I was just thinking that if the use of an impossible mathematical quantity ie the square root of -1 can be used to solve real world problems, does that imply that what we can perceive and measure in the universe is only half of what is really happening, and the other half, although imaginary to us has actual effects on how we all operate in our ‘reality’.

And therefore it could be said “we don’t know the half of it”, and any model we make of the cosmos will forever be incomplete.

Just a thought.

rumpole said:

I was just thinking that if the use of an impossible mathematical quantity ie the square root of -1 can be used to solve real world problems, does that imply that what we can perceive and measure in the universe is only half of what is really happening, and the other half, although imaginary to us has actual effects on how we all operate in our ‘reality’.

And therefore it could be said “we don’t know the half of it”, and any model we make of the cosmos will forever be incomplete.

Just a thought.

If something has real effects, then it is real, even if we can’t access it directly. For instance, gravity is real even though we can’t manipulate it directly, and don’t know how it works.

It is of course quite possible (in fact near certain) that there are real things that have no effect on us, directly or indirectly. We can know nothing of these things though.

None of this has anything to do with the square root of -1, which is not really “imaginary” at all.

“None of this has anything to do with the square root of -1, which is not really “imaginary” at all.”
==========
So why is it called an imaginary number ?

Although Greek mathematician and engineer Heron of Alexandria is noted as the first to have conceived these numbers, Rafael Bombelli first set down the rules for multiplication of complex numbers in 1572. The concept had appeared in print earlier, for instance in work by Gerolamo Cardano. At the time, such numbers were poorly understood and regarded by some as fictitious or useless, much as zero and the negative numbers once were. Many other mathematicians were slow to adopt the use of imaginary numbers, including René Descartes, who wrote about them in his La Géométrie, where the term imaginary was used and meant to be derogatory. The use of imaginary numbers was not widely accepted until the work of Leonhard Euler (1707–1783) and Carl Friedrich Gauss (1777–1855). The geometric significance of complex numbers as points in a plane was first described by Caspar Wessel (1745–1818).

wiki

rumpole said:

“None of this has anything to do with the square root of -1, which is not really “imaginary” at all.”
==========
So why is it called an imaginary number ?

It seemed like a good idea at the time I suppose!

I actually struggle with the abstract side of maths, and I make little use of complex numbers in my work.

Have a go at sibeen when he turns up. He uses them all the time!

rumpole said:

I was just thinking that if the use of an impossible mathematical quantity ie the square root of -1 can be used to solve real world problems, does that imply that what we can perceive and measure in the universe is only half of what is really happening, and the other half, although imaginary to us has actual effects on how we all operate in our ‘reality’.

And therefore it could be said “we don’t know the half of it”, and any model we make of the cosmos will forever be incomplete.

Just a thought.

People (and electrical engineers) essentially use complex numbers because it makes the form many differential calculus problems much easier.

It’s really just a domain transformation.

rumpole said:

“None of this has anything to do with the square root of -1, which is not really “imaginary” at all.”
==========
So why is it called an imaginary number ?

The numbers are called ‘imaginary’ because they sit on a number line that is orthogonal to the ‘real’ number line.

there is something about it, though not sure of complex numbers, I’m a number dunce, the idiot part of savant, the dysmathpergic

what do know is take a cat sitting on a mat, it excludes all other cats sitting on mats in that location, or any overlapping location, but it also diplaces mats, cats and a whole host of other possibilities, so in a sense that cat sitting on that mat are so by all they displace.

that that isn’t, seems to contribute to possibility space

weird as it seems it’s not so different to looking at the shadow cast of your body by a light source and stating you would not exist without your shadow

insane as it sounds it is in fact true

diddly-squat said:

rumpole said:

“None of this has anything to do with the square root of -1, which is not really “imaginary” at all.”
==========
So why is it called an imaginary number ?

The numbers are called ‘imaginary’ because they sit on a number line that is orthogonal to the ‘real’ number line.

I’m suggesting that there may be a universe or an extension of our universe that is orthogonal to our own reality. Hard to prove of course, but it aligns with the concept of complex numbers.

Calling the number ‘imaginary’ has caused all sorts of grief, as has the malarky of using the letter ‘i’ to represent it :)

rumpole said:

diddly-squat said:

rumpole said:

“None of this has anything to do with the square root of -1, which is not really “imaginary” at all.”
==========
So why is it called an imaginary number ?

The numbers are called ‘imaginary’ because they sit on a number line that is orthogonal to the ‘real’ number line.

I’m suggesting that there may be a universe or an extension of our universe that is orthogonal to our own reality. Hard to prove of course, but it aligns with the concept of complex numbers.

what you are saying doesn’t make any sense, mathematics is just a construct that we use to describe things, that’s all…

>what you are saying doesn’t make any sense, mathematics is just a construct that we use to describe things, that’s all…

not entirely sure that is true as a generalization, if math is seen as part or as tools of ‘computation’ loosely speaking.

Take “1” for example, it is both quantity and quality that exists indepenedent of human concepts and constructions.

“what you are saying doesn’t make any sense, mathematics is just a construct that we use to describe things, that’s all…”
=============
OK fair enough, so we accept that mathematics is arbitary, not absolute, and a race of aliens may have different mathematics and therefore a different description of ‘reality’ ?

diddly-squat said:

rumpole said:

diddly-squat said:

The numbers are called ‘imaginary’ because they sit on a number line that is orthogonal to the ‘real’ number line.

I’m suggesting that there may be a universe or an extension of our universe that is orthogonal to our own reality. Hard to prove of course, but it aligns with the concept of complex numbers.

what you are saying doesn’t make any sense, mathematics is just a construct that we use to describe things, that’s all…

I wouldn’t say that. It is possible for a “construct that we use to describe things” to align with a physical reality.

>I wouldn’t say that. It is possible for a “construct that we use to describe things” to align with a physical reality.

representations do what they do

thing is the universe is totally full of examples of “1”(this or that, real things), and the quality of each is also a quantity.

OK fair enough, so we accept that mathematics is arbitary, not absolute, and a race of aliens may have different mathematics and therefore a different description of ‘reality’ ?

i wouldn’t say the mathematics is arbitrary, just the counting system. so we could understand alien mathematics just like we can understand foreign languages.

transition said:

>I wouldn’t say that. It is possible for a “construct that we use to describe things” to align with a physical reality.

representations do what they do

So does everything

ChrispenEvan said:

OK fair enough, so we accept that mathematics is arbitary, not absolute, and a race of aliens may have different mathematics and therefore a different description of ‘reality’ ?

no

oi!

>So does everything

Was wondering of Humpty’s ‘neither more nor less’, if it applies.

while on the subject of Humpty Dumpty, probably what of math this person didn’t much like maybe of interest.

http://en.wikipedia.org/wiki/Lewis_Carroll

transition said:

probably what of math this person didn’t much like maybe of interest.

Nice sentencing.

dv said:

transition said:

probably what of math this person didn’t much like maybe of interest.

Nice sentencing.

Puts up hand, “please sir what is a “Math”?”

I’d guess the real world of things (forces inclusive) that exists and ever existed are in fact representations of what were eliminated from possibility space. On the face of it this sort of conceptually spins things around, but likely the nearest-truth is something like this.

Has some similarities to the idea that a pregnant woman doesn’t have to worry about getting pregnant – pregnancy as a form of contraception.

complex numbers are each just two real numbers in order;

SCIENCE said:

complex numbers are each just two real numbers in order;

Indeed, but the second of those is a multiple of the square root of -1 , a number which cannot exist in our mathematical system.

rumpole said:

SCIENCE said:

complex numbers are each just two real numbers in order;

Indeed, but the second of those is a multiple of the square root of -1 , a number which cannot exist in our mathematical system.

A number which does in fact exist in our mathematical system.

sibeen said:

rumpole said:

SCIENCE said:

complex numbers are each just two real numbers in order;

Indeed, but the second of those is a multiple of the square root of -1 , a number which cannot exist in our mathematical system.

A number which does in fact exist in our mathematical system.

I’m with the Electrical Engineer on this one.

Practical application of imaginary numbers in fields like electrical engineering aren’t dissimilar in concept of how Fourier Transforms are used in signal processing.

rumpole said:

I was just thinking that if the use of an impossible mathematical quantity ie the square root of -1 can be used to solve real world problems, does that imply that what we can perceive and measure in the universe is only half of what is really happening, and the other half, although imaginary to us has actual effects on how we all operate in our ‘reality’.

And therefore it could be said “we don’t know the half of it”, and any model we make of the cosmos will forever be incomplete.

As others have said, the square root of -1 is a perfectly valid mathematical entity. I’m not sure what would qualify as an impossible mathematical quantity…

You have acknowledged that it can be used to solve real world problems, but I feel obliged to have a mini-rant on the topic of complex numbers. :)

As Sibeen complained, the term “imaginary” is an unfortunate historical legacy; I’d prefer it if imaginary numbers were called orthogonal numbers, or sideways numbers. A few years ago, while chatting with one of my young nephews who’s interested in maths, I introduced the topic of sideways numbers. He was only 10 at the time, but he had no trouble understanding what I was talking about. Whereas many high school students are intimidated when they are first introduced to imaginary numbers, and I mainly blame the terminology for that.

True, imaginary numbers can seem a bit strange at first. You can’t have 3_i_ oranges, but you can’t have -3 oranges, either (although I suppose you might owe somebody 3 oranges, and choose to represent that situation mathematically by saying you have -3 oranges). That doesn’t mean imaginary numbers are impossible, just that they aren’t useful when counting things like oranges.

But that doesn’t mean they are merely a mathematical abstraction, since they are very useful in all sorts of very physical situations, particularly when rotations in a plane are involved. It has already been mentioned that they are useful in electronics, since fundamental equations that apply to simple direct current can be applied to alternating current (and more complicated waveforms) if you use complex numbers.

Diddly-Squat said that “mathematics is just a construct that we use to describe things”. I’d take that further, and say that mathematics is the science of structure. Many of the structures that mathematics can describe are quite abstract, with little bearing on the physical world; OTOH, much of the development of mathematics has been spurred on in the quest to model physical things. And there are plenty examples of maths that was once though to be totally abstract but which later found practical application, including much of the core maths of relativity, quantum mechanics, and the maths used in the modern encryption systems that allow information to be securely transported over the Internet.

Diddly-Squat mentioned that complex numbers are used in Fourier analysis: the mathematics of representing a complex periodic wave as a combination of a series perfect sine waves. The transformation that is used in Fourier analysis to convert an arbitrary wave into a set of sine waves relies heavily on complex numbers. Amazingly, the formula used to convert an arbitrary wave into a set of sine waves is almost identical (apart from a change of sign and a scaling factor) to the formula used to reverse that transformation.

Fourier analysis and related parts of mathematics are extremely useful in signal analysis and transformation, since it allows us to handle all sorts of weird & wonderful waves once we know how to handle the simple sine wave. One mundane (but exceedingly common) application is in data compression: JPEG & MPEG image compression use a close relative of the Fourier transform (the Discrete Cosine Transform) to dramatically reduce the number of bytes needed to store image data.

In Quantum Mechanics, we use waves to describe the position and momentum states of systems; it can be shown that there is a Fourier relationship between position and momentum waves, so Fourier analysis is very important in Quantum Mechanics.

…

But to get back to the main point of your question. Yes! There may be more to the universe than meets the eye, and it’s quite possible that there are other spatial dimensions that are orthogonal to the ones we’re familiar with. And although we cannot perceive these dimensions directly, we may still be able to indirectly detect their effects on the visible world.

This is actually a fairly major area of current mathematical physics research, although it has its modern origins in the 1920s with Kaluza-Klein theory. Kaluza wanted to unite
Maxwell’s laws of electromagnetism with Einstein’s new theory of gravity. He realised that if he applied General Relativity to a 5 dimensional spacetime, Maxwell’s equations almost “fell out” as if electromagnetic phenomena were some kind of higher dimensional motion. Roughly speaking, in Kaluza’s model, conservation of electric charge is conservation of momentum in this extra dimension of space.

People were rather impressed by this unification, but they wondered that if this model is true, then why do we not have other evidence for the existence of this extra dimension. Klein had the bright idea that the extra dimension is compactified: the universe is so thin in the 5th dimension that we just don’t notice it like we notice the other spatial dimensions. Also, the 5th dimension is closed, like a circle, so if you travel a tiny distance in that direction you get back to where you started from.

Although the model works mathematically, most physicists thought it was so weird that it got put on the back burner for a couple of decades (they had enough problems handling the arcane maths of normal General Relativity, and the even more arcane maths of Quantum Mechanics). And in the meantime, the Strong and Weak nuclear forces were discovered, so Kaluza-Klein was no longer a theory uniting all the known forces. But in recent decades, mathematical physicists have revived Kaluza-Klein by adding extra compact dimensions to handle the symmetries of those nuclear forces. The resulting theory is more complex to work with than the original Kaluza-Klein theory, but it shows much promise.

PM 2Ring,

Thank you so much for taking the time to write that, it was fascinating and will lead me off into new directions.

In fact I asked the question because I’m doing a course on signal processing (elementary) and of course complex numbers came up.

The reason I say j (or i) is an impossible mathematical entity is simply that you can’t take the square root of -1 and come up with a number as an answer, so the concept is abstract, but happens to work in special situations.

Are here other examples in maths that define similar abstract identities ?

// Are here other examples in maths that define similar abstract identities ?

Absolutely, pun not unintended.

Consider the natural “counting” numbers, {1, 2, 3, …}.

Adding two of these numbers is fairly well-defined. We could claim that 4+7 = 11.

Once we have addition, we can define subtraction: A minus B means find a number to add to B to give A.

For example, 11 – 4 is what we need to add to 4 to get 11. We saw above that this is 7, and it is one of our counting numbers.

Please find the counting number that is 4 – 11.

rumpole said:

PM 2Ring,

Thank you so much for taking the time to write that, it was fascinating and will lead me off into new directions.

In fact I asked the question because I’m doing a course on signal processing (elementary) and of course complex numbers came up.

The reason I say j (or i) is an impossible mathematical entity is simply that you can’t take the square root of -1 and come up with a number as an answer, so the concept is abstract, but happens to work in special situations.

Quite the contrary. The number i (or j) is, by definition, a number, and an answer to the square root of -1 (not the answer: there are actually several.) I think the difficulty you’re having stems from the emphasis most maths teachers put on the set of real numbers, and the students’ assumption that a real number is, by definition, “real” (i.e. has some corporeal existence.)

Here’s an exercise for you: draw a picture of the number four. Now write it in Roman numerals. Now in Chinese. Now Hebrew. All these representations look very different to each other, but all represent the same abstract concept, the number four. All numbers are imaginary; their only existence is in the human mind. Adding another superset of numbers – which we’ve been doing since we discovered the cardinal numbers couldn’t account for negative, rational, or irrational numbers – is no logical stretch.

rumpole said:

Are here other examples in maths that define similar abstract identities ?

All mathematical entities are abstract. We can use our imagination to apply them to real-world problems and obtain meaningful, measurable results.

‘e probably means that general numbers in |C are harder to immediately conceive real links for than numbers in |R, or in |Q, or in |Z, or in |N.

rumpole said:

PM 2Ring,

Thank you so much for taking the time to write that, it was fascinating and will lead me off into new directions.

No worries, rumpole.

> In fact I asked the question because I’m doing a course on signal processing (elementary) and of course complex numbers came up.

Excellent.

> The reason I say j (or i) is an impossible mathematical entity is simply that you can’t take the square root of -1 and come up with a number as an answer, so the concept is abstract, but happens to work in special situations.

Well, the square root function isn’t defined on the negative real numbers, so in that sense it’s impossible. But that doesn’t mean that imaginary numbers are merely “a weird trick” that happens to work in special situations.

Humans learned about real numbers long before we discovered complex numbers, but in a sense, the complex plane was always there, lurking in the background. Functions “live” in the complex plane and trying to understand them by just looking at the real number line is a lot like trying to understand the world when your only view is through a narrow gap between fence palings. Thus, the early literature discussing quadratic equations was unnecessarily complicated. Back then, they not only didn’t have complex numbers, they didn’t even accept that negative numbers were proper numbers, so quadratic theory had to handle a bunch of special cases. The modern approach simplifies the picture considerably by showing us the underlying unity.

> Are there other examples in maths that define similar abstract identities ?

I suppose a similar situation arose when irrational numbers were first discovered. It’s pretty easy to show that sqrt(2) is irrational, so to some Ancients sqrt(2) wasn’t a proper number. But it has to be a legitimate number, since it’s the length of the diagonal of a unit square. The traditional resolution of this apparent paradox was to say that the laws of geometry are obviously correct, but algebra is suss. :) This attitude was still prevalent in the time of Newton, so he gave all of his proofs in Principia Mathematica in geometric form, even though algebraic proofs would’ve been much simpler in many cases. It took several centuries after Newton for real numbers (and complex) numbers to be given an adequate theoretical foundation.

Modern mathematics tends to look at mathematical stuff in terms of sets. So you have a set of entities, and a bunch of operations on those entities that take one or more of those entities as input and return an entity from the set as output. Obviously, we can cast traditional arithmetic in that format, but it’s also possible to come up with all sorts of sets of entities and associated operations that have interesting (and useful) structures.

A very important application of this is the mathematics of symmetry .

As mentioned earlier complex numbers can be a useful way to handle 2D geometry algebraically. Various mathematicians tried to extend the complex numbers to a system of triplets of numbers to handle 3D geometry, but they were not successful. An Irish physicist & mathematician, Hamilton , finally stumbled across a solution that works, but it uses sets of 4 numbers, known as quaternions .

Again very helpful and interesting.

I think I’ll go away now and do some more reading.

Thanks to all for your responses.

PM 2Ring said:

An Irish physicist & mathematician, “Hamilton , finally stumbled across a solution that works, but it uses sets of 4 numbers, known as quaternions .

You’re making me a little tense, tensor, tensest with all that, PM.

Rumpole, in electrical engineering the term ‘j’ is really just a rotation within the xy plane. So that 1 means a 1 to the right across the real line ‘x’. -1 means 1 to the left across the real line. 1j means rotate the 1 90 degrees anti-clockwise, so that it represents 1 northwards along the ‘y’ axis. -1j means rotate the 1 90 degrees clockwise, so that it represents 1 southwards along the ‘y’ axis.

A similar operator within EE is the alpha ‘a’ operator. In this case the rotation is 120 degrees, so that 1a will rotate the 1 in an anti-clockwise direction 120 degrees around the Cartesian plane.

I really do hate the term ‘imaginary’ :)

PM 2Ring said:

An Irish physicist & mathematician, “Hamilton , finally stumbled across a solution that works, but it uses sets of 4 numbers, known as quaternions .

sibeen said:

You’re making me a little tense, tensor, tensest with all that, PM.

Sorry, I didn’t mean to phase you, sibeen.

sibeen said:

Rumpole, in electrical engineering the term ‘j’ is really just a rotation within the xy plane. So that 1 means a 1 to the right across the real line ‘x’. -1 means 1 to the left across the real line. 1j means rotate the 1 90 degrees anti-clockwise, so that it represents 1 northwards along the ‘y’ axis. -1j means rotate the 1 90 degrees clockwise, so that it represents 1 southwards along the ‘y’ axis.

A similar operator within EE is the alpha ‘a’ operator. In this case the rotation is 120 degrees, so that 1a will rotate the 1 in an anti-clockwise direction 120 degrees around the Cartesian plane.

I really do hate the term ‘imaginary’ :)

FWIW…

We can do α here, but I suppose it’s a PITA to type.

Mathematicians often use ω to represent an arbitrary complex nth root of unity , i.e., a number on the unit circle of the complex plane. Your α is of particular interest; see Eisenstein integer .

PM 2Ring said:

Your α is of particular interest; see Eisenstein integer .

I immediately read that as Einstein, until I went to the link. I then crossed across to the page on Eisenstein.

quote from wiki

He suffered various health problems throughout his life, including meningitis as an infant, a disease that took the lives of all five of his brothers and sisters.

quote

Fuck me. I wished I lived in the olden days.

• Fuck me. I wished I lived in the olden days.

Not a fan of your siblings?

furious said:

• Fuck me. I wished I lived in the olden days.

Not a fan of your siblings?

I suspect your sarcasm detector needs calibrating, furious :)

• I suspect your sarcasm detector needs calibrating, furious

Can’t afford none of that fancy new clap trap, nothing that you can’t sort out with a hammer and a shovel…

nice thread guys… special thanks especially to sibeen and PM 2Ring…

Thanks, Diddly!

diddly-squat said:

nice thread guys… special thanks especially to sibeen and PM 2Ring…

Not to mention the OP.

:)