• Quick note - the problem with Youtube videos not embedding on the forum appears to have been fixed, thanks to ZiprHead. If you do still see problems let me know.

For the Math people ...

Just thinking

Philosopher
Just thinking
Joined
Jul 18, 2004
Messages
5,169
Well, everyone else is also certaily welcomed to comment too:

We all know that the accepted answer to the square root of negative one is i; and that this begins a numberline perpendicular to the Real Number numberline (Imaginary Numbers). But just exactly what is defined as being the square root of i? (Or -i?) Is this too an imaginary number, or does it constitute constructing yet a third numberline perpendicular to the other two? And thus require a new field (and name) of numbers? Has a problem in the real world ever required the value of the square root of i? And then what happens if we look for the square root of that new number? (I'll stop now as my head is about to explode.)
 
Just thinking said:

But just exactly what is defined as being the square root of i? (Or -i?) Is this too an imaginary number, or does it constitute constructing yet a third numberline perpendicular to the other two?
Click to expand...

Great question!

Note that we can write 2i as: (1+i)(1+i)

So sqrt(2i) = 1+i

So sqrt(i) = 1/sqrt(2) + i/sqrt(2)
 
Can Heaviside be far behind? What about approaching the left or right side of the square root of "i"?
 
*whistles non-chalantly, backs away slowly from thread for a couple seconds, runs away as fast as possible*
 
Here's an alternate, geometric picture to get the same answer. You can write any complex number (a + bi) (ab and b real) as c(cos(d) + i sin(d)), where c is the absolute value, and d is the angle in the complex plane. This is also equivalent to c*exp(id). Now let's multiply two complex numbers:

[c*exp(id)]*[f*exp(ig)] = (c*f)*exp[i(d+g)]

Multiplication in the complex plane essentially is multiplying the absolute values of the two numbers and adding their angle in the complex plane. So if you want to find the square root of a number, you take the square root of the absolute value, and you take half the angle in the plane. Of course, the angle isn't unique: for -1, you could use either 180 degrees or -180 degrees (or even 540, etc). Half of that gives you either + or - 90 degrees, corresponding to i or -i. You always end up with two possible roots (180 degrees apart) this way. Taking the square root of i, we get something with an absolute value of 1, and an angle of 45 degrees (or -135 degrees) in the complex plane. A little simple geometry will then give you the same answers as other posters have given, but you can visualize the answer without even doing any calculations.
 
Ziggurat said:
Multiplication in the complex plane essentially is multiplying the absolute values of the two numbers and adding their angle in the complex plane. So if you want to find the square root of a number, you take the square root of the absolute value, and you take half the angle in the plane. Of course, the angle isn't unique: for -1, you could use either 180 degrees or -180 degrees (or even 540, etc). Half of that gives you either + or - 90 degrees, corresponding to i or -i. You always end up with two possible roots (180 degrees apart) this way. Taking the square root of i, we get something with an absolute value of 1, and an angle of 45 degrees (or -135 degrees) in the complex plane. A little simple geometry will then give you the same answers as other posters have given, but you can visualize the answer without even doing any calculations.
Click to expand...

Two Short Planks

Find some way to square the circle.
Feet slipping, sliding on the level.
Connect to reason, is there anybody there?
Drum it in to me now if you dare.

Triangles by Isosceles.
Principles by Archimedes.
Amo, amas; even amat
make for one less way to skin the cat.

Two short planks –
Try my luck on another day
Must be thick as
two short planks –
That’s about all that I have to say.

Two short planks –
Pop the question: I sit the test
Must be thick as
two short planks –
Spin me round till I come to rest.

They say truth comes flooding if you let it.
But what happens if I just don’t get it?
I’m blissful in my sweet ignorance
and delight in my incompetence.

Two short planks –
Try my luck on another day
Must be thick as two short planks –
That’s about all that I have to say.

Two short planks –
Pop the question: I sit the test
Must be thick as
two short planks –
Spin me round till I come to rest.

[From "Rupi's Dance" by Ian Anderson (Jethro Tull)]


BillyJoe.
 
On the i vs. -i debate:
The complex numbers are symmetric in this respect. If you took any true or false equation, and replaced every number with its conjugate, it would remain true or false. Thus, there's no way to tell whether i or -i is the "true" i. Your professors might be using the "wrong" one.

First time I ever used complex numbers for something useful was in electronics. (Note that in electronics they use 'j' instead of 'i')

Static circuits can be analyzed using

Resistors: Voltage = Current * Resistance
Capacitors: Current = 0
Inductors: Voltage = 0

For non-static circuits, these become very awkward:
Resistors: Voltage = Current * Resistance
Capacitors: Current = Capacitance * dVoltage/dt
Inductors: Voltage = 1/Inductance * dCurrent/dt

By allowing complex numbers, all of these can be unified into one equation:

Voltage = Impedance * Current

A "complex voltage" is a voltage that alternates like a sine wave, and a "complex current" is similar.
The impedance of a resistor is simply its resistance. The impedance of a capacitor is j * capaictance, and the impedance of an inductor is -j / inductance.

I apologize for any minor errors here; I was writing quickly, and from memory.
 
I got into a fight once over this topic. I thought it was stupid that we just invent something when we can't make sense of an observation, or a formula.

What's one plus one?

Hmm, I'm not sure. Let's just call it 'b' and be happy with that.

Ok.

I was being a smart-arse (I do understand the value of doing it...but was in a fightin' mood). But it still seems strange to me.

Athon
 
phildonnia said:
On the i vs. -i debate:
The complex numbers are symmetric in this respect. If you took any true or false equation, and replaced every number with its conjugate, it would remain true or false. Thus, there's no way to tell whether i or -i is the "true" i. Your professors might be using the "wrong" one.

First time I ever used complex numbers for something useful was in electronics. (Note that in electronics they use 'j' instead of 'i')

Static circuits can be analyzed using

Resistors: Voltage = Current * Resistance
Capacitors: Current = 0
Inductors: Voltage = 0

For non-static circuits, these become very awkward:
Resistors: Voltage = Current * Resistance
Capacitors: Current = Capacitance * dVoltage/dt
Inductors: Voltage = 1/Inductance * dCurrent/dt

By allowing complex numbers, all of these can be unified into one equation:

Voltage = Impedance * Current

A "complex voltage" is a voltage that alternates like a sine wave, and a "complex current" is similar.
The impedance of a resistor is simply its resistance. The impedance of a capacitor is j * capaictance, and the impedance of an inductor is -j / inductance.

I apologize for any minor errors here; I was writing quickly, and from memory.
Click to expand...

You got the impedances wrong:
The impedance of an inductor is X<sub>L</sub> = j*ω*L
The impedance of a capacitor is X<sub>C</sub> = - j/(ω*C)
where ω is the frequency of the sinusoid, L the inductance and C the capacitance.
 
You must log in or register to reply here.

Back
Top Bottom