With light pollution everywhere, the last resort to watch a black sky is from the sea, far away from land.

And the bad news is Bio-luminance, the oceans are full of it, which means you can make out things when you are forward watch etc. The second is it rocks which makes taking a star sight harder than it could be.

Few people actually realise the gut wrenching reality of shiping containers and logs in the water, yet most know about RMS Titanic and what caused that. If you are in a small to moderate sized sail boat doing five-six knots (~9-10.8KPH) single handedly with "self stearing gear" you are hoping that there are not just no other vessels in the area, but logs or significant flotsam like logs or shipping containers to rip your hull open faster than a rin can meeting a rip saw... That's why I used to sleep a couple of hours here and there during the day and hope any vessels had a forward lookout.

It's when I truly discovered the meaning of "ignorance is bliss" and "Knowledge leads to the slow madness" that is fear induced insomnia...

I had a chat once with someone back in the 90's who had been in the Vietnam war and had been shot down. He told me that what caused him the most fear was not trying to hide from patrols, but that he might fall asleep thus snore and wake up with a slit throat... It was still giving him nightmares a quater of a century later. Sometimes true bravery is being able to make it through another day...

]]>OMG you were "moonlighting" ;-)

More like being mooned by the sky!

You did not say if it was "business with a little pleasure"

Pure business. Trip was a late decision. The stars were aligned, so to speak.

Whilst I don't mind sailing

With light pollution everywhere, the last resort to watch a black sky is from the sea, far away from land. Never tried that. Probably Sinai, the Sahara or Siberia are the best places now. They all start with an 'S' for 'Sky', eh?

]]>So I had to go to Arizona for some work... ...I looked at the moon phase and saw an opportunity to look at dark skies in Arizona.

OMG you were "moonlighting" ;-)

You did not say if it was "business with a little pleasure" or the other way around, which your description makes it sound like :-)

But I hope you did manage to get atleast a change if not some relaxation whilst you were there.

Sadly as I've mentioned my globe trottting traveling days are over as aircraft and mountains, the quacks have told me are verboten.

Whilst I don't mind sailing, spending upto a month getting somewhere and a month getting back is not exactly productive the business people tell me.

The thing is around Europe the Ferry/coach option from the UK is still the least expensive way to get from A to B but it does have a human cost if you can not sleep on coaches.

]]>I still owe @Anura something

In electrical engineering, (as in other domains of engineering since you end up solving the same set of differential equations) we have two notations for some quantities: Phasor notation and Complex notation. For example, an ideal capacitor has zero resistance but has > 0 Reactance. The reactance is the relationship between the current (through the capacitor) and the voltage (across the capacitor.) The current and voltage are out of phase, and this can be represented as an Imaginary component. Same for an inductor, but with current phase lagging the applied voltage.

Engineers are lazy folk. They like to make things easier. That's why they'd rather solve some set of algebraic equations (by using a Laplace transform of some signal) rather than solve a set of differential equations. With a Fourier Transform (Laplace and Fourier transforms are related) one works in the frequency domain rather than in Time domain, which is easier for certain operations (filtering for example, or spectrum analysis.) Complex numbers appear in all these operations.

Complex notation makes it easier to make calculations, and in some sense, it makes sense. Measurement equipment, like you said above (**my not-so-humble way of saying you are correct**,) can represent this reactance in complex form by making some calculations as the following links show:

Measuring Impedance of a capacitor, Inductance

@Clive Robinson,

The harsh mistress is your mistress's twin. So you know she works overtime. I'll try, but this is no vacation, my friend.

So I had to go to Arizona for some work (in my opinion, Arizona is the most beautiful state in the United States -- it should be called "land of enchantment", instead of new Mexico ;) ) I looked at the moon phase and saw an opportunity to look at dark skies in Arizona. So... Over the weekend I drove a thousand miles (started with an Uber ride -- not a step) to visit Grand Canyon, Sedona, Horseshoe Bend, and Antelope Canyon. Went to Navajo Point away from city-light pollution and well, the moon was a bit showing, and there was city light pollution, but I got to see the Milky Way -- nothing compared to what I saw in the Sinai Peninsula (Land of the Turquoise,) but worth the trip. And to be fair, during the Sinai visit, there was no moon, no city lights, I was between two mountains, and the eyes were in better shape. Antelope Canyon was a disaster! I could not visit it because now one needs a guide (it's in Navajo Nation) and all tours were sold out (both upper and lower canyons, so I did not make it -- bummer.) Next time ...

Horseshoe bend is a dangerous place. Saw people walking on the edge (and drinking.) People cannot asses risk very well (there... a security plug) and the death toll is significant between Grand Canyon and Antelope Canyon, although Grand Canyon claimed more lives (a little short of 300 since they started recording.)

I'll book-mark the rest for later :)

Strange! I don't know what I was going to say... I had something in mind at the time, but it escaped my skull. Can't be *important*, I guess.

And you make sure you send that hash mistress of yours on a similar length vacation, shes obviouslt been woeking over time on you ;-)

The harsh mistress is your mistress's twin. So you know she works overtime. I'll try, but this is no vacation, my friend.

I'll book-mark the rest for later :)

]]>Count me out for a couple of weeks

And you make sure you send that hash mistress of yours on a similar length vacation, shes obviouslt been woeking over time on you ;-)

With regards the Five-Eyes, it's a long story so the "Letts Notes" version.

Towards the end of WWII a couple of people at Bletchly Park realised that Britain would be a busted flush after the war due to the fact the war effort had knocked the bottom out of everything. So they decided to put together a post war Intel plan which after a relativly short period of time became a two page MOU that became the basis of the BRUSA agreement and later the UKUSA agreement.

In essence Britain would provide specialist knowledge and development in return for US industrial capability. They also due to Britain being the heart of the subsea cables gave access to most diplomatic traffic etc.

As the demands for other parts of the world became a requirment it was --and still is-- British Empire / Commonwealth that was stradling cable nodes in other parts of the world. The US however was "paranoid" about Spys and foreign agents, thus they only wanted WASP nations to be involved.

So Australia, Canada and New Zeland were brought in as tier two members, hence "Five sets of eyes on the problem" shortened to Five-Eyes.

Since then many other nations including many Commonwealth nations have signed up. As have some Northern European nations.

It is this peculiar arangment and early "give it all away" agreaments that UK politicians refere of when they talk of the "Special Relationship".

I can assure you that the only thing special about it is how the UK has kow towed to US interests all through to the 1980's much to the detriment of the UK's economic position...

Before you do, your complaint is about "imaginary numbers"

Nobody's complaining about anything! It started here. The discussion started (or took an inflection point) here:

But it's ok to use in a discussion. We often use mathematical entities that we know don't exist in real life such as the square root of -1 in imaginary numbers. The use of infinity, etc...

No complaints: just giving examples. I am not complaining! re-read the context.

which raises a question aboutwhat you thing of "negative numbers"?

What do I think about negative numbers? I think of them all the time when I look at my bank account balance.

Butif you assumenegative numbers are not real, then the issue of imaginary numbers does not arise.

Conditional statement is false.

Happy now?

More puzzled than happy; I feel like I passed to this blog through a membrane from another reality! I need to find some challenging security related problems to open for discussion. I must be in the twilight zone!

Or if you insist on discussing numbers and tie the discussion to the thread topic: why Five eyes? are we talking about Cyclope-nations (as in Cyclopes) here? I mean five nations, shouldn't we call it "9-eyes" or something?

I am not discussing this any more, except maybe for giving an example to someone here about measurements of *complex* numbers, which was another digression.

[Y]ou might find interesting Stephen Grossberg’s work on neural nets.

It looks like I need a trip to the library ;-)

]]>I'll probably say something in a couple of weeks or so.

Before you do, your complaint is about "imaginary numbers", which raises a question about what you thing of "negative numbers"?

Negative numbers in real tangible form do not exist, they are not "natural". That is whilst you can have +7 apples you can not have -3 apples to pick up and hold or even eat, they make no sense in the real world (the same as "negative frequencies" and similar). But are essential to make sense of the real world[1].

So why are you apparently comfortable with "negative numbers" but not "imaginary numbers"?

Oh imaginary numbers exist in the same way negative numbers do because the basic laws of natute as we currentky understand them says we need "reversability of information".

If you consider multiplying two numbers together from just their sign bits you find that the truth table you get is the equivalent of the XOR truth table.

However if you consider it by binary value not sign then the truth table you get is the same as the AND gate.

Neither truth table is fully reversable. In fact the only thing you can conclude about the inputs is when the output of the AND gate is True, and all it tells you is that both input values were not zero.

You need to somehow retain the sign and value of one input if you are to get reversability thus your energy back.

The easiest way is just to pass on one of the two inputs --as long as it's not zero-- to the output. Thus from a multiplication you would do a division or second multiplication in a field etc.

Now it just so happens with the square route function you can determine the value of the inputs but not the sign from just the output. To get the sign bit for full reversability you would still need to pass it forwards...

Have a think about a "halving function"[2] it too will recover the two input values to an add, provided they are equall but because the sign bit rules follow a different rule set you can determin the sign.

But if you assume negative numbers are not real, then the issue of imaginary numbers does not arise.

Happy now?

[1] As the Romans found things like basic surveying do not work with only the natural numbers and especially without the concept of a zero. How much this might have held back their mathmatics and thus science nobody realy knows but it took the west a millennium or so to catch up again.

[2] One of the issues with high level laguages that does not occure in most assembly languages is the question of shifting with sign bits. In assembly you usually get a variety of shifts that are due to having the sign bit "inband" with the value. This shows up when people try to build their own data structures for "extended precision maths" such as using two 32bit values to make a 64bit value. You have to be aware that only one of the two values should be a signed value and get the right one for your CPU architecture. Whilst easier for addition multiplication can become a bit of a nightmare. As for division you can end up with four valid output formats depending on what you want to do next...

no such feeling of intuition

In this regard, if you haven’t looked at it, you might find interesting Stephen Grossberg’s work on neural nets. Grossberg based his designs on the physiology and even on insights from the psychology of natural neural systems, abstracting what they do, and then treating it as a dynamical system. In many cases this results in a system that learns without “external” supervision, but rather by recursive feedback iteration. As Grossberg put it, the usual layered back propagation approach is analogous to your mother teaching you to read by directly tweaking your synapses, whereas actual learning in real systems proceeds because the system dynamics result in steering to a desired behavior.

]]>Lotfi Zadeh's death was just last year,

I have some of his books and papers in my dead-tree cave.

Back in the hey day of 8bit MCUs in the 80's & 90's Fuzzy logic was a way to get things done with machines and motion that you could just not do with the usuall engineering maths.

Also after you build a fuzzy logic system you got this feeling of it being an intuative way to do things.

It was partly his work and the feeling of intuition that made me start to look at neural networks. Unfortunatly no such feeling of intuition there :-(

]]>Hazy Logic

*Hazy* :)

Grrr, typaws: “to get your more بٓنگ for your buck.” (Not for sale yet, though.)

How much is the big one, dawg?

]]>I guess I can’t show you thecomplexnumbers actually really exist. :-)

You have got to be freaking kidding me! Imaginary, dawg! **imaaaaaginary**. Oh, well. Seems like a dead-end discussion. I still owe @Anura something.

I can tell you a crash leads to a “bong dump” that allows my QA people to investigate what the programmers were smoking. (Needless to say, the system is a work in progress.)

Lol. Thank you. Your clever wits made me smile on an otherwise miserable day.

]]>Anyway, I’m out. さようなら!

]]>Final comment:

I still don’t know what is required for numbers to “really exist”. The latest criterion seems to be that operations on numbers are single-valued, but this isn’t a universal property of operations on elements of ℝ either. Another criterion is that numbers that “exist in the real world” express cardinality, which disqualifies anything beyond even ℕ_{0}, say. Despite all this, ℝ “really exists” and ℂ… obviously *doesn’t* because, dude, criteria.

I guess I can’t show you the complex numbers actually really exist. :-)

(Not that I think any numbers “really exist,” but if the reals do, so do all the others. I just can’t show it.)

@Spookmaster Bong,

*That so-called C++ 11 of yours, what's it supposed to do?*

No C++, Your Excellency. (In fact, if you change *LDBL_DECIMAL_DIG* to *DECIMAL_DIG* it’s valid C99, I think.) Have a look, it might actually work.

*First he does a #1, then a #2 in my *device*!*

A misunderstanding, Your … Highness. I’m environmentally conscious and only use 100% *grass*-green energy. As a result, I’m limited to computing devices based on Hazy Logic, applying lessons from Lotfi Zadeh’s (sometimes unpublished) work, but turning it up to eleven to get you more بَنگ for my buck. Without going into implementation details, I can tell you a crash leads to a “bong dump” that allows my QA people to investigate what the programmers were smoking. (Needless to say, the system is a work in progress.)

Each dimension needs to be measured independently. Whatever device does the measuring, is going to pick up information as real numbers; if it's useful to represent that as a complex number, you can do so, but you are going to be taking two separate measurements.

I guess the penny has not dropped yet...

You folks are thinking about it in the wrong direction.

Depending on just who you listen to we live in a multidimension space, that is N-Dimensional, where N ranges from 3 to 11. We can only directly experience 3. That is every object we observe is three dimensional.

But we know that any real point in N-Dimensional space exists in some way in each dimension.

As I noted above you can represent a 3D object in 2D but you lose information and get ambiguity in return. Thus the 3D helix when you draw it in 2D produces a circle in only one axis coaxial to the helix.

In another axis orthagonal to that you will get what aproximates a triangle wave, and at any point inbetween a complex series of loops. At no point has the real points in the helix changed, all the changes are due to the way you project through the object onto the 2D surface.

Now having obtained your two dimensional image you can project it down again to a one dimensional image. Again you will lose information and gain ambiguity. However if you use a series of orthagonal projections all the information can be recovered.

At no point however has the real points in the helix changed, nor has their relationship to other points in the helix.

Are you all with me at this point?

Thus the point in N-Dimensional space only makes sense if you account for all the N dimensions. How you measure and record them is of little consequence as long as you account for every one of those N dimensions.

So you could use a distance from the point of origin for all dimensions and a series of angles for N-1 dimensions, or you could use N distances from the apropriate one dimension origin. The thing that all those measurments have is that they are one dimensional. That is they have been stripped of all other dimensional information.

You might want to think over that again, because the angles are one dimensional measure even though we think of them in a two or more dimensional way.

However at no point has the real point changed it still has a presence in all N-Dimentions.

It is we humans that chose to throw away information by projecting down to one dimensional measurments. We do infact often use two dimensional measurments call them images or technical drawings they are slices through a three dimensional object. Providing we follow the rules we can reconstruct a three dimensional object from them.

Now the important bit, you can take a series of N-Dimensional points as an N-Dimensional object and project it down to an N-1 image, that an N-Dimensional being will recognise as an N-Dimensional object. However an N-1 Dimensional being can also see it but as a complex N-1 dimensional object. They might infer it has more dimensionality but they can not see it as such. The usuall example given is a four dimensional cube drawn in wire diagram form as a little cube inside a larger cube with verticees joining the outer corners to the inner corners. Another way to project it is as we do when we open a paper cube out flat and end up with an image of six squares arranged as a T or upright cross with three squares across and four squares down with one square shared. In both projections however all volumetric information is lost or distorted. Thus with a four dimensional opened out projection we end up with three dimensional cross that we could see in our mind folding up. But we would have lost the coresponding information that would be four dimensional volume, but by following a set of mathmatical rules we could reconstruct it.

However whilst our mathmatics works quite happily to invert the projections only if we do not try to bend it. Thus we can use one dimensional or N-x Dimrntional measurments to record every point inside the four dimentional object in what form we take those dimentional measurments and record them does not matter in the slightest.

What we can see more easily from one dimensional measurments is there are points where some of the measurments are zero. If we pick out all the points that have the same dimensional measurment at zero then they can be plotted as an N-1 dimensional projection with no loss of information for those points.

So the question you should be asking your self about the square root function is "Is it a complete function or a sub function" and the answer is it's a sub function. It's only part of the required point transformation function, as you all should know.

Talking about the square route function outside of it's full function kind of does not make sense. It's like talking about the single digit subtract mapping when doing multi digit subtraction. That is you need the "borrow" if you want the result to make sense in the way we normally use digits in multi digit subtraction.

I guess none of you have written maths libraries in assembler... Or designed CPUs arithmetic instructions using bit slice parts. Otherwise you'ld know the difference between an XOR, half adder, and full adder, and be able to relate the AND function and it's limitations in all three circuits to the square root function in multidimentional transformations and functions that can never have negative inputs.

]]>Just don't confuse the input with the output, it tastes terrible!

I've done worse! First he does a #1, then a #2 in my *device*! He thought it's a bonsai tree for the first act, but I don't know what it looked like to him in the second!

(The empty line in the C code above actually contains a single space to prevent everything after it from coming outdouble-spaced.) Now you know.

I struggled with this in the past, but now I know. Thanks, dawg! That so-called C++ 11 of yours, what's it supposed to do?

]]>Not quite!

I afraid I’ll have to give you a citation for unrolling phase in a principle value zone. It’s a mandatory fine of of 20 hours community service at the Riemann laundry ironing sheets.

]]>Just don't confuse the input with the output, it tastes terrible!

There speaks the voice of experience folks :-S

]]>Just don't confuse the input with the output, it tastes terrible!

]]>First of all: We were talking about pure imaginary numbers -- the example I gave originally.

There’s also a difference between 0.666666 and 2/3. Please fully expand 2/3.

It's **single-valued**

Anyway, both cos(ln(2)) and sin(ln(2)) are ℝeal numbers that supposedly actually really exist and I assume that all works.

Not quite! 2^{i} has **more than one value**.

In fact, the same thing is true even when a is a real number. Technically speaking, the expression a^(b+ic) has infinitely many possible values (except when b and c are both rational numbers),

Because instead of doing the calculation writing a = e^{(ln a)}, you could also do it by writing a = e^{(ln a) + n π i}. Do you see what you missed when you did your "expansion"? There is a lot more to it than this but it's way OT.

both cos(ln(2)) and sin(ln(2)) are ℝeal numbers that supposedlyactually really exist

Okay, we agree. **Now show me** that "i" actually really exists ;) -- my original request (I stated my request in more than one way, but you evaded answering the question.)

Other way ’round: numerals are symbols used to reprent numbers.

My bad, you're correct.

さようなら

]]>Segmentation fault (bong dumped).

Oh, cry me a river.

💧💧💧💧💧💧💧💧💧💧💧💧💧💧💧💧💧💧💧💧💧

One can posit i, j, k, complex, quaternion, octonion, tensors, etc. as entities with algebraic rules, and see that everything works and even has practical applications, and everyone gets along fine, but still be mystified and lack scientific understanding as to what these things really are, because they have not been defined in a way that relates them to the first principles of mathematics i.e. quantity abstracted from real things.

I am a lazy bad student and haven’t read these subjects historically as is required from the beginning, but I have never with one or two exceptions seen a treatment that does that. E.g. one sees several explanations of complex numbers, but they seem to be particularized and don’t seem to reach the appropriate and commensurate universal.

]]>I made another typaw: “puppy eyes at full b**L**ast.”

*There is a difference between: *0.666666* and […]*

There’s also a difference between 0.666666 and 2/3. Please fully expand 2/3. :-P

Anyway, both *cos(ln(2))* and *sin(ln(2))* are ℝeal numbers that supposedly actually really exist and I assume that all works. All I do is stick them in a pair, like a fraction but with a “plus i times” instead of the slash. I’m just a dog, dawg!

This should work (untested C11; no formal anything used to create these bugs):

#include <complex.h> #include <float.h> #include <stdio.h> int main(void) { long double complex z = cpowl(2.0, I); printf("2^i = %#.*Lg + i %#.*Lg\n", LDBL_DECIMAL_DIG, creall(z), LDBL_DECIMAL_DIG, cimagl(z)); return 0; }

Let’s see:

$ cc thing.c $ ./a.out Segmentation fault (bong dumped).

Guess not…

@Anura,

*Now, if anyone can tell me why my C++ comes out double-spaced and my assembly single-spaced...* (remember?)

That happens when your *<pre>…</pre>* contains empty lines. (The empty line in the C code above actually contains a single space to prevent everything after it from coming out double-spaced.) Now you know.

Thatdidn’t seem to bother youwhen I picked it as an example: of course 2/3 is a number that actually

Not at all!

There is a difference between:

0.666666

And

Y = sin (x + ...n)

Out for real this time. I'll pick it up later ;)

]]>*It doesn't count as a *single* number if you complete the expansion, slick!*

Ehm… 2/3 isn’t a single number. (That’s why you need the solidus.) That didn’t seem to bother you when I picked it as an example: of course 2/3 is a number that actually exists in the real world.

*I gotta take a break as i have a living to make.*

Yeah, there’s that too. ;-)

]]>No, I did not read the book ("...Palle Yourgrau

book on Godel, A World without Time....").

You may be interested in this paper by Don Hotson:

https://groups.google.com/forum/#!topic/diracwasright/sEZQnMAwzII

I found it mind-blowing.

------------

NOTE: It's getting hard to find online, so I made a dummy group for it.

. .. . .. --- ....

]]>Final thing for now.

Does the complex number above count as a number? *puppy eyes at full bast*

No! It doesn't count as a *single* number if you complete the expansion, slick! Don't get sneaky on me and show me a half-*ankle* solution, yes?

Look what I found under the couch! 2i = e i ln(2) = cos(ln(2)) + i sin(ln(2)) [...] Who’s a good pup?

Excellent! Complete the expansion. There is a 2 pi n term somewhere, *dawg!* I gotta take a break as i have a living to make.

Whatever device does the measuring, is going topick up information as real numbers; if it's useful to represent that as a complex number,

You answered yourself. I'll probably say something in a couple of weeks or so.

]]>*Numbers are representations of numerals.*

Other way ’round: numerals are symbols used to reprent numbers.

*You have not made the distinction between numbers and numerals, so your question was ambiguous.*

I asked if you thought that «mathematical entities like 1 and 2/3 **do “exist in real life”**», in contrast to what you described as “mathematical entities that we know **don't exist in real life** such as the square root of -1 in imaginary numbers” in an earlier comment.

*Real numbers represent the cardinality of real world objects: two apples, three pens, whereas Imaginary numbers don't represent any cardinality of real world objects.*

Only nonnegative whole numbers (“counting numbers” if you count from zero) express cardinalities of finite sets. Where does that leave 2/3 and all the other elements of ℝ \ ℕ_{0}?

(Or maybe the number of real-world objects is countably infinite, in which case we can add א_{0} but even then 2/3 is right out.)

*Imaginary numbers are mathematical models of entities that don't exist.*

Points in two-dimensional space exist. I model them using complex numbers. (Also see *Battleship* above.) How doesn’t this example demonstrate that by your criteria complex numbers “exist in real life”?

*By definition there is no solution to "square root of -1". -- it's undefined, because all squares are non-negative numbers.*

Isn’t this circular? (It’s like saying ℚ doesn’t “exist in real life” because 2 / 3 has no solution in ℤ and we all know only ℤ exists.) In ℂ there are solutions: i^{2} = (-i)^{2} = -1.

*Find the value of * 2^{i}

Ooooh, this Wael is turning out to be loads of fun! I love looking for things and playing fetch! Now, where do humans keep their values of 2^{i} …?

Look what I found under the couch! 2^{i} = e ^{i ln(2)} = cos(ln(2)) + i sin(ln(2)).

Who’s a good pup?

*Does *2^{i}* evaluate to a number?*

Does the complex number above count as a number? **puppy eyes at full bast**

*Why do you like to keep me awake man? I need to count imaginary canaries now. And the next two weeks... I'll be out of pocket.*

Heh. Sorry ’bout that. :-)

]]>See my other response. Each dimension needs to be measured independently. Whatever device does the measuring, is going to pick up information as real numbers; if it's useful to represent that as a complex number, you can do so, but you are going to be taking two separate measurements.

]]>"Being 'funny'. Saying that Caesium should work when everything else ceases to move."

I never saw that coming.

Hee-Haw: "Eeuuu ... that's corny."

Complex numbers are not measurable, but that doesn't mean they don't represent real things in nature

Do you want to rethink that a little bit?

Whilst true that complex numbers do not fall on the one dimensional "number line". They are however found in the two dimensional plain and are in many practical subjects fairly easy to measure (think voltage and phase in AC systems for instance)

As you go up in dimensions N you get similar N-1 etc issues thing for instance how a three dimensional object like a helix can be represented in two dimensions and measured two dimensionally. That is a unity circle is a way to measure a helix quit accuratly as a projection when the z axis is coaxial to it.

The limitation is not the points in N-Dimensional space or their movment in that space, but the limitations of humans and their rulers numbered in a single dimension.

]]>I'll do that later... finally feeling sleepy.

]]>Can you give examples of where we measure a complex number? You measure real numbers, and compose them into complex numbers.

Why do you insist on thinking of things in terms of quantity? Just because things aren't quantity, doesn't mean they don't represent real things. When you use complex numbers, the real and imaginary parts both represent different measures of the same object. They are measured separately and then composed. It doesn't make sense to talk about imaginary numbers alone; it's simply the second component of a two dimensional number.

]]>@pup socket,

Why do you like to keep me awake man? I need to count *imaginary* canaries now. And the next two weeks... I'll be out of pocket.

Sure there is: i. By definition.

Fine! 2+2 = 4, by definition?

Complexnumbers are not measurable, but that doesn't mean they don't represent real things in nature;

They are measurable! There's a huge difference between complex and **imaginary**! Complex number = real part +/- imaginary part. We're discussing the "imaginary" part only! I read the first third of the link. Pretty interesting but not related to this discussion. Show me a pure imaginary quantity. What does it mean if I:

1) Ask you five questions

2) Ask you -1^{1/2} questions

Explain points 1 and 2 in **plain English** to me. If you convince me, I'll transfer 1000,000i dollars to your account -- you can then "imagine" that you're a million dollars richer ;)

**Anura: **I'll pay you i x 90k for this Tesla

**Dealer: **Cool! And you can imagine that you drove it out of the store!

By definition there is no solution to "square root of -1". -- it's undefined, because all squares are non-negative numbers.

Sure there is: i. By definition.

Natural numbers represent cardinality, while real numbers represent things that are measurable. Complex numbers are not measurable, but that doesn't mean they don't represent real things in nature; both quaternions and complex numbers show up in quantum mechanics, and work is being done to extend octonions to quantum mechanics as well.

https://www.quantamagazine.org/the-octonion-math-that-could-underpin-physics-20180720

]]>You might be interested to look at Armand Maurer,

That'll have to wait a few weeks at best. I haven't finished half the books I have. The question: what's 2^{i} I asked @pup socket to look at wasn't a challenge; it was posed to demonstrate the difference between real and imaginary in the context of this discussion. 2^{4} is a number. Does 2^{i} evaluate to a number?

Then again, what does the number: square root of -1 evaluate to, and what numeral does the evaluation output represent?

]]>> Real numbers

You might be interested to look at Armand Maurer, “Division and methods of the sciences”, Fourth edition, 1986, PIMS, a translation of some writings of Thomas Aquinas on what the speculative sciences are, and in particular what mathematics is and the nature of the mathematicals.

In a word, the mathematicals are those things that exist in matter and motion but do not contain matter in their definition. They are obtained by considering real things leaving out all sensory and qualitative aspects.

If number and continuous quantity are “real”, things like imaginary and complex numbers have to be regarded as sharing in that reality, since they basically just deal with relations between those things.

]]>"Compilers are not required to implement crypto algorithms. Simple interpreters and assembler on 8bit machines ... pen and paper"

That's not my point and I'm sure the Five Eyes don't want to ban compilers, nor do they expect to be able to ban encryption (in products and SSL) or prevent experts from using encryption.

Instead, they are almost certainly focused on simpler everyday cases such as "how to unlock a suspect's phone during an everyday criminal investigation to view messages and photos". Most average criminals are not going to use custom or pen-and-paper encryption because (a) it's extra work and (b) you need to send messages to other people, and some of those people will not have super-strong opsec.

Now I agree that if the SigInt people really want to target someone they will be able to succeed via insecurities in the end user device, targeted phishing, etc. Although not so easy if all they have is a locked iPhone with no way to get it to fetch content over the network.

Hence why I think they do not expect to ban custom encryption generally but just push back against easy targets and on-by-default encryption. For example not banning disk encryption but preferring it not to be on by default for consumers. My only comment was about not needing to ban compilers.

In any case I agree the Five Eyes request is bad because there's no way to give access to law enforcement without also giving access to evil maids and foreign countries people may visit, amongst other things.

numbers, not the numerals

Numbers are representations of numerals.

Your original inquiry was:

... if mathematical entities like 1 and 2/3 do “exist in real life.

You have not made the distinction between numbers and numerals, so your question was ambiguous. Now, as you know, one cannot point out an abstract concept as a concrete real world object. So back to the difference between real and imaginary numbers (talking representations, now.) Real numbers represent the cardinality of real world objects: two apples, three pens, whereas Imaginary numbers don't represent any cardinality of real world objects. Imaginary numbers are mathematical models of entities that don't exist. By definition there is no solution to "square root of -1". -- it's undefined, because all squares are non-negative numbers.

In various fields of engineering sinusoidal signals can be represented in terms of e^{jΘ} or a complex sinusoid; x(t) = cos(t) + j sin(t), and they are related by Euler's formula... I know you know all this. But you actually don't pay for the imaginary component of the electric power you receive at your house: you pay for the real component (if I remember correctly.)

Back at ya: Find the value of 2^{i}

(Hint: this is a "Complex Analysis" topic) -- I may have forgotten how to solve it, but I kinda remember the technique -- it's been over 20 years since I looked at this topic.

Caution: pup socket is a tough customer. I thought I split hairs, but he'll split them ten times ;)

]]>We really need to come up with new names for number sets. Natural numbers, real numbers, and complex numbers are all poor terms. Or we should only use octonions and just stop wasting our time with lesser numbers.

]]>*[1 and 2/3] are Real numbers.*

Yes, they are ℝeal numbers.

*Of course they exist! I ate one bagel today, the moon was at 2/3th size on such and such date and time*

- My guess
*(0, 1)*just sank your carrier in a game of*Battleship*. (You might know this position as “A2” and I secretly call it “i”. Because I’ve used an imaginary number in a description of everyday reality, imaginary numbers now apparently “exist in real life”.) - You say of course the mathematical entities 1 and 2/3 exist. Where do they exist? Can you point to them? (The
*numbers*, not the*numerals*.)

*Imaginary numbers are called imaginary for a reason!*

They are called that because centuries ago otherwise smart mathematicians picked a mocking name to describe them and it stuck.

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