rational numbers

[hanhan]

I don't understand this at all. Could anyone shed some light on it?
The rational numbers were invented for measuring lengths. Since we can
transduce things like voltages and times to lengths, we can measure other
things using the rational numbers, too.

 

[Pommie]

I'm guessing that they mean that there first real use was in measuring length. However, I'm sure one of the first uses was in weighing things - I'll have 2½ pounds of gold.

As for transducing voltage and time to length, not in my experience. Voltage and time have always been expressed as themselves and as decimal numbers. How long is a volt?

Mike.

 

[Ratchit]

anhnha,

I don't understand this at all. Could anyone shed some light on it?
The rational numbers were invented for measuring lengths. Since we can
transduce things like voltages and times to lengths, we can measure other
things using the rational numbers, too.

Rational numbers were not invented to measure lengths. They are used to describe lengths, just as any other number is. If you want to measure something, then you need to use some physical standard to compare the quantity to be measured (think of a ruler). Numbers were not "invented", they were defined, collated, and codified according to their natural sequence. Do you know what a rational number is?

Ratch

 

[hanhan]

Thanks Pommie and Ratch,

Do you know what a rational number is?

Yes, a rational number is any number that can be expressed as the quotient or fraction p/q of two integers, with the denominator q not equal to zero.

Rational numbers were not invented to measure lengths.

The above is taken from "An Introduction to Complex Analysisfor Engineers" by Michael D. Alder

If Complex Numbers had been invented thirty years ago instead of over
three hundred,they wouldn't have been called`Complex Numbers' at all.
They'd have been called`Planar Numbers', or`Two-dimensional Numbers'
or something similar, and there would have been none of this nonsense about
`imaginary' numbers.The square root of negative one is no more and no less
imaginary than the square root of twr two itself, for that matter.All of
them are just bits of language used for various purposes.
`Two' was invented for counting sheep. All the positive integers (whole numbers)were invented so we could count things, and that's all they were invented for. The negative integers were introduced so it would be easy to
count money when you owed more than you had.
The rational numbers were invented for measuring lengths. Since we can
transduce things like voltages and times to lengths, we can measure other
things using the rational numbers,too.
The Real numbers were invented for wholly mathematical reasons: it was
found that there were lengths such as the diagonal of the unit square which,
in principle,couldn't be measured bythe rational numbers. This is of not
the slightest practical importance, because in real life you can measure only
to some limited precision,but some people like their ideas to be clean and
cool, so they went off and invented the real numbers,which included the rationals but also filled n the holes.

 

[Ratchit]

anhnha,

Yes, a rational number is any number that can be expressed as the quotient or fraction p/q of two integers, with the denominator q not equal to zero.

Good.

The above is taken from "An Introduction to Complex Analysisfor Engineers" by Michael D. Alder

You will have to decide whether numbers were "invented" or defined. You will also have to decide whether you measure a board by a number or a ruler. In a previous post addressed to you, I gave you a a link to a good definition of a complex or duplex number. What part of that definition did you not understand?

Ratch

 

[hanhan]

Ratch,

In a previous post addressed to you, I gave you a a link to a good definition of a complex or duplex number. What part of that definition did you not understand?

At first, I liked your understanding of considering j as an operator but then I got stuck in explaining Euler's formula using j as an operator.
e^jθ = cosθ + jsinθ
In this example, θ is a real number. We can express it as (θ, 0) then jθ = j (θ, 0) = j (θ + j0) = jθ is a point in imaginary axis.
To me e^jθ then also has to be an pure imaginary number. It is something like this: 3 is a real number then e^3 also a real number. Similarly, jθ is an imaginary number then e^jθ also an imaginary number.
However, that is wrong according to Euler's formula.

Another question:
With real number, for example 3 can mean that 3 sheep, 3 tables, 3 pens,...
Then what does 3i means here?

PG1995 ,

A complex quantity in the physical world means that a quantity has 2 or 3 parts. One part is designated as the reference, the second part is 90 degrees out of phase with the reference, and the third part if it exists is 90 degrees out of phase with the first and second parts. Complex power, voltage and current are good examples of 2 part (duplex) quantities.

Ratch

In mathematics with Cartesian coordinate system I can know if two parts are orthogonal. But in real life, for example, with AC voltage we have two parts : amplitude and phase. How can you know that they are orthogonal and then you can model them as a complex number?

First of all, many math books say or imply that a root of a negative number has a complex esoteric value not conceivable in the physical world. Whereas in fact, complex numbers have a finite value. They should be called "duplex" numbers instead of "imaginary" numbers. Duplex numbers have a real part and an orthogonal part. The symbol "i" or "j" does not mean √-1. "i" or "j" is a mathematical operator, not a finite value. For instance, 7i does not mean i + i + i + i + i + i + i. It means instead, "perform the mathematical operation of rotating the number 7 by 90 counterclockwise (CCW). It is true we get correct results by treating i or j as an arithmetic constant, but that only works because of its conformal similarity.

I can see that 7i means rotating the number 7 by 90 counterclockwise (CCW) makes sense. However to me 7i = i + i + i + i + i + i + i also works.
This link gives a good link about complex number: https://mathforum.org/library/drmath/view/53809.html
According to the link i = (0, 1), then i + i + i + i + i + i + i = (0, 1) + (0, 1) +(0, 1) +(0, 1) +(0, 1) +(0, 1) +(0, 1) = ( 0, 7) = 7(0, 1) = 7i

 

[Ratchit]

anhnha,

At first, I liked your understanding of considering j as an operator but then I got stuck in explaining Euler's formula using j as an operator.
e^jθ = cosθ + jsinθ

You are looking for a proof of Euler's formula, which would explain how the "j" operator affects the result. Look at this link. https://en.wikipedia.org/wiki/Euler's_formula . Toward the end of the article are listed proofs of Euler's formula. Study the easy to understand power series proof, and see how the j rotation operator is incorporated. Ask if you have any questions about the proof.

In this example, θ is a real number. We can express it as (θ, 0) then jθ = j (θ, 0) = j (θ + j0) = jθ is a point in imaginary axis.
To me e^jθ then also has to be an pure imaginary number. It is something like this: 3 is a real number then e^3 also a real number. Similarly, jθ is an imaginary number then e^jθ also an imaginary number.
However, that is wrong according to Euler's formula.

You reasoning is faulty, and your conclusion is not based on mathematical identities like the power series proof of Euler's theorem is. A real number raised to a complex power is not necessarily a purely orthogonal number.

With real number, for example 3 can mean that 3 sheep, 3 tables, 3 pens,...
Then what does 3i means here?

As explained in the link I submitted in an earlier post, it means rotate 3 by 90° CCW. "i" is really an operator, not a algebraic number. To avoid confusion, it should have been designated something like qr(3), meaning quadrature rotation of 3, instead of 3j, which implies an algebraic term.

In mathematics with Cartesian coordinate system I can know if two parts are orthogonal. But in real life, for example, with AC voltage we have two parts : amplitude and phase. How can you know that they are orthogonal and then you can model them as a complex number?

Because amplitude and phase are simply polar coordinates. Polar coordinates can always be alternatively represented as Cartesian coordinates.

I can see that 7i means rotating the number 7 by 90 counterclockwise (CCW) makes sense. However to me 7i = i + i + i + i + i + i + i also works.
This link gives a good link about complex number: https://mathforum.org/library/drmath/view/53809.html
According to the link i = (0, 1), then i + i + i + i + i + i + i = (0, 1) + (0, 1) +(0, 1) +(0, 1) +(0, 1) +(0, 1) +(0, 1) = ( 0, 7) = 7(0, 1) = 7i

As explained in the link I submitted in an earlier post, treating "i" as a algebraic quantity works because of its conformal similarity . You would not be able to do that if "i" meant rotate a number by 45°.

Ratch

 

[misterT]

To me e^jθ then also has to be an pure imaginary number. It is something like this: 3 is a real number then e^3 also a real number. Similarly, jθ is an imaginary number then e^jθ also an imaginary number.
However, that is wrong according to Euler's formula.

in equation: j*j = -1, you multiply two purely imaginary numbers, but end up with a real number. What is happening in e^jθ is not easy to see, but that is not necessarily a pure imaginary number. Depends what the value of θ is.. if θ=0 then you clearly have e^0 = 1.
When θ = PI, then you have: cos(PI) + jsin(PI) = -1
When θ = PI/2, then you have: cos(PI/2) + jsin(PI/2) = j

Note that the length e^jθ is always 1, only the phase angle of the complex number changes. That is very convenient in many applications.

 

[steveB]

Another question:
With real number, for example 3 can mean that 3 sheep, ...
Then what does 3i means here?

Those are the imaginary sheep we count when we are trying to fall asleep.

 

[hanhan]

Thanks all,
Ratch,
I read the article and these proofs of Euler's formula by using power series and using calculus. Now I want to find a physical representation for complex number.
For example, with negative number where there are two opposite directions, right and left on a line, north and south latitude, east and west longitude, future and past, assets and liabilities, etc., there may be application of the negative number;
-30 can mean that I owe you 30 dollars.
Now is there a similar physical representation for complex number?

7i means that the real number 7 rotated 90 degree CCW but what is the physical representation of this?
I see that we usually represent amplitude and phase of AC voltage by a complex number. A complex number contains two parts that are orthogonal with each other. But in reality, how can you know that amplitude and phase are orthogonal, thus they can be represented by a complex number?

In what cases a physical phenomenon can be represented by a complex number?

 

[Ratchit]

anhnha,

Now I want to find a physical representation for complex number.
For example, with negative number where there are two opposite directions, right and left on a line, north and south latitude, east and west longitude, future and past, assets and liabilities, etc., there may be application of the negative number;
-30 can mean that I owe you 30 dollars.
Now is there a similar physical representation for complex number?

Where have you been? Haven't you noticed that the inductive and capacitive reactances are 90° or at quadrature with the the resistance? So isn't that something that can be represented by a complex number?

7i means that the real number 7 rotated 90 degree CCW but what is the physical representation of this?

It means that the orthogonal 7 number is 90° out of phase with a real number.

I see that we usually represent amplitude and phase of AC voltage by a complex number. A complex number contains two parts that are orthogonal with each other. But in reality, how can you know that amplitude and phase are orthogonal, thus they can be represented by a complex number?

Wasn't this question answered in post #7 of this thread?

In what cases a physical phenomenon can be represented by a complex number?

Whenever a physical quantity is 90° out of phase with another quantity.

Ratch

 

[HowardP]

hi anhnha,

one of the problems with " number systems " is that they are just that, systems - you can apply them however they fit best. As an earlier poster replied, it is a bit unfortunate that names such as complex and imaginary were given to the defining elements of these systems. Back then, they couldn't believe that a negative number could even have a square root - imagine that ! And, wowie, did that make things complex . One thing that helped me get past these mental hurdles was to see that things like the four dimensions of space-time were nothing more than a 4-dimensional array. Or think in terms of matrix math, as a graphics programmer would when using 4-dimensional vectors, homogeneous coordinates, and quaternions, etc. It's all in the underlying definition. Whether or not such systems reflect, mimic, model, or ultimately, approximate something physical really depends on how well-coupled the modeling system is to the particular phenomenon in question. In other words, for instance, if you use complex numbers, do so only because they are the easiest to work with.

I see that we usually represent amplitude and phase of AC voltage by a complex number. A complex number contains two parts that are orthogonal with each other. But in reality, how can you know that amplitude and phase are orthogonal, thus they can be represented by a complex number?

Here the most thorough question really should be: "Does representing AC amplitude and phase relationships (the 'real system') using a complex-valued model (the 'approximate system') adequately represent all the factors which affect the real system, which matter to understanding the system in a practical way ?" In AC, in a theoretical setting, the specific answer is perhaps yes. In a "real-world" setting, perhaps no.

Why in the world do we need four dimensions in 3D graphic representations ( which are presented on a flat (2D) screen) ? Why not only three ? ... If you look up the term ' gimbal lock ' you'll see that 3 variables does not adequately yield all the phenomena that can happen in a 3D space.

Please don't get caught up in the names of things - look at what they do. Math is ultimately like a programming language - only it is so sophisticated that it defines itself in what ever ways the mathematician can functionally achieve. (And aside to the earlier poster who hinted at the " is math discovered or invented " question, I'd like to propose that answering that question doesn't matter for practical work such as electronics. It does matter if epistemology (" how do / can we know what we know ?" ) is a burning question in your life ... and even then, we might realize that ultimately we are still "representing" things with math - representing a thought or an object - and a Platonist would argue that thought IS an object ! --- just not a 'physical' one as we would ordinarily think of it )

[...] is there a similar physical representation for complex number?

Yes ... and this is not meant condescendingly ... draw an Argand diagram on a piece of paper. There, you'll have a physical representation for complex numbers ! ;-P On the other hand, if you want complex numbers to represent a physical process, then you're out of luck.

(Huh ?)

... because you'll only be partially representing, approximating the physical process. Physics is the science of measuring things. But when you take a 'thing' and want to measure it, both the thing itself and the measuring device (or system) contain too much information. In other words, it is impossible to represent nature completely. (And before someone disagrees, please note the emphasis on "completely"... to date, we can represent only pieces, not the dynamic totality.)

Physics measurements eliminate things that are not important to the observation. Does the temperature of a specific iron-ore rock on Mars affect my oscilloscope's two channel measurement of AC phase shift ? Yes, a thorough understanding of Einstein's and Maxwell's field equations show that it does ! But who gives a rat's behind for such a profoundly minuscule quantity such as the temperature-affected, E.M. eddy current of that rock in proportion to my well-shielded o-scope probes ??? Ditto AC wave form representation --- use only what you need.

- H

 

[WTP Pepper]

Hmmm Some very complex explanations. Complex numbers allow you not only to point a fishing rod of sufficient length, but to take into account the angle you point it at a target.

 

[HowardP]

nice, terse explainer, WTP... but what about the temperature needed to fry that fish ?

 

[WTP Pepper]

Ha Ha. That's when 3rd order polymorphic differential equations come into it. It not only includes the fish temperature, but the cooking appliance, fish type, lake Ph value and number of breadcrumbs/squared cm.

Back on topic, my old Electrical Enginnering lecturer once said after we had a near fatal shock in one of our labs. There is nothing imaginary about j1000Volts across the fingers...Classic!

 

[hanhan]

Thanks all and epecially HowardP!
I finally have found my answer about complex number in this book: **broken link removed**
I can only read part of it but it really helped.