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. :-)