Imaginary Numbers Could Be Needed To Describe Reality, New Studies Find (livescience.com) 179
InfiniteZero writes: Imaginary numbers are necessary to accurately describe reality, two new studies have suggested. Imaginary numbers are what you get when you take the square root of a negative number, and they have long been used in the most important equations of quantum mechanics, the branch of physics that describes the world of the very small. When you add imaginary numbers and real numbers, the two form complex numbers, which enable physicists to write out quantum equations in simple terms. But whether quantum theory needs these mathematical chimeras or just uses them as convenient shortcuts has long been controversial. In fact, even the founders of quantum mechanics themselves thought that the implications of having complex numbers in their equations was disquieting. In a letter to his friend Hendrik Lorentz, physicist Erwin Schrodinger -- the first person to introduce complex numbers into quantum theory, with his quantum wave function -- wrote, "What is unpleasant here, and indeed directly to be objected to, is the use of complex numbers. quantum wave function is surely fundamentally a real function."
Schrodinger did find ways to express his equation with only real numbers alongside an additional set of rules for how to use the equation, and later physicists have done the same with other parts of quantum theory. But in the absence of hard experimental evidence to rule upon the predictions of these "all real" equations, a question has lingered: Are imaginary numbers an optional simplification, or does trying to work without them rob quantum theory of its ability to describe reality? Now, two studies, published Dec. 15 in the journals Nature and Physical Review Letters, have proved Schrodinger wrong. By a relatively simple experiment, they show that if quantum mechanics is correct, imaginary numbers are a necessary part of the mathematics of our universe. "The early founders of quantum mechanics could not find any way to interpret the complex numbers appearing in the theory," lead author Marc-Olivier Renou, a theoretical physicist at the Institute of Photonic Sciences in Spain, told Live Science in an email. "Having them [complex numbers] worked very well, but there is no clear way to identify the complex numbers with an element of reality." To test whether complex numbers were truly vital, the authors of the first study devised a twist on a classic quantum experiment known as the Bell test. The test was first proposed by physicist John Bell in 1964 as a way to prove that quantum entanglement -- the weird connection between two far-apart particles that Albert Einstein objected to as "spooky action at a distance" -- was required by quantum theory.
Schrodinger did find ways to express his equation with only real numbers alongside an additional set of rules for how to use the equation, and later physicists have done the same with other parts of quantum theory. But in the absence of hard experimental evidence to rule upon the predictions of these "all real" equations, a question has lingered: Are imaginary numbers an optional simplification, or does trying to work without them rob quantum theory of its ability to describe reality? Now, two studies, published Dec. 15 in the journals Nature and Physical Review Letters, have proved Schrodinger wrong. By a relatively simple experiment, they show that if quantum mechanics is correct, imaginary numbers are a necessary part of the mathematics of our universe. "The early founders of quantum mechanics could not find any way to interpret the complex numbers appearing in the theory," lead author Marc-Olivier Renou, a theoretical physicist at the Institute of Photonic Sciences in Spain, told Live Science in an email. "Having them [complex numbers] worked very well, but there is no clear way to identify the complex numbers with an element of reality." To test whether complex numbers were truly vital, the authors of the first study devised a twist on a classic quantum experiment known as the Bell test. The test was first proposed by physicist John Bell in 1964 as a way to prove that quantum entanglement -- the weird connection between two far-apart particles that Albert Einstein objected to as "spooky action at a distance" -- was required by quantum theory.
I wonder (Score:2)
Random thought - is there a chance the fundamentals of the math we learn as children ins missing something important?
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According to the article it's missing the imaginary component.
Re:I wonder (Score:5, Insightful)
According to the article it's missing the imaginary component.
Students learn about complex numbers in algebra class, taught in 8th and 9th grade when most would still consider them "children".
My kids learned earlier because I made them do an hour of Khan Academy math lessons every time they back talked to their mom.
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Sounds like a wonderful way of encouraging your kids to hate math...
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I saw on facebook the other day someone ranting about pi and how we should just replace it with 3 and "update circles", wtf that means. I'm sure pythagoras would have approved (The pythagoreans where an actual cult who placed all sorts of weird and mystical meanings on numbers. They found the idea of numbers that arent fractions, aka "irrational numbers" heretical)
Though I guess if actual adults can believe the world is flat, then I guess people can believe the most banal of stupidities. What a world.
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I saw on facebook the other day someone ranting about pi and how we should just replace it with 3 and "update circles
The laws of mathematics (and physics) are easily amendable by voice vote in Congress, didn't you know that?
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Why not reverse the question and make it more general in nature?:
Is there a reason why brains that evolved to use tools and participate in larger social groups should even be able to comprehend the fundamentals of reality?
Or is all of our math and physics just what we're capable of doing with our brains, and a fundamental understanding of reality will forever be out of our grasp?
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Yes.
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I think this is a question that's been asked before, and gave us the New Math of the 60s and 70s. Having been exposed to it as a child, I'm left with the impression that however well intentioned, revising K-6 math pedagogy is a road fraught with peril. I didn't learn how to do long division until I transferred to a private school that did it the "old fashioned" way. Once you get a little older, there seems to be a lot of evidence that you're more capable of handling advanced concepts. The brain really d
Re:I wonder (Score:4, Insightful)
It just seems like New Math got rolled out nationwide with way too many bugs.
New Math [wikipedia.org] was designed to shift from basic arithmetic to more abstract concepts such as set theory, modular arithmetic, inequalities, bases other than 10, and symbolic logic.
The problem is that there are two types of students:
1. Students who understand the concepts in the first five minutes.
2. Students who never understand.
So group #1 is bored while the teacher repetitively and futilely tries to get group #2 to understand.
There are two solutions:
1. Separate the smart kids from the dumb kids.
2. Give up and go back to arithmetic.
Solution #1 is politically unacceptable, so we went with solution #2.
Today, there is a third solution: Self-paced online learning.
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Yeah, I remember being taught about the alligator for less than, greater than, and a few other things that might be considered "abstract", but the guess and check method they taught us for long division has always stuck out as so fucking brain dead. If they wanted to push us harder, they could have taught us the effective method for division and *then* taught us why it works.
In my public schooling, they made no qualms about separating smart from dumb students. They had the whole "gifted and talented" thin
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Can you suggest another manual method [wikipedia.org] (i.e., paper and pencil; no electronic calculator involved) for division that should be taught to children?
Division is hard to implement on computers, too. We take it for granted that it's a solved problem, but it often takes several times as many clock cycles to divide than to multiply. And even then, it can have problems [wikipedia.org].
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I liked some of the new stuff added in the 2000s.
Less time grinding away at what can be done on a calculator and more time on estimation and concepts.
I don't need long division to figure out 324,057,324,508 / 29,384,029,374, it's either somewhere around 11 or I need a calculator.
I'm sure people got upset when they stopped teaching square root with pencil and paper too, but it seems like a better use of time to focus on math and not algorithms to be a calculator.
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Richard Feynman had some interesting things to say about the New Math. Worth reading even now, IMO.
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When your rality is based on fantasy you have to invent things to explain it.
Looking at the TG community here.
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Anything beyond that is probably completely useless to four standards of deviation of the population, plus or minus 0.05. The rate of increase by which it becomes increasingly useless as calculato
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Kids absolutely need enough math to take sum up their income, multiply by some arbitrary percentage and pay that much to the government.
That can all be done in software.
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Yes, there's a simplified income tax form with two lines:
Line 1: How much money did you make last year?
Line 2: Send it all in.
Re: I wonder (Score:3)
Years ago in NY you could send in a blank, signed state tax return, and the state would figure out how much you owed and send you a bill. Then HR Block complained...
Re:I wonder (Score:4, Insightful)
Kids absolutely need enough math to take sum up their income, multiply by some arbitrary percentage and pay that much to the government.
Except they don't because
1) That's not how income taxes are levied anywhere in the world that has tax brackets. Instead you pay X% on the amount of income you make within tax bracket 1, Y% on the amount you make within bracket 2, etc,etc,etc.
2a) In the US at least you don't have to actually do all that mildly complicated math because the manual that comes with the tax form includes a lookup table where you just look up your adjusted gross income and it tells you how much tax you owe. And there are lookup tables for almost everything more complicated than simple addition and subtraction.
2b) In many other countries you don't have to do any math at all for your taxes because the government simply sends you pre-filled tax form that tells you how much to pay/what your refund will be, and you only have to make changes if you are in the tiny percentage of the population that has income or exemptions not reflected in the tax information already filed about you by your employer(s), bank(s), etc. Luckily here in the land of the free (to overpay) we're protected from such convenience by extensive lobbying by a handful of large tax-filling companies who want to protect our freedom (to pay them to fill out all that paperwork for us).
Also, computation is not math, it's arithmetic. They're vaguely related, but
Math is the theory and language used to express arbitrarily complex quantitative relationships in a concise and completely unambiguous manner that simultaneously encompasses all possible values every variable could have.
Arithmetic is how you apply such a relationship to a specific situation to get a specific numerical result.
I'm dubious about the value for most people of being able to perform arithmetic in their head, or even on paper, in a world where calculators are cheap and ubiquitous.
However, math is the language of science and finance, and if you don't understand the language well enough to judge even relatively simple claims, then you're just a sucker waiting for some scammer to rob you blind or otherwise manipulate you with vaguely plausible sounding stories about how the world works.
Kids need to know spreadsheets (Score:2)
To be able to build a simple model of how well their lemonade store is likely to do. Or a hundred other simple models.
They are not taught at school to any degree. Even advanced kids that can solve simple differential equations do not know how to build a non-trivial model.
The one useful skill.
The other, for the advanced kids, would be basic statistics. How to use ANOVA etc.
As to arithmetic, it is difficult to factor a quadratic if you cannot recognize, for example, that 42 = 6 * 9.
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Can't argue there. Building/describing models is what math *is* - we need a lot more of that, and a lot less rote memorization of how to perform calculation a calculator could do faster and more reliably. Even just a massive increase in word problems would be a huge improvement. All the calculation skills in the world are worthless if you can't first describe your problem mathematically. While if you *can* describe it mathematically, a calculator can grind the numbers just fine. Once upon a time you n
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Algebra vs Calculus (Score:3, Insightful)
I'm not a mathematician but what I understood from my time in college was, Complex numbers are just a way to describe 2D reality using Calculus, when you could very well just use vector algebra. On top of that, when studying all this math and physics I was convinced that continuous functions are probably not the most accurate way of describing reality, as there isn't really anything that you can or want to infinitely subdivide. All of the world is quantizable, there is a minimum amount of time and space that realistically make sense. It's like that joke when they model a cow with a perfectly round sphere in a vacuum or whatever it was.
Ultimately, there should be a simpler way of describing reality. Maybe one day we will achieve it. I have found that people who use a lot of complex language and handwaving don't really understand what they're saying, and people who know absolutely everything about a subject can be perfectly concise.
In what concerns the search for a grand unified theory of reality, it seems to me that we may well be at the "handwaving" stage of evolution and could likely stumble upon something more insightful and elegant one day.
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Ultimately, there should be a simpler way of describing reality. Maybe one day we will achieve it. I have found that people who use a lot of complex language and handwaving don't really understand what they're saying, and people who know absolutely everything about a subject can be perfectly concise.
I stepped on a Lego this morning. Shit got real.
Re:Algebra vs Calculus (Score:5, Insightful)
Complex numbers and vectors have different properties and work differently in situations where you're doing more than trivial arithmetic operations. While additions or multiplying by a scalar value is superficially similar. Even multiplying two vectors versus multiplying two complex number should make the difference quite apparent between the systems. I mean if they were the same then maybe we would have simplified it down to one system instead of teaching it twice ;-)
Ultimately, there should be a simpler way of describing reality.
The complexity of a succinct description of reality will be proportional to the number of dimensions it has. I assure you that it has more than 1 or 2 dimensions, so all the fun and easy equations we like to use to describe 2D (or even 3D) space are inadequate to describe fundamental aspects of reality. (assuming, philosophically, such a thing is even possible)
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It's the perpetual problem with physics. Newtonian Mechanics seemed the answer, until astronomers and mathematicians started to find oddball cases where Newton's theory didn't fit, and that ultimately lead to Special and General Relativity, which themselves model reality rather well, except for even more oddball things that come out of Quantum Mechanics. It's more a feeling among physicists that a lot of the complexity comes from the fact that we don't have an "ultimate theory of physics", and that if and w
Re:Algebra vs Calculus (Score:5, Insightful)
Physics has the problem that it must make models on what is observed in the physical world (hence the name). If they didn't observe it yet then the model they create can only be an approximation at best.
Re:Algebra vs Calculus (Score:4, Informative)
Actually that's not where the name "physics" comes from. The discipline "physics" is named after Aristotle's Physics (one of the many books he wrote). In Greek, "phusis" or "physis" [wikipedial] [wikipedia.org] means how a thing behaves or is (the nature of a thing).
Over time, the meaning shifted and in the 1750--1850's when natural philosophy moved out philosophy and became what we now call "the sciences," the word "physics" (now with a 'c') was applied to those of us who carry on Aristotles study the physical world and how it changes.
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Perhaps the act of discovering the correct model is what spawns the creation of new corner cases it can't describe?
Re: Algebra vs Calculus (Score:3)
Like Godels proof says.... hmmm
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Ah, but, if you use Number Theoretic Transforms [wikipedia.org] you get the same effect provided that the prime you pick is sufficiently large. The finding if a proper prime is a nontrivial problem but that can be compensated for by-
ah, screw it. I will just skip the build-up and say it. "Your momma's fat!"
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"multiplying two vectors versus multiplying two complex number": I know they're different, but why?
If someone says it doesn't make sense to ask why--they're simply defined differently, and both are consistent--then I'd like to know when it's appropriate to use one or the other, whether there's a way to map results from one to the other, etc.
Re:Algebra vs Calculus (Score:5, Informative)
The general idea here is that addition and subtraction is trivial in Cartesian coordinates. Multiplication and division is a lot more trivial in polar or circle coordinates where it changes the length of the vector and rotates the angle. This form has also interesting results for derivation and integration of complex numbers where you need to keep in mind that i^2 = -1 when you 'pull' down that i from the exponent of e.
But complex numbers are more than just two dimensional vectors because of the properties of i, which offer higher dimension solutions for some problems that otherwise don't have a solution in their current 'real dimensions'.
Veritsium on YT had an interesting video about the history last month: https://www.youtube.com/watch?... [youtube.com]
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I am a physicist, and well, not really. Yes, you can map from 3-D real spaces to complex 2-D spaces rather naturally in field theories [SO(3) to SU(2)], but if you're going to describe spatial positions, you would never use complex numbers to do so in any kind of ordinary mechanics problem.
Apart from quantum mechan
Not a physicist either (Score:2)
I'm not a mathematician but what I understood from my time in college was, Complex numbers are just a way to describe 2D reality using Calculus, when you could very well just use vector algebra.
It's easy to see how wrong this is if you think about things at a very fundamental level. The Higgs field, which gives particles mass, is a complex scalar field (actually a doublet) while the electromagnetic field is a vector field that gives us electrical and magnetic forces. So the properties of complex numbers give us a quantum field that creates mass while the properties of vectors give us a quantum field that creates a force.
Re:Algebra vs Calculus (Score:5, Informative)
The complex numbers C are not just a fancy way to express 2dimensional real space. It is more like a coincidence that we can visualize complex numbers this way, but actually they are still one dimensional - just in complex, not real dimensions and in fact have some surprising (at least at the time of their discoveries) consequences for the real numbers R.
Algebraically it makes more sense to think of C as an extension of R, and not as a R^2, and one thinks of it more as C = R[i], so like, the extension of the real numbers by one additional algebraic thing i.
The main feature of course is that you can solve x^2 - 1 = 0 over C, but not in R, which is equivalent to saying the polynomial decomposes into linear factors, i.e. x^2 - 1 = (x - i)(x + i) = 0, so you can read off the solutions from the linear decomposition, and this is then the reason why one calls C also the algebraic closure of R.
While this first just seems to some algebraic nitpicking, this actually has some real consequences.
For example in the case of power series, even though all our formulae only involve real numbers, the function sometimes behaves weirdly at points one would not expect.
Everyone favorites example for this would be the taylor series of f(x) = 1/(1 + x^2), which obviously has no singularities in R and is a very nice function, but the taylor series sum n= 0 to infinity (-1)^n x^(2*n) only converges for |x| But the answer is that in C there are singularities at +-i, and in fact, any radius of convergence can actually be expressed as the radius of the disk on which the function would be holomorphic, a complex property, if we were to allow complex arguments.
This also also shows why holomorphic functions, so complex differentiable and not only real differentiable, are much more restricted than real differentiable functions and in some sense boring, but I guess I digress and on the latter there are probably some people who might disagree. ;-)
Another fun thing is that in complex vector spaces, all matrices are diagonizable, because eigenvalues are the roots of some polynomial and well, we can find all those roots, but we can not do the same thing in real vector spaces.
However, when we think about how complex multiplication works, multiplying by i is the same as rotating by pi/2 on the plane represantation, so what we actually figure out by this is that whenever we have complex eigenvalues, there is some rotation going on if we think of it in the real world.
Starting from this thought then one can figure out how there are maybe some ways to represent these complex numbers as real matrices (e.g. i as a rotation by pi/2 can be very simply epxressed as a 2x2 matrix) and this leads you down the path of things like Jordan Normal Forms.
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Ultimately, there should be a simpler way of describing reality
Why? Why "should"?
Almost every time someone resorts to the word "should", I imagine David Hume rolling on his grave saying "is/ought!"
There's nothing that says reality "should" have a simpler description, or even have a guarantee that a "simpler" description (with simpler as unifying) is as comprehensible as what we have today.
Maybe there is, maybe there isn't. And the world surely doesn't yield to the notion that things "should" be simpler.
This is news? (Score:5, Informative)
We've known that complex numbers are needed for more than 100 years. /. is really going downhill.
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Indeed. This is really a big WTF?, nothing else.
Dirac Equation (Score:2)
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Actually, even floating-point calculations are, in the end, integer calculations. We just approximate real numbers with rational numbers in computers.
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How does one do "exp(i*pi) = 1" with real numbers?
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I see what you're getting at. However, it is perfectly possible to define sine and cosine as infinite series [wikipedia.org] using only real numbers.
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We call i imaginary. We treat it as such. When we solve equations in science and engineering, we only care about the real roots. This result suggests that i is not just a mathematical convenience but is an actual scalar value that can quantify a physical property.
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You beat me to it, but since I bothered to write it out, and I give some examples that might help some people, I'll still post it.
A common use of complex numbers is to simplify the solution of math problems often in engineering. In these cases, complex numbers are not used to model the fundamental elements of physics but are just used to make it easier to solve the equations. In the end, any imaginary solutions are thrown out. Just like a grade school problem were one throws out negative solutions be
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Well, if I plot a quadratic equation with imaginary roots, imaginary x is y and reflection in y. So if I solve x^2+1, I get +/- i. If I swap ix for y, that gives me a minimum of +/-1 along the y axis. The solution still means something, even if it's not obvious at first.
Would it be reasonable, then, to say that you can perform a similar swap in electronics (an imaginary solution to current corresponds to a real value for something else in any hypothetical situation where the real value solution to current d
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I'm not an expert on complex analysis, but I agree that the complex solutions will tell you a lot about the behavior of real valued functions. For polynomials, they are just a different way to represent the function.
In the context of engineering, they typically deal with differential equations and use things like Laplace transforms to convert the differential equations to simple equations that are solved with root finding. These are things I had to memorize as an engineering undergrad, so I don't really
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Or perhaps the number line should be two-dimensional, which is how we treat imaginary numbers right now - as extending the real number line to the complex plane. A complex number is really a two dimensional number, and how we use it depends on whic
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Imagine if the request "Please bring me 5i apples" could be meaningfully granted...
Doesn't surprise me (Score:3)
Ever since the current..."unpleasantness" has been happening, reality has gone tits up, and I think we need some new ways to define/understand what has happened to our society.
Imaginary numbers seem like as good as place as any to start.
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Well, virtually all the news reads like the OctOnion.
Raises hand ... (Score:2)
I submitted the same article earlier this morning-- looks like earlier than this submission, but that's okay -- but, for some reason my submission got tagged "SPAM"? Um... why?
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We suspect the editors have been replaced with iguanas wearing fake neck beards. Don't worry about it, but that model does explain a thing or two.
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We suspect the editors have been replaced with iguanas wearing fake neck beards. Don't worry about it, but that model does explain a thing or two.
Okay. I'd buy that. :-)
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Very interesting. I also had an interesting and informative article, about NFTs, from the Communications of the ACM, the flagship journal of the ACM, inexplicably marked as SPAM as well.
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Probably because it is spam. This must be the least useful article published in PRL this year.
Okay, then why wasn't this one similarly flagged?
Prejudice against imaginary numbers (Score:3)
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If I solve the quadratic equation y=x^2=2x+2, then I get the solutions of -1 +/- i. If I replace ix with y, I get two solutions, the points (-1, 1) and (-1, -1). (-1, 1) does indeed correspond to the minimum for this equation. (-1, -1) corresponds to a reflection of the curve in x.
(Since a reflection of the curve in x, if I only have real solutions, gives me exactly the same real solutions, I'm quite happy to delude myself into thinking that the second solution is always for the reflection in x, so I'm deal
what do you mean "could be needed"? (Score:5, Insightful)
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Yes, imaginary numbers get a bad wrap simply because of their name.
A name which is entirely arbitrary, of course. And that arbitrary name hurts doubly because the term "imaginary" is often used in contrast to the "real" of real numbers.
In fact, real numbers are no more or less "real" or "imaginary" than imaginary numbers are. They are both logical constructs and both are used as a model for reality in various ways.
If the behavior of numbers - real, imaginary, complex, positive whole number, integer, rati
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It is a bit tricky. Of what are they abstractions? One might try to see natural numbers as abstractions, e.g., from groups of cows. It is trickier when it comes to infinity, reals, or complex numbers.
The cool thing here is your claim that reals are no more or less real (different sense of "real", I take it) than complex numbers. Say you want to be a realist about science, and physics, in particular. That is, you believe that it attempts to describe reality, rather than, for example, it telling an inter
Re:what do you mean "could be needed"? (Score:5, Informative)
Reals and complex numbers are closures [wikipedia.org], which are basically ways of extending our metaphors to ensure that they can express answers to questions that arise from our metaphors when they are not so extended.
So when the natural numbers are understood using a "rocks in a box" metaphor, we suddenly get the question "what number do we use when the box is empty?" And thus we invent zero as the first natural closure of the natural numbers. More interesting is that with this metaphor, subtraction is meaningful, and we have a clear answer for "what happens when I have three rocks in the box and I take two of them out?" You have one rock in the box. So subtraction fits the metaphor nicely. But what happens when you have three rocks in the box and you take FIVE out??? For that, we must extend our metaphor to include negative numbers. Thus, negative numbers are a closure of subtraction on natural numbers.
What do negative numbers represent? It doesn't make sense that you could "have" negative two rocks in a box. But with a little thought we find it very naturally describes such concepts a debt, or "overshoot," for example. These things are themselves abstract concepts, not real objects, but they are nonetheless extremely useful concepts that apply directly to engineering or economics, so negative numbers are very meaningful even when they don't directly correlate to some simple real entity.
Anyway, reals (being ratios of two numbers) are a natural closure of the operation of division whenever that operation doesn't result in an integer. (How many groups of 5 rocks are there in this box of 10 rocks? Easy: Two. How many groups of 4 rocks are there in this box of 10 rocks? We need a new closure for that so we can represent it as two fifths). That closure is THE answer to your question "of what are they abstractions?" They are extensions of abstractions, which may or may not become useful for other purposes.
Same for complex numbers. Its a closure of roots. The square roots of all positive numbers require the introduction of the irrational number closure, but once we want to root negative numbers we need YET ANOTHER closure. Negative numbers already were a closure of natural numbers, so we have a closure on top of a closure. There isn't an obvious natural object that serve as the metaphorical foundation of a negative root, but as a closure of a closure it makes perfect sense.
Incidentally, when numbers are visualized as points on a line, instead of rocks in a box (the metaphor being drawing a literal line in the sand and marking out even intervals thereof), we can use "rotation" as a metaphor for visualizing the negative numbers. If you draw your line on a wheel, out from the radius to an edge, and put your numbers on that line....you can then perform a 180 degree rotation to land on the negative number. The act of multiplying by negative 1 is how one performs this rotation. Well, i is the square root of negative 1, so you have to multiply by i TWICE to do that 180 degree rotation. If you multiply by i only once, that would be a 90 degree rotation, and suddenly your number line is now a grid with a vertical axis. And...voila! Complex numbers, which are now represented as points on a 2d graph rather than points on a line. SO, if you really need them to correspond to something, that series of metaphors is the best we've got.
Oh and "infinity" is just a way of capturing the boundlessness of the set (there is no greatest number). So it isn't a number itself, its a property of a set of numbers. It doesn't need to correspond to any real thing in the real world in order to be understood as indicating that a set of numbers has no limit.
Also, you don't need imaginary numbers (Score:4, Insightful)
Let's say that you have some physical quantity that is described best as a pair of numbers <a,b>. You find that there are two reactions between pairs of these quantities.
One reaction between quantities <a,b> and <c,d> results in <a+c, b+d>.
The other reaction results in <ac-bd, ad+bc>.
Call the first one "addition" and the second one "multiplication" and you have an equivalent mathematical description of complex numbers without a concept of "imaginary" involved. (Instead of the square root of -1, you simply have the property that <0,1> * <0,1> = <-1,0>.)
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Your example uses integers, which are much less abstract than real numbers.
Even integers are problematic when applied to macroscopic objects like apples. What constitutes an apple? What if you had left one to rot until it was a pile of goo; is that still an "apple". If not, exactly when did the transition happen? At what instance in time does the apple wink out of existence so you can update your integer tally?
Re: what do you mean "could be needed"? (Score:2)
How "one" is defined to physical objects is philosophy, not maths. However if I cut that apple in half then I have 2 halves of an apple so fractions are real too, in as much as anything is.
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How "one" is defined to physical objects is philosophy, not maths.
That's exactly why all numbers are abstractions that don't neatly map onto physical reality.
Heap of Sand (Score:2)
If I have a heap of sand, and I remove just one grain, I still have a heap of sand, obviously ...
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Your example does not prove that "real numbers are not an abstraction," as it uses a real number as an abstract representation of quantity.
The apple is real. How many of them there are is a fact. The number 2 is an abstraction used to describe that fact.
To make it simple: you can put two apples in a bucket. You cannot put "the number 2" in a bucket.
Re: (Score:2)
Using that logic then I can't put an apple in a bucket either because "apple" is nothing more than a word, not the physical object itself. You're confusing language with reality.
Imagine that! (Score:2)
Re: (Score:3)
If you begin by drawing a distinction between "real things" and "concepts that explain real things," then the question is easily answered. Math is clearly and obviously a concept used to explain real things. You can drop two rocks on your foot, but you cannot drop "the number two" on your foot. So simple a child can understand it, but this distinction tends to trip up our most brilliant minds precisely because the human brain is so good at multi-level processing of abstract concepts that people who do it
If you call Ice Fire will it become cold? (Score:3)
OK you called it imaginary, so does it mean it can not exist in reality? What about negative numbers? Do they exist in reality? So negative number is just absence of something? We could have named positive numbers Presence Numbers and negative numbers Absence Numbers? Then what about fractions? Irrational numbers? Trancendental numbers? Quarternions?
It is unfortunate they were called imaginary with lots of connotations in ordinary plain language. It would have been better if we had named it something else, Polar number or Orthogonal numbers or Euler/Newton/Lagrange/any_mathematician_scientist Numbers to make them no different from the other numbers mentioned when it comes it existence.
as consumer tech advances (Score:2)
This may well be an ad for metaverse or the Matrix Resurrection
So who is planning on showing off their NFT's artwork in the metaverse?
Don't-cha just gotta wonder who's gonna grab the name PlayerOne Or maybe DejaVuCat
In the digital world, everything is real... even imaginary numbers.
When you reach an imaginary number (Score:2)
Re: (Score:2)
Then you get this:
"The number you have dialed is negative. You suck. You should have turned it the other way."
So if imaginary numbers... (Score:2)
Complex numbers are used in classical mechanics (Score:4, Insightful)
to describe waves among other things. Unless you're reading this site on paper you're using systems that were built using a model of reality that uses complex numbers. For some reasons electrical engineers use "j" instead of "i" to mean "sqrt(-1)" (maybe because they use "i" for current?), but it's pretty basic to describing radio circuits and alternating current. Once you get to the level of differential equations in calculus, it becomes completely natural to describe any kind of periodic phenomenon in terms of complex numbers.
Sure, imaginary numbers seem weird, but if you look into *real* numbers, those turn out to be weird too. A culture that only used numbers of measure things would never have come up with real numbers. You only need them when start to you pose rather abstract, theoretical questions (e.g., what is the ratio of the diagonal of a unit square to its sides?). In fact we only use the concept of real numbers to do math; we never actually use real numbers in any practical context like measurement or computation, only some rational approximation. So reals are just as "imaginary" in a cognitive sense as imaginary numbers are.
Negative numbers were the devils work (Score:2)
How on earth can you have -2 sheep!
Appropriate Connection (Score:5, Informative)
Veritasium traces this line of thought from the "completing the square" solution to the quadratic equation - where you are literally manipulating pieces of paper representing areas of a square - through a similar technique applied to the cubic. In these, a solution with an imaginary number involves negative areas or negative volumes. What is the length of the side of a square that has negative area? Such a thing doesn't completely make sense in the physical world, but does yield valid solutions to the equation. Until this breakthrough was made, many solutions to the quadratic or cubic couldn't be calculated - even some solutions that were themselves not complex
In making this breakthrough, we gained not only new tools for manipulating equations, but all of the insights that those tools later permitted - up to and through quantum mechanics.
Complex numbers needed in this formulation (Score:5, Informative)
what they say is:
"Our main result applies to the standard Hilbert space formulation of quantum theory, through axioms (1)–(4). It is noted, though, that there are alternative formulations able to recover the predictions of complex quantum theory, for example, in terms of path integrals13, ordinary probabilities14, Wigner functions15 or Bohmian mechanics16"
This is interesting in a mathematical physics sense, but it doesn't mean you cannot in any way calculate quantum mechanics without complex numbers. (nor do the authors claim that )
Re: Complex numbers needed in this formulation (Score:2)
Indeed . Complex numbers are just a mathematical shortcut (or even hack) because if you couldnt do the same using real numbers youd never be able to calculate a result either with a computer or on paper.
N J Wilderberger reconstructs Imaginary Numbers (Score:2)
Imaginary Numbers were a conceptualization used to solve quadratic equations.
I never bothered to deeply study it, but there's a math professor with a series on YouTube (N J Wilderberger) who made an entire mathematical system that -- is quite rigorous -- and I think he did it at least in part to demystify "imaginary" numbers.
This is according to my memory and I may be wrong, but my recollection is he was asking, "How can we solve this problems, but without asserting that there are square roots of negative n
complex mechanics by professor c d yang (Score:2)
https://arxiv.org/pdf/2103.109... [arxiv.org]
professor yang has been working on this for years: he uses complex numbers inserted into classical equations, to derive quantum mechanics equivalents, as a way to teach his engineering-trained students about quantum mechanics.
Stop Press! Divide by Zero needed describe physics (Score:2)
But, take a step back and look at it again. Most of the physica
Vector multiplication. (Score:2)
It means something is missing (Score:2)
When your calculations of a solution to a set of equations gives "imaginary" numbers, as in "the square root of a negative number", this means that your equations are missing something.
Usually it is missing a "dimension" or a "degree of freedom". Probably something in the specifications of the units, as used in the equations.
For instance, using calculations meant for DC electricity on an AC circuit can give imaginary expressions. This is because the full equations need to have a time (or "phase") expression
Imaginary numbers are real numbers. (Score:2)
There is nothing imaginary about our quaternion number system, complex numbers are as real as any other number. Itâ(TM)s better to think of as lateral numbers, and just because you teachers wrote the equations on the board using shorthand notation which left off n + 0k doesnâ(TM)t make them magically disappear. E = mc^2 is actually (E)^2 = (mc^2)^2. c^2 = t^2 + s^2, which is functionally equivalent to k^2 = i^2 + j^2.
42 (Score:2)
42 isn't an imaginary number...