Some remarks on complex numbers

[ClamShell]

I just finished reading the "Reality Bits" in a recent copy of NewScientist.
It discusses attempts to purge mathematics of the need for complex numbers.
Started me thinking(danger, danger) of not how to get rid of the square
root of negative one, but, more easily, simply find out where it enters
the mathematical picture. The Pythagoreans were probably the first who
wondered about this;

$a^2 + b^2 = c^2$

We sure know how to represent the square root of $c^2$, it's $c$.
But can we write an expression for the square root of say, $1 + x^2$?
Try as we must, it doesn't seem to exist. But wait, if we supplement
our real number system with an imaginary number system, it is at least
factorable into,

$(1 + xi)(1 - xi) = 1 + x^2$

That's where it enters the mathematical picture; by the simple act of
trying to factor a sum. Anybody wish to add to this? Or even to point
out other places it enters the mathematical picture, other than the
factoring of sums? Have any ideas of your own, on how to "purge"
complex numbers from your physics homework?

I'm adding the words UBIT and U-BIT because I can't figure out how
edit my tags.

 

[Tobias Funke]

Why would anyone want to purge them from mathematics? They're useful theoretically and practically. If something better comes along then sure, they'll be forgotten, but I don't expect that to happen anytime soon.

They entered the mathematical picture, historically speaking, when they became useful. Look into the cubic formula for more information. Very briefly, by manipulating square roots of negative numbers as if they "really" existed, mathematicians arrived at results that turned out to be correct. Even then, they still weren't really understood or fully trusted until people came up with a way to picture them geometrically.

But we don't have to follow this same development though. Usually you'll see their origin explained by the reasoning "now, we still can't solve x+1=0, so we invent negatives, but then we still can't solve 2x=1, so we invent rationals, but...etc,etc." This is a nice, clean, and frankly very misleading explanation unless one makes it clear that this is not how it happened historically.

 

[Mentallic]

But can we write an expression for the square root of say, $1 + x^2$?
Try as we must, it doesn't seem to exist.

Isn't \sqrt{1+x^2} good enough?

 

[ClamShell]

Isn't \sqrt{1+x^2} good enough?

Probably, but only for government work. But try to factor it into
two equal numbers...yes, "$c$" is the answer, but that's no fun.

 

[ClamShell]

They entered the mathematical picture, historically speaking, when they became useful.

Perhaps there are some history buffs out there that can tell us when
\sqrt{-1} became "useful". "Necessity is the mother of
invention", and all that tommyrot.

 

[SteamKing]

Perhaps there are some history buffs out there that can tell us when
\sqrt{-1} became "useful". "Necessity is the mother of
invention", and all that tommyrot.

Don't ask a history buff, ask an electrical engineer, specifically one who works on alternating current.

BTW, it is said that the Pythagoreans were so horrified to find that SQRT(2) was irrational, they swore themselves to secrecy about this unsettling knowledge. If they could even contemplate imaginary numbers, their heads would probably have exploded.

 

[ClamShell]

Don't ask a history buff, ask an electrical engineer, specifically one who works on alternating current.

Yah, who was that EE who first applied "phasors" to analysis of AC motors?
Not Tesla, the other guy; not Edison either; he was a DC nut.

 

[Mentallic]

Probably, but only for government work. But try to factor it into
two equal numbers...yes, "$c$" is the answer, but that's no fun.

Well (1+ix)(1-ix) isn't exactly factoring it into two equal numbers either. The expression isn't a perfect square in terms of elementary functions, sure, but I don't see how complex numbers changes any of that.

 

[ClamShell]

Yah, who was that EE who first applied "phasors" to analysis of AC motors?
Not Tesla, the other guy; not Edison either; he was a DC nut.

Charles Proteus Steinmetz (AKA Karl August Rudolph Steinmetz).
The patron saint of the GE motor business.

 

[ClamShell]

Well (1+ix)(1-ix) isn't exactly factoring it into two equal numbers either. The expression isn't a perfect square in terms of elementary functions, sure, but I don't see how complex numbers changes any of that.

Maybe the next best thing to finding a perfect root, is at least
finding representable factors?

 

[SteveL27]

I just finished reading the "Reality Bits" in a recent copy of NewScientist.
It discusses attempts to purge mathematics of the need for complex numbers.

I haven't seen the original article you referenced. But this strikes me as the idea of someone who was the unfortunate victim of bad math teaching.

The number i is a gadget that represents a counterclockwise quarter turn of the plane. It's just a notation. i is a quarter turn to the left; i^2 is two one-quarter turns, which brings you to a point facing opposite of the way you started. i^3 goes another quarter turn, now you're pointing down. And i^4 brings you back to where you started.

If you can conceptualize the ordered quartet (east, north, west, south), then complex numbers are just a notation that says "here's the angle to turn, and here's how far out you travel."

Complex numbers are just a notation for rotations and stretches of the plane. This is so elementary that the invention of the notation of complex numbers was inevitable.

It's a historical tragedy that complex numbers were discovered algebraically, and that certain complex numbers were labelled "imaginary." It's led to generations of students thinking they are unnatural; when in fact i is as natural as turning left. How could you ever abolish the notion of standing up and turning yourself in a circle, and looking near and then far? Those are the complex numbers.

 

[ClamShell]

I haven't seen the original article you referenced. But this strikes me as the idea of someone who was the unfortunate victim of bad math teaching.

The number i is a gadget that represents a counterclockwise quarter turn of the plane. It's just a notation. i is a quarter turn to the left; i^2 is two one-quarter turns, which brings you to a point facing opposite of the way you started. i^3 goes another quarter turn, now you're pointing down. And i^4 brings you back to where you started.

If you can conceptualize the ordered quartet (east, north, west, south), then complex numbers are just a notation that says "here's the angle to turn, and here's how far out you travel."

Complex numbers are just a notation for rotations and stretches of the plane. This is so elementary that the invention of the notation of complex numbers was inevitable.

It's a historical tragedy that complex numbers were discovered algebraically, and that certain complex numbers were labelled "imaginary." It's led to generations of students thinking they are unnatural; when in fact i is as natural as turning left. How could you ever abolish the notion of standing up and turning yourself in a circle, and looking near and then far? Those are the complex numbers.

Truly fascinating...why didn't my high school algebra teacher put it that way?

 

[homeomorphic]

If you want the ultimate explanation of complex numbers, you should read Visual Complex Analysis by Tristan Needham. Best math book ever. He explains the origins of complex numbers and their evolution over time.

You can get excepts from it free online if you google it and go to the book's webpage.

 

[SteamKing]

I just finished reading the "Reality Bits" in a recent copy of NewScientist.
It discusses attempts to purge mathematics of the need for complex numbers.
Started me thinking(danger, danger) of not how to get rid of the square
root of negative one, but, more easily, simply find out where it enters
the mathematical picture. The Pythagoreans were probably the first who
wondered about this;

$a^2 + b^2 = c^2$

We sure know how to represent the square root of $c^2$, it's $c$.
But can we write an expression for the square root of say, $1 + x^2$?
Try as we must, it doesn't seem to exist. But wait, if we supplement
our real number system with an imaginary number system, it is at least
factorable into,

$(1 + xi)(1 - xi) = 1 + x^2$

That's where it enters the mathematical picture; by the simple act of
trying to factor a sum. Anybody wish to add to this? Or even to point
out other places it enters the mathematical picture, other than the
factoring of sums? Have any ideas of your own, on how to "purge"
complex numbers from your physics homework?

I'm adding the words UBIT and U-BIT because I can't figure out how
edit my tags.

A reference to the particular article you are discussing is always appreciated.
If everyone has a chance to read it, the discussions of it might be fuller.

 

[ClamShell]

A reference to the particular article you are discussing is always appreciated.
If everyone has a chance to read it, the discussions of it might be fuller.

"Reality bits" in January 25-31, 2014 of NewScientist

 

[ClamShell]

Personally, I don't think "i" is purge able. It's just a notation in mathematics
following the rules:

$i^1$ times a quantity puts the quantity onto the positive imaginary axis.

$i^2$ times a quantity puts the quantity onto the negative real axis.

$i^3$ times a quantity puts the quantity onto the neqative imaginary axis.

$i^4$ times a quantity puts the quantity onto the positive real axis.

and repeating,

$i^5$ times a quantity puts the quantity onto the positive imaginary axis.

$i^6$ times a quantity puts the quantity onto the negative real axis.
.
.
.

Can't think of a different way to do it, but my "thinking" or lack thereof,
shouldn't be considered an obstacle.

 

[DrClaude]

"Reality bits" in January 25-31, 2014 of NewScientist

I can't access the article (for some reason, there is a 30 day embargo imposed on our library for the electronic access to New Scientist), but it appears to be about the "u-bit":

Antoniya Aleksandrova, Victoria Borish, William K. Wootters
Real-Vector-Space Quantum Theory with a Universal Quantum Bit
Phys. Rev. A 87, 052106 (2013)
Arxiv preprint: http://arxiv.org/abs/1210.4535

http://pitp.ca/videos/ubit-model-real-vector-space-quantum-theory

 

[D H]

Your URL was incorrect, DrClaude. I fixed it.

 

[ClamShell]

On another point...why does this message system keep adding
tags that are essentially meaningless?

IE, the tags "complex" and "remarks" and "numbers".

 

[jasonRF]

They entered the mathematical picture, historically speaking, when they became useful. Look into the cubic formula for more information. Very briefly, by manipulating square roots of negative numbers as if they "really" existed, mathematicians arrived at results that turned out to be correct. Even then, they still weren't really understood or fully trusted until people came up with a way to picture them geometrically.

That is my understanding as well. A nice fun book on the subject is "an imaginary tale" by Nahin (who was an EE prof, by the way).

As an EE I use complex numbers all the time. If mathematicians completely drop the subject engineers will keep using (and teaching) them for a long time.

jason

 

[1MileCrash]

One might note that (1+xi) and (1−xi) have the same absolute value. One might also note that i and -i are qualitatively equivalent. I think that (1+xi) and (1-xi), while not algebraically or numerically the same, may be regarded as qualitatively the same and so they are, in a way, a "complementary" square root of the relevant expression.

 

[ClamShell]

That is my understanding as well. A nice fun book on the subject is "an imaginary tale" by Nahin (who was an EE prof, by the way).

As an EE I use complex numbers all the time. If mathematicians completely drop the subject engineers will keep using (and teaching) them for a long time.

jason

I agree...but there is this nagging suspicion that if serious attempts are being made
to figure out real number techniques for quantum mechanics, that seem to be
"modified" or "controlled" by a complex "reality bit", then that UBIT could actually be
the reason why capacitors and inductors have their imaginary AC characteristics to
begin with; a much deeper discovery or suggestion.

IE, there may be no need to change the way EEs do the calculations, just a deeper
understanding of why resistors, capacitors, and inductors even exist.

 

[ClamShell]

One might note that (1+xi) and (1−xi) have the same absolute value. One might also note that i and -i are qualitatively equivalent. I think that (1+xi) and (1-xi), while not algebraically or numerically the same, may be regarded as qualitatively the same and so they are, in a way, a "complementary" square root of the relevant expression.

Very good observations. Someone above said that it "was unfortunate we had to discover imaginary numbers via algebra",
and I guess at first looking unequal could have delayed mathematicians interest in them.

I'm thinking that I need to know more rules of squares and square roots in order to
appreciate the "equivalence" of the two factors.

Neither is equal to "c" though...maybe some type of "averaging" could yield a value equal to "c". Eg, if x=1, then the "average" would need to equal 2; or, (1+i)*(1-i) = 2 = sqrt(2) *sqrt(2).

EDIT: Something like having a reason to say: (1+i) is "equivalent" to sqrt(2) is "equivalent" to (1-i);
not exactly equal, but only "equivalent" in some sense.

 

[1MileCrash]

Neither of those are equivalent to root 2 at all, they are qualitatively equivalent to each other.

 

[ClamShell]

Neither of those are equivalent to root 2 at all, they are qualitatively equivalent to each other.

We know that "1" and "i" are perpendicular and form a right triangle whose
hypotenuse is the sqrt(2), nes pa?

 

[HallsofIvy]

They each have modulus \sqrt{2} so are "equivalent" using the equivalence relation "have the same modulus". Since |\sqrt{2}|= \sqrt{2}, those numbers are all "equivalent" to \sqrt{2}. (The "equivalence classes" are circles centered on 0.)

 

[1MileCrash]

OK, under that type of equivalence relation, sure.

 

[ClamShell]

We know that "1" and "i" are perpendicular and form a right triangle whose
hypotenuse is the sqrt(2), nes pa?

All we need is the Pythagorean Theorem...no fancy math jargon :-)

 

[ClamShell]

So are we all in agreement that,

$(1 + i) = √2 = (1 - i)$ ?

 

[Integral]

So are we all in agreement that,

$(1 + i) = √2 = (1 - i)$ ?

No. Real numbers do not have a non zero imaginary component. Or did you forget the magnitude bars?

 

[ClamShell]

They each have modulus \sqrt{2} so are "equivalent" using the equivalence relation "have the same modulus". Since |\sqrt{2}|= \sqrt{2}, those numbers are all "equivalent" to \sqrt{2}. (The "equivalence classes" are circles centered on 0.)

Where do you want me to put the dag-nab bars? @Integral

 

[Integral]

So are we all in agreement that,

$|(1 + i)| = √2 = |(1 - i)|$ ?

You are talking about the magnitude or modulus of the imaginary number so the imaginary numbers need the modulus operator.

Please get a copy of a Complex Analysis text. This a huge rich field there is much to know and learn. It is best done from a good text. I like Complex Variables and Apps by Churchill and Brown

 

[ClamShell]

You are talking about the magnitude or modulus of the imaginary number so the imaginary numbers need the modulus operator.

Please get a copy of a Complex Analysis text. This a huge rich field there is much to know and learn. It is best done from a good text. I like Complex Variables and Apps by Churchill and Brown

I've have "INTRODUCTION TO THE GEOMETRY OF COMPLEX NUMBERS"
by ROLAND DEAUX collecting dust.

I'm beginning to see the error of my ways...Thanks...no sarcasm intended.

 

[LCKurtz]

We know that "1" and "i" are perpendicular and form a right triangle whose
hypotenuse is the sqrt(2), nes pa?

I think you mean "n'est ce pas". It's a French expression which could be translated as "isn't that so".

 

[ClamShell]

I think you mean "n'est ce pas". It's a French expression which could be translated as "isn't that so".

Moocho grassyass, I hope you don't want me to enclose it in modulus bars too. :-)

 

[lendav_rott]

Are you doing this on purpose? :D

 

[ClamShell]

Are you doing this on purpose? :D

No, I spells 'em like I hears 'em.

 

[ClamShell]

While those interested parties wait for the blackout of the "Reality Bit" article
in the Jan, 25-31, 2014 issue of New Scientist...here are some teasers:


http://www.newscientist.com/article...o-u-searching-for-the-quantum-master-bit.html

http://arxiv.org/abs/1210.4535

http://en.wikipedia.org/wiki/U-bit

 

[ClamShell]

BTW, it is said that the Pythagoreans were so horrified to find that SQRT(2) was irrational, they swore themselves to secrecy about this unsettling knowledge. If they could even contemplate imaginary numbers, their heads would probably have exploded.

Rumor has it that Hippasus, a student of Pythagoras, was murdered by the Pythagoreans,
by heaving Hippasus into the Mediterranean with a stone collar, for making the conjecture
that sqrt(2) is irrational. Guess they thought Hippasus couldn't keep his mouth shut.

Closer to the truth is probably that Hippasus got fired and couldn't find a job other than
tending a flock of sheep. And the square root of a flock of sheep is merely a leg of lamb.

 

[ClamShell]

As an EE I use complex numbers all the time. If mathematicians completely drop the subject engineers will keep using (and teaching) them for a long time.

jason

Complex numbers aren't really needed to analyze AC circuits. If you want to
see for yourself, it's only a more "compact" way of viewing the problem.
Actually looks like phase between voltage and current is more easily handled
using complex arithmetic.

See,

http://www.animations.physics.unsw.edu.au//jw/AC.html

if you are interested in the "real" method to analyze AC circuits.

Looks like Dr. Bill Wootten, is trying to do the same with Quantum
Mechanics.

EDIT: And concerning General Relativity...I always silently cringe
when Dr. Hawking refers to "imaginary time" as if "ict", the "imaginary"
space distance, means time can be "imaginary". Time is the real part of
the imaginary part, it's not the imaginary part, nes pa? I mean n'est ce pas?

In short, I am a true believer that "i" is a purely notational construct and
does not have actual physical significance other than to organize equations
into more compact form. And that the absolute operator is an example
of how to recover what the real method already knows. This idea is not
like trying to prove 1 = 0, it's just a total rejection of mystical superstition
that so many fall prey to. Like Pythagoras's fear of irrationals.

 

[lurflurf]

I am a true believer that "17" is a purely notational construct and
does not have actual physical significance other than to organize equations
into more compact form.
What is it with all this talk of physical significance?
Complex numbers are used to make things easier. If someone wants to avoid complex numbers they can. Often it is a bit silly because the "real" method is equivalent as when we introduce a matrix equation like
$$\left[ \begin{array}{ccc}
0 & -1 \\
1 & 0 \end{array} \right]^2=-\left[ \begin{array}{ccc}
1 & 0 \\
0 & 1 \end{array} \right]$$
to avoid using i.

 

[smize]

The complex numbers are a very important aspect of mathematics. They are utilized often in Analysis (obviously), Mathematical Physics, Algebra, and Number Theory (I am not certain about Geometry/Topology).

There was a problem that was solved in the 19th century: Can one construct a square with the same area as a given circle with a straight edge and compass.

The answer is no. And it deals with the fact that ∏ is transcendental. To prove this, they had to use numbers in the complex plane.

Also, if you know of the beloved i, j, k notation of vectors...they were originally defined as:
i², j². k² = -1. And another notation is gone.

Complex numbers are wonderful. There are cases where I'd rather work with complex number that reals, since it makes things simpler (by making it complex). It has its uses.

 

[ClamShell]

I am a true believer that "17" is a purely notational construct and
does not have actual physical significance other than to organize equations
into more compact form.
What is it with all this talk of physical significance?
Complex numbers are used to make things easier. If someone wants to avoid complex numbers they can. Often it is a bit silly because the "real" method is equivalent as when we introduce a matrix equation like
$$\left[ \begin{array}{ccc}
0 & -1 \\
1 & 0 \end{array} \right]^2=-\left[ \begin{array}{ccc}
1 & 0 \\
0 & 1 \end{array} \right]$$
to avoid using i.

YES, I agree 100%. That's what Dr. Wootten is trying to do with
the standard model, he wants to agree with the standard model.
He wants to reformulate the standard model and at the same time
"avoid" imaginary numbers. But I suspect that he is having a bit
of trouble because so many theorems are formulated in complex
form.

It is my conjecture, that real number reformulations have a much
better chance of revealing the physics that is going on and the
complex number formulations have a better chance of "clouding" the
physics that is going on. I also have no doubt that real number
formulations can be much "uglier".

Complex numbers and their properties ARE wonderful, especially
since some "avoidance" techniques could make proofs millions of
lines long and be too much for even modern computers to handle.

 

[jasonRF]

Complex numbers aren't really needed to analyze AC circuits. If you want to
see for yourself, it's only a more "compact" way of viewing the problem.
Actually looks like phase between voltage and current is more easily handled
using complex arithmetic.

See,

http://www.animations.physics.unsw.edu.au//jw/AC.html

if you are interested in the "real" method to analyze AC circuits.

Yes, for very simple circuits you can do this the hard way and it isn't such a big deal. Now do that style of analysis with dozens of circuit elements. Not so much fun.

For my work it is the complex representation of baseband signals in communications systems that is most useful (I never do circuits). Here we end up estimating correlation matrices (Hermitian), inverting them, and doing all sorts of computation where using complex representation saves work. Sure, you could do this the hard way too ...

But in the end, engineers have been using complex numbers for many decades. If we stop using and teaching it, do we throw away 60+ years of literature and just start from scratch? Do we have our research engineers spend a bunch of years simply translating the complex results into the "new" method instead of doing actual new research? Do we graduate power system engineers that do not even understand the documentation that goes along with our power grid, or communications engineers that do not understand how modern systems internally represent signals?

C
In short, I am a true believer that "i" is a purely notational construct and
does not have actual physical significance other than to organize equations
into more compact form. And that the absolute operator is an example
of how to recover what the real method already knows. This idea is not
like trying to prove 1 = 0, it's just a total rejection of mystical superstition
that so many fall prey to. Like Pythagoras's fear of irrationals.

Agreed. But why tie one hand behind my back just because it is possible to live my life with one hand?

Do I think that this Ubit research is worthwhile - yes! But attempting to purge complex numbers (and by extension complex analysis) from our bag of tools seems silly to me. That is all I am saying.

Thanks for the links, though. Interesting stuff!

jason

 

[ClamShell]

For my work it is the complex representation of baseband signals in communications systems that is most useful (I never do circuits). Here we end up estimating correlation matrices (Hermitian), inverting them, and doing all sorts of computation where using complex representation saves work. Sure, you could do this the hard way too ...

Ahhh, Communications Theory, love that stuff; especially that the entropy(information)
packets come in -Probability *Log(Probability) and it becomes Information Theory.

 

[ClamShell]

The complex numbers are a very important aspect of mathematics. They are utilized often in Analysis (obviously), Mathematical Physics, Algebra, and Number Theory (I am not certain about Geometry/Topology).

There was a problem that was solved in the 19th century: Can one construct a square with the same area as a given circle with a straight edge and compass.

The answer is no. And it deals with the fact that ∏ is transcendental. To prove this, they had to use numbers in the complex plane.

Also, if you know of the beloved i, j, k notation of vectors...they were originally defined as:
i², j². k² = -1. And another notation is gone.

Complex numbers are wonderful. There are cases where I'd rather work with complex number that reals, since it makes things simpler (by making it complex). It has its uses.

Agreed

 

[ClamShell]

It is my conjecture, that real number reformulations have a much
better chance of revealing the physics that is going on and the
complex number formulations have a better chance of "clouding" the
physics that is going on. I also have no doubt that real number
formulations can be much "uglier".

Pretty weak conjecture, if I do say myself.

Does no one have an objection to this?

 

[homeomorphic]

In short, I am a true believer that "i" is a purely notational construct and
does not have actual physical significance other than to organize equations
into more compact form.

I am a true believer that "17" is a purely notational construct and
does not have actual physical significance other than to organize equations
into more compact form.
What is it with all this talk of physical significance?

No, no, no, no...

Math is NOT just a bunch of equations. There are IDEAS. And complex numbers are involved in thinking about those ideas. They have geometrical and physical meaning. Does nature use them explicitly? Maybe not. But do we use them to think about what nature is doing? Hell, yes. And not just in equations. In ideas. Physical and geometric.

And yes, 17 has physical significance. If I told you to give me 17 apples, and you gave me 10, that would be wrong. Those are physical actions. It's true that 17 is just a symbol. But it does stand for an abstraction of stuff that is actually out there. We take all collections of 17 objects that we have ever seen and sort of put an equivalence relations on them. And at some point, it gets wishy-washy as to what constitutes an object or something, but we know what I am saying if you tell me to give me 17 apples, so yes, in my book, that's physical significance.

Another point of view is that 17 is the set containing the empty set, set containing the empty set, set containing that...

So, we can define 17 without reference to physical things, but when we talk about 17, I don't think we really have that in mind. We have in mind different things, depending on the context. Sometimes discrete sets of objects like the apples, other times maybe 17 continuous units of something, like time. Sometimes, maybe the set theory definition. It really means something different depending on context. Different, but closely related meanings.

 

[ClamShell]

No, no, no, no...

Math is NOT just a bunch of equations. There are IDEAS. And complex numbers are involved in thinking about those ideas. They have geometrical and physical meaning. Does nature use them explicitly? Maybe not. But do we use them to think about what nature is doing? Hell, yes. And not just in equations. In ideas. Physical and geometric.

And yes, 17 has physical significance. If I told you to give me 17 apples, and you gave me 10, that would be wrong. Those are physical actions. It's true that 17 is just a symbol. But it does stand for an abstraction of stuff that is actually out there. We take all collections of 17 objects that we have ever seen and sort of put an equivalence relations on them. And at some point, it gets wishy-washy as to what constitutes an object or something, but we know what I am saying if you tell me to give me 17 apples, so yes, in my book, that's physical significance.

Another point of view is that 17 is the set containing the empty set, set containing the empty set, set containing that...

So, we can define 17 without reference to physical things, but when we talk about 17, I don't think we really have that in mind. We have in mind different things, depending on the context. Sometimes discrete sets of objects like the apples, other times maybe 17 continuous units of something, like time. Sometimes, maybe the set theory definition. It really means something different depending on context. Different, but closely related meanings.

Thankyou Homeomorphic, I was trying to think of a way to defend the significance of "17",
you put it into a nutshell...thanks again.

 

[ClamShell]

@Homeomorph, now you need to explain the significance of 42.