imagine an imaginary number (-/+) "inf" for divisions by number approaching zero from left/right, yet the algebra would not be possible to define properly because you can approach zero with a different rate.
The undefined cases, where the left/right limits are not equal coulf get imaginary number "shrug" because it would be even less useful.
Of is anyone able to define a useful algebra for these? I'm really curious.
yes, check out Yaglom's "Complex Numbers in Geometry" for a fairly complete treatment of this topic. You can extend the dual numbers and double numbers with their own special infinities, similarly to how the complex numbers can be extended to include a single point at infinity.
It is! Calling them 'imaginary' seems like a misnomer, they're just a different _kind_ of number. It becomes much clearer when you plot it in 2D space.
The number line is just one dimension of the possibly infinite dimensions you can plot a number in.
Apart from the waffle the actual news [0] was that some people claim that they need complex quantum mechanics, quantum mechanics over C instead of just R. For some reason our universe might require complex structure.
Dubious. Any Computer Scientist can tell you that any Turing complete system can emulate any other such system. Perhaps the Quantum aspect of reality makes the difference, but I deeply suspect it’s the same.
Quote: "...or millions of times colder than the insides of your fridge."
I smiled. Eh, modern journalism, just a few order of magnitudes wrong here but who's counting anymore, yes?
It's funny that the article uses quantum mechanics to argue that complex numbers are "real", since I would argue that quantum mechanics shows that irrational numbers aren't "real".
Quantum mechanics belies the idea of space bring a continuum and so you cannot actually physically manifest numbers like sqrt(2) and pi, because Euclidean right triangles and circles do not actually exist.
Space is quite continuous in quantum mechanics, it's only the interactions between particles which are quantised (in fact the governing equations of quantum mechanics are almost all continuous. It's only when you try to solve them that quantisation appears, in a similar way that a guitar string is continous but only certain shaped standing waves can exist on it).
"That this subject [imaginary numbers] 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." — Gauss
Yes, I hope the consensus eventually moves away completely from "imaginary". Complex numbers is already ok, so it doesn't need to change, only the terminology for that i axis.
So what can we contribute, do you know any authors that have already found some new and better terminology and promoted it?
[+] [-] mkl|3 years ago|reply
The use of the word "imaginary" to describe parts of complex numbers is quite unfortunate [1], because it misleads people into thinking there's something fake or weird about them, leading to confusion and entire articles trying to convince people they're valid. They're a totally normal, expected, and widely used feature of number systems as we understand them today.
The name they've got comes from how surprising and foreign they seemed to the people who first encountered them. Irrational numbers were similarly surprising and foreign to the people who first encountered them. Today we're perfectly comfortable working with irrational numbers like π, but the original reaction lives on in other meanings of "rational" and "irrational": based reasoning, and not based on reasoning, respectively.
"Imaginary" seems like a tougher hurdle, and while the article says "imaginary numbers" many times, mathematicians and physicists don't really. We talk about complex numbers, and describe complex numbers with 0 real part as "pure imaginary". "Imaginary" and "real" are just labels for the axes of the complex plane, and neither is literal in this context.
[1] As is the word "complex", which most commonly now means "complicated", but is used here with another meaning: made up of multiple parts.
[+] [-] wruza|3 years ago|reply
Doesn’t it simply mean “cannot be written as a ratio”?
Edit: the parent is correct, they were called absurd numbers originally.
[+] [-] TMWNN|3 years ago|reply
I'm not sure whether <https://scp-wiki.wikidot.com/scp-033> and <https://scp-wiki.wikidot.com/scp-1313> are evidence for, or against, your proposition.
[+] [-] trainfromkansas|3 years ago|reply
On the way to changing my mind, I learned that the Fundamental Theorem of Algebra only works for complex numbers (and not "real" numbers), the beauty and simplicity of rotations in the complex plane, but maybe most convincing to me was a history lesson about quaternions.
Quaternions are an extension of the complex numbers, but they're not typically taught in higher math education these days, which contradicted my resentment that mathematicians were just obsessed with getting more and more abstract and general for the sake of it. Of course, they were in vogue in the 19th century (Maxwell's equations were originally written down using them), but mathematicians soon realized they just weren't as useful or as "nice" philosophically as complex numbers and just about anything you can accomplish with quaternions were better accomplished with vectors of complex numbers.
That story played a big part in persuading me that there really is something special about complex numbers -- that maybe they're the most "natural" or "real" numbers of them all.
[+] [-] DavidSJ|3 years ago|reply
Contrary to legend, they weren't discovered out of a desire by mathematicians to have roots to all quadratics such as x^2 + 1 = 0. It's perfectly sensible for an equation like that to just lack a solution: this just means the standard parabola never drops below zero. Analogously, if we calculate a rocket's payload mass to Low Earth Orbit and the answer is negative, we don't feel a need to find some deep meaning behind negative mass: we just say the rocket can't get to orbit at all. Simple.
It's cubics for which complex numbers were introduced. Cubics (with real coefficients), unlike quadratics, always have real roots, since one arm goes to +∞ and the other to -∞, so it has to cross the x axis somewhere in between. But when the cubic formula was finally discovered, it had this strange property that you frequently had to take square roots of negative numbers, then add those weird square roots to "regular" numbers, and if you just shut up and calculated, the weird parts would always cancel out and you'd get a "regular" number that solved the original equation. That is, you had to pass through the complex numbers in order to find the real solutions.
[+] [-] dontlaugh|3 years ago|reply
[+] [-] evanb|3 years ago|reply
Nowadays we usually write 4 vector equations, or 2 in the language of differential forms.
[+] [-] anyfoo|3 years ago|reply
You mention the Fundamental Theorem of Algebra, I would also add analytic signals. Those are complex by nature, and yet they make so much more sense than real signals ("real" in both senses: non-imaginary and "real world"). In fact, it turns out, real/real world signals are better represented as the sum of two analytic signals.
It's so weird how real numbers, the numbers we consider to be the normal ones, are the special case in a universe that seems to favor complex numbers for the most fundamental things.
[1] Of course also physics in general, but physics can be theoretical, while engineering is almost always rooted in practicality.
[+] [-] est|3 years ago|reply
What about octonions? Are they even less useful?
[+] [-] blooalien|3 years ago|reply
[+] [-] siraben|3 years ago|reply
[0] https://math.unm.edu/~jvassil/Recurrence%20Relations.pdf
[+] [-] amai|3 years ago|reply
Imaginary numbers were invented too early, that is why the usual explanations following the historical development are so mysterious. The notion of Sqrt(-1) does not help to explain what they actually are.
In fact imaginary "numbers" are not numbers. They cannot be ordered by size. Imaginary "numbers" are also not vectors in the geometrical sense. This is because you can multiply two geometrical vectors and get a scalar as result (scalar product). But if you multiply two imaginary "numbers" you get another imaginary "number", so this is different to a geometrical vector. In fact imaginary "numbers" are transformations or operators, because they multiply like operators/matrices and they act on geometrical vectors like rotation matrices. Group theory makes all that very clear: Imaginary "numbers" are just the elements of the group SO(2), the group of rotation operations in 2D (multiplied with a scaling factor).
Regarding quantum mechanics what the scientists showed is, that a theory which does break SO(2) rotation symmetry (so called real quantum mechanics) does not describe nature. That should surprise no one as the SO(2) symmetry (or equivalently called U(1) symmetry (https://en.wikipedia.org/wiki/Circle_group#Isomorphisms)) is necessary to describe conservation of the electric charge. (https://en.wikipedia.org/wiki/Charge_(physics)#Abstract_defi...)
[+] [-] xyzal|3 years ago|reply
https://www.youtube.com/watch?v=ALc8CBYOfkw
[+] [-] LgWoodenBadger|3 years ago|reply
[+] [-] jacobolus|3 years ago|reply
https://en.wikipedia.org/wiki/Dual_number
https://en.wikipedia.org/wiki/Automatic_differentiation#Auto...
Quantities which square to zero are also implicit in spacetime. A "lightlike" vector which has equal displacements in space and time between two spacetime “events” (e.g. the displacement between two points along the path of a photon) has a squared length of 0, compared to "timelike" vectors with negative squared length and "spacelike" vectors with positive squared length. (conventions about signs vary from source to source)
https://en.wikipedia.org/wiki/Spacetime#Spacetime_interval
* * *
In other contexts it makes sense to define 1/0 to be the quantity ∞. There are two relevant models here, with different practical applications. One model where we add a single number ∞ which is also equal to –∞, and makes the number line "wrap around" into a circle. Another model where we add two separate numbers +∞ and –∞ at the two ends of the number line.
https://en.wikipedia.org/wiki/Projectively_extended_real_lin...
https://en.wikipedia.org/wiki/Extended_real_number_line
[+] [-] greendream17|3 years ago|reply
The closest thing to what you describe is the the dual number (that together with the imaginary and hyperbolic numbers make the geometric numbers), which has zero dividers, and is defined as k^2=0 (where k is not in R).
This is a very interesting number and it can help clean up a lot of problems when using complex numbers.
[+] [-] yk|3 years ago|reply
[+] [-] vletal|3 years ago|reply
The undefined cases, where the left/right limits are not equal coulf get imaginary number "shrug" because it would be even less useful.
Of is anyone able to define a useful algebra for these? I'm really curious.
[+] [-] jqgatsby|3 years ago|reply
[+] [-] unknown|3 years ago|reply
[deleted]
[+] [-] ahmadmijot|3 years ago|reply
[+] [-] exabrial|3 years ago|reply
[+] [-] unsafecast|3 years ago|reply
The number line is just one dimension of the possibly infinite dimensions you can plot a number in.
We're just labelling numbers wrong.
[+] [-] foxes|3 years ago|reply
[0] https://physics.aps.org/articles/v15/7
[+] [-] amadeuspagel|3 years ago|reply
[1]: https://www.youtube.com/watch?v=T647CGsuOVU&list=PLiaHhY2iBX...
[+] [-] transfire|3 years ago|reply
[+] [-] unnouinceput|3 years ago|reply
[+] [-] ardit33|3 years ago|reply
Imaginary numbers are functions... not numbers!
Functions are real if you can reproduce their output, even though you don't know the inward working of them.
[+] [-] amai|3 years ago|reply
[+] [-] ubj|3 years ago|reply
e^{i \pi} + 1 = 0
Or even better, raising i to the ith power:
i^i = e^{-\pi / 2}
That looks pretty real to me.
[+] [-] drdec|3 years ago|reply
Quantum mechanics belies the idea of space bring a continuum and so you cannot actually physically manifest numbers like sqrt(2) and pi, because Euclidean right triangles and circles do not actually exist.
[+] [-] rcxdude|3 years ago|reply
[+] [-] edgyquant|3 years ago|reply
Yep the ancient philosophical math debate about what real even means beyond some points on a grid.
[+] [-] TeeMassive|3 years ago|reply
[+] [-] alangibson|3 years ago|reply
[+] [-] cal85|3 years ago|reply
[+] [-] game-of-throws|3 years ago|reply
[+] [-] unknown|3 years ago|reply
[deleted]
[+] [-] superb-owl|3 years ago|reply
[+] [-] Chinjut|3 years ago|reply
[+] [-] kzrdude|3 years ago|reply
So what can we contribute, do you know any authors that have already found some new and better terminology and promoted it?
[+] [-] pvg|3 years ago|reply