Square root is positive or negative?

[Ratchit]

Vlad777,

Maybe it was like this:

If you consider your number real then root is positive. (operating in real set)
If you consider your number complex (although it has no imaginary part)
then result is both negative and positive.
(operating in complex set)

But I still don't understand why?

I sense that you are bewildered, and you can't get a handle on your above wonderings. So let's get some things straight. We will only consider integer roots, i.e. 0,±1,±2 ... . Next, believe it that the number of the integer root determines how many roots a number has. If you are trying to find the fifth root of a number, whether positive, negative or complex, that number will have 5 roots. Take that to the bank.

Now, if you are going to do a lot of rooting around, you will want to become pals with a French mathematical genius whose name is Abraham de Moivre. Specifically, his theorem, which is called de Moivre's theorem. https://en.wikipedia.org/wiki/De_Moivre's_formula Attached are 4 examples that show how it it applied.

Ratch

Ratch

 

[vlad777]

Ratchit,

What you posted is valuable to me because it touches on what I know and builds on it only a little.
I can't analyze it wright now, but I'm saving everything.

 

[Ratchit]

Vlad777,

You are welcome. Can you find the mistake in equation #2 of example 2? De Moirve's theorem will also make a believer out of you, in that i or j is a rotational operator, not a constant. I was trying to trumpet that fact in the previous post I linked to.

Ratch

 

[vlad777]

Something like that can derive from material in post: #7 link: rootofi.

Is the mistake pi/2 when it should be pi?
Because cos(pi) is -1 which then gives -4.

 

[Ratchit]

vlad777,

Is the mistake pi/2 when it should be pi?

Yes, you found it.

Ratch

 

[vlad777]

So to take a root of a number you represent it in complex trigonometric form.
This includes modulus Z that multiplies a phase represented as cos(ang)+i*sin(ang).
If the number is negative, minus sign is represented in the phase part so that
modulus is always positive.

So to take a root you divide the angle ang and root the modulus. (This is complex rooting. Multiple roots.)

But mind that you still had to root the positive modulus, and this is real rooting which result is always positive and single.

 

[Ratchit]

vlad777,

But mind that you still had to root the positive modulus, and this is real rooting which result is always positive and single.

The modulus is defined as the absolute value of the root, so it is always positive. Every root of a particular number has the same absolute value, so you only have to find root of one positive real number to get the modulus.

Ratch

 

[april]

There are always 2 square roots. But if we are working with lengths, ages, area etc the negative square root is discarded because it does not make sense.

A lot of people go nuts with numbers and do not relate it back to the problem they are solving, what they are modeling. They come up with nonsense answers that had the preserved the units and did some thinking they would see are clearly wrong.

Now isn't that the truth
Of course when we number the pins of the chips we use we call the first pin "0" because it is never really there and this must make sense or we would not label it so.
Additionally when electrons flow from the negative terminal of a battery we say the current runs from positive to negative because it is never really there and we invent a hole for where it should be to explain the sense of why we do this .
It all makes perfect sense to me.

Thank God retirement is here at last

 

[Ratchit]

april,

Of course when we number the pins of the chips we use we call the first pin "0" because it is never really there and this must make sense or we would not label it so.

Must not confuse a number used as a label with the count of something. Those are two different mutually exclusive uses of numbers.

Additionally when electrons flow from the negative terminal of a battery we say the current runs from positive to negative because it is never really there and we invent a hole for where it should be to explain the sense of why we do this .

Holes only exist in bipolar semiconductors. A hole cannot exist in a conducting wire because it would be instantly annihilated by the sea of electrons present within. Assuming the charge flows from the positive voltage terminal to the negative is just a mathematical artifice. It is physically the opposite in a metal wire, but is is correct in P-type material or some electrochemical reactions involving positive ions. If necessary to know the correct physical direction of the charge, just take note of the polarity the charge carrier, and reverse the calculated value if the charge carrier is negative.

Ratch

 

[april]

No my friend . When you study Electronics at Uni you will be told that electrons flow in one direction and holes in the other . It was a totally farsical idea in the first place but that's what they were teaching in 1991. I hope its different now.

With the pins Why not label them 1 to 8 then? No they insist on 0 to 7 so it becomes the 0th pin or the zero pin instead of readily accepted English of the first pin or the eighth pin

 

[DerStrom8]

No my friend . When you study Electronics at Uni you will be told that electrons flow in one direction and holes in the other .

I've studied Electronics at Uni, and they NEVER said that there are holes anywhere besides semiconductor devices. You have to learn the chemistry behind semiconductors in order to understand the theory of "holes". They do not exist in wires or any other solid material.. How about you read up on how electronics work, what current is, how it flows, and while you're at it you should look up semiconductors and learn what holes really are. I don't mean to be rude, but you're sounding absolutely clueless right now. I'm afraid you don't know what you're talking about.

Anyway, as mentioned before, square roots always have two solutions, except where the inside value is equal to zero. The sqrt(9) equals 3 and -3, since either of those values, when squared, will equal 9. Complex numbers come into play when you try taking the square root of a negative number.

 

[april]

I've studied Electronics at Uni, and they NEVER said that there are holes anywhere besides semiconductor devices. You have to learn the chemistry behind semiconductors in order to understand the theory of "holes". They do not exist in wires or any other solid material.. How about you read up on how electronics work, what current is, how it flows, and while you're at it you should look up semiconductors and learn what holes really are. I don't mean to be rude, but you're sounding absolutely clueless right now. I'm afraid you don't know what you're talking about.
r.

Hmmf well good for you . I too went to uni for EE -it was fed to me there repeatedly
To give a current example (pun intended) Forrest M Mims 111 was very active teaching my generation and is still reproduced by such as Master Publishing and sold by Sparkfun today. He is deservedly highly respected !

below are three bits of his work ammended since his original publications and the hole theory is getting less used and comes with more explanations these days but is still pretty much balderdash to cover a reluctance to show current (electron) flow correctly

 

[DerStrom8]

Hmmf well good for you . I too went to uni for EE -it was fed to me there repeatedly
To give a current example (pun intended) Forrest M Mims 111 was very active teaching my generation and is still reproduced by such as Master Publishing and sold by Sparkfun today. He is deservedly highly respected !

below are three bits of his work ammended since his original publications and the hole theory is getting less used and comes with more explanations these days but is still pretty much balderdash to cover a reluctance to show current (electron) flow correctly

Why are you showing me the theory behind diodes? I know there are electron "holes" in the P-type semiconductor in diodes. In fact, I'm sure I said it:

I've studied Electronics at Uni, and they NEVER said that there are holes anywhere besides semiconductor devices. You have to learn the chemistry behind semiconductors in order to understand the theory of "holes". They do not exist in wires or any other solid material.

I've got a newsflash for you: Diodes are semiconductors.

 

[Ratchit]

april,

... hole theory is getting less used and comes with more explanations these days but is still pretty much balderdash to cover a reluctance to show current (electron) flow correctly

Where appropriate, holes cannot be ignored anymore than electrons can. In fact, at the quantum level for semiconductors, holes have the same status, and are thought of as particles. Before I correct the misconceptions propagated by Forrest Mims, let me say a few things about charge flow. It is wrong to define mathematically, current direction by the direction of charge movement. That is because there are two kinds of charges that move in response to a voltage field. It is true that in metals the charge carriers are electrons. But in other materials like p-type semiconductors or +ions in electrochemical reactions, positive charge carriers predominate. So the engineers, not Benjamin Franklin have developed a convention which defines a positive current as emitting from the positive terminal of a voltage source and returning to the negative terminal regardless of the polarity of the charge carrier. This is a mathematical artifice that was invented so as to calculate curcuit parameters without regard to charge carrier polarity. If really necessary to know the current direction after the calculations, the direction is reversed if the charge carriers are negative. As you must know, all semiconductor manufacturers and ammeters are marked to show current as if it consisted of positive charge carriers.

Now let's take on Mims' statements. In the first page you submitted, he says that electrons travel at the speed of light. Not true. The electric field that drives them does, but the electrons travel a a much slower rate called the drift velocity. This is true for electrons in wires or a vacuum tube. The analogy is that of a hose filled with marbles. Push a marble in at one end, and a different marble pops out at the opposite end. After many successive insertions, it takes a long time for the first inserted marble to exit out other end due to its slow drift velocity.

Next he says that electrons are attracted to "holes" caused by positive ions. Well, those positive ions do not possess holes. Those ions are deficient in electrons because a voltage swept them away, as in a capacitor. The electrons are attracted to the positive voltage that caused the ions to form. The difference between the ions in a capacitor and p-type material is that p-type material has a hole built into its ionic struction by dopants, and is electrically neutral. If it were not neutral, there would be a current existing in a slab of p-type material, which there is not. But there is current existing between the plates of an energized capacitor if a conduction path is provided.

On the second page Mim supposedly shows how a diode works, but he doesn't. He only shows how to hook a diode up for forward or reverse bias, not the reason why it works that way.

On the third page is explains current direction by saying Franklin made a mistake. If he labeled a electron as positive, then the holes and ions would be negative. So Franklin was right or wrong no matter which polarity he chose for the electron.

His last paragraph says that the semiconductor labeling was done to accomodate the hole direction. Not true. It was done to be consistent with the mathematical convention of current exiting out the positive terminal of a battery. An ammeter is marked the same way.

Ratch

 

[april]

Now let's take on Mims' statements. In the first page you submitted, he says that electrons travel at the speed of light. Not true. The electric field that drives them does, but the electrons travel a a much slower rate called the drift velocity. This is true for electrons in wires or a vacuum tube. The analogy is that of a hose filled with marbles. Push a marble in at one end, and a different marble pops out at the opposite end. After many successive insertions, it takes a long time for the first inserted marble to exit out other end due to its slow drift velocity.
Ratch

Hmm Thank You .I'm interested in this . How do you see it as a small change in time scenario ? I often thought along the lines of hanging steel balls. lift and drop one and the free to move last ball gets the momentum. How long do you reckon it takes for the force to traverse a long chain of these as steel balls . Speed of light or otherwise (steel balls only)?

In your assertion for electron insertion and the push imparted are you saying the next field is compressed on one side and this forces an expansion of that field on the other side which compresses the next and so on to propogates down the metal strand? How exactly do you think this works?

 

[DerStrom8]

Hmm Thank You .I'm interested in this . How do you see it as a small change in time scenario ? I often thought along the lines of hanging steel balls. lift and drop one and the free to move last ball gets the momentum. How long do you reckon it takes for the force to traverse a long chain of these as steel balls . Speed of light or otherwise (steel balls only)?

It travels considerably slower than the speed of light. Watch this: https://www.youtube.com/watch?feature=player_detailpage&v=Do1lm9IevYE#t=304s

The example is different but the theory is the same.

 

[Ratchit]

april,

Hmm Thank You .I'm interested in this . How do you see it as a small change in time scenario ? I often thought along the lines of hanging steel balls. lift and drop one and the free to move last ball gets the momentum. How long do you reckon it takes for the force to traverse a long chain of these as steel balls . Speed of light or otherwise (steel balls only)?

In your assertion for electron insertion and the push imparted are you saying the next field is compressed on one side and this forces an expansion of that field on the other side which compresses the next and so on to propogates down the metal strand? How exactly do you think this works?

My description of a marble filled hose is an analogy which illustrates a principle. Specifically, it shows that something that moves very slowly can collectively cause an effect a long distance away. My analogy is not a model for charge carrier flow in a conductor. I don't know what your description of momentum conservation is supposed to illustrate. Anyway, the speed of the individual electrons (drift velocity) in a typical wire is about the same as cold molasses, but the electrons exit the wire at almost the same instant they entered the wire. The electric field that drives the electrons moves at the speed of light, but the electrons do not. The electrons that first exit the wire are not the first electrons that enter the wire. Forrest Mims does not seem to know that.

Ratch

 

[MrAl]

Hi,


The electrons in a wire travel at a speed of roughly 1x10^6 meters per second. However, few discussions of speed are complete without also considering the *direction*, which brings us to the vector of velocity. The magnitude of the velocity of an electron is 1x10^6, but the direction is random. That is, none of the electrons tend to speed down the length of the wire any more than speed across the wire or even backwards down the other way. They stay in the wire as they are attracted to the bound ions in the wire. Their motion is random at least until a potential difference is applied to the ends of the wire.
Once that occurs, the field is set up at close to 300x10^6 meters per second (the speed of light), and that exerts a force on each electron in the wire which influences their direction which tends to cause them to move along the wire with a rather slow velocity jumping around still with some randomness but with a direction that now slightly favors the direction along the length of the wire, which can be 1x10^-4 meters per second for example, which is far slower than their actual speed which was already 300 times slower than the speed of light.
But if we observe an electron entering one end we also might observe an electron exiting the other end, which seems to imply that the electron got out really fast, but it's just that one electron entered one end and another electron exited the other end. And it has nothing to do with one electron pushing on another (although that is an often used analogy for a rough idea what is going on) it is the field that does all the pushing with a force of F=q*E. The field exists as photons. So the steel balls being pushed through the pipe move for an entirely different reason, their motion being controlled by phonons not photons. This requires one ball to push on the next until the one pops out the end and can take much longer depending on the properties of the balls material.

Even though when one electron entered one end of the wire and another different one exited the other end, it doesnt matter to any application (so far devised) because each electron has the property (at least until recently and still holds for the most part) of non identity. So our electronics work just fine and dont care which electrons went through them.

For a wire that wrapped around the Earth one time, it would take about 0.0675 seconds to see an electron at the center of the wire start to move after a potential difference was applied to the ends of the wire.

 

[april]

april,My description of a marble filled hose is an analogy which illustrates a principle. Specifically, it shows that something that moves very slowly can collectively cause an effect a long distance away. My analogy is not a model for charge carrier flow in a conductor. I don't know what your description of momentum conservation is supposed to illustrate. Anyway, the speed of the individual electrons (drift velocity) in a typical wire is about the same as cold molasses, but the electrons exit the wire at almost the same instant they entered the wire. The electric field that drives the electrons moves at the speed of light, but the electrons do not. The electrons that first exit the wire are not the first electrons that enter the wire. Forrest Mims does not seem to know that.Ratch

Thanks - I was really just trying to see how you see it in your mind's eye. Rather than analogies how do you think it might be happening . What do you personally think might be going on rather than repeating what you might have learnt verbatim/parrot style..?Most people have suspicions of what they think is really going on and I am enquiring as to yours.

 

[Ratchit]

april,

What do you personally think might be going on rather than repeating what you might have learnt verbatim/parrot style..?Most people have suspicions of what they think is really going on and I am enquiring as to yours.

Going on with respect to what? I believe I and others have sufficiently explained what current is and how it exists, along with its particle speed.

Ratch