Imaginary numbers

[eric]

Is it just me, or is this whole imaginary number stuff just a bunch of crap? I learned this in Algebra II class the other week. Now, my teacher is trying to apply it. What meaning will this have in my life? Don't get me wrong, I understand the concept...but, why do I need to know this?

[fafalone]

Well since you want to be a pharmacist, you'll need it to get into graduate school, but then won't need it at all :P

[mister_me]

Imaginary #'s are imaginary. They don't exists so who cares?

[JaKiri]

It depends what you do. For instance, imaginary numbers are rather vital in certain areas of electronics.

You might as well say this of any topic that you won't directly use in your life (assuming you don't use them).

Bear in mind, of course, that the "imaginary" part is just the name and only idiots misunderstand this to mean that they don't exist.

[Mastermold]

Imaginary numbers are important for all reasons mentioned.. AND to better understand the theory of mathematics. Believe it or not... there is a REAL problem when we cannot resolve a number because of the paradox it posses. My favorite (and most famous) imaginary number is the square root of -1, which is represented by a lower case i. what's interesting is that i can be manipulated mathmatically and make real numbers, but i is imaginary. And you find multiples of i to be non-real roots of many quadratic equations. So solving these requires knowledge of imaginary numbers.. and that means they are essential to elements of Calculus (applied to curves with imaginary solutions) and Algebra (the whole quadratic thing). Imaginary numbers will not help you in most professions, but then again.. there are a LOT of impractical things you are forced to learn.

[bucks]

a root or a quadratic is where the line touches or crosses the x axis, if the quadratic has imaginary roots, it doesnt actually touch the x axis. so why the hell do we have to say it has a root? i didnt know there was equal rights for polynomials.

[fafalone]

Because imaginaries are solutions to the quadratic equation.

[bucks]

i know there solutions, but if the quad doesnt touch the x axis, it doesnt really need solutions, why not just say the quadratic formula cannot be factorised?

[fafalone]

Because that would not be accurate.

[bucks]

if imaginary numbers had never been created, then it would be accurate, and a hell of a lot more logical. and i wouldnt have had so much work to do in yr 12

[Mastermold]

It's true that it would make straighforward quadratics easy... but if you remember the "Determinant", which is b^2 plus 4*a*c... where a,b,c are all three numbers of the quadratic. The determinant shows how many roots of the quadratic exist, and if you get a negative number, you know there are two imaginary root and none others... so that makes your job a lot easier because you can't solve the equation. If it weren't for imaginary numbers, you would not be able to determine this and would instead work tirelessly on the solution. OR when you used the Quadratic "formula" (the whole b squared plus or minus the square root of....) you will see the solutions.. and sometimes you get imaginary answers.

[JaKiri]

Originally posted by bucks
i know there solutions, but if the quad doesnt touch the x axis, it doesnt really need solutions, why not just say the quadratic formula cannot be factorised?

It does if you have a 3 (or 4) dimensional graph with each set of axes being represented by an Argand Plane.

[NSX]

Originally posted by MrL_JaKiri
It does if you have a 3 (or 4) dimensional graph with each set of axes being represented by an Argand Plane.

What's that?

Also, I've read somewhere on the Theory of Relativity, and faster than light travel.

Apparently travelling faster than light is imaginary as well:D

[JaKiri]

Originally posted by NSX


What's that?

Also, I've read somewhere on the Theory of Relativity, and faster than light travel.

Apparently travelling faster than light is imaginary as well:D

Oh god, this sounds like Hawking. IGNORE HIM HE'S A MORON.

an argand plane is where one axis represents real numbers and the other imaginary. Just have 2 of those instead of 2 lines for the axes.

[NSX]

Originally posted by MrL_JaKiri


Oh god, this sounds like Hawking. IGNORE HIM HE'S A MORON.


lol
Why's that?

Originally posted by MrL_JaKiri
an argand plane is where one axis represents real numbers and the other imaginary. Just have 2 of those instead of 2 lines for the axes.

So do the axes intersect each other?

[fafalone]

Why wouldn't 2 axes intersect?

[JaKiri]

Originally posted by NSX


lol
Why's that?



So do the axes intersect each other?

Well, hawking isn't really all that respected because he dumbs all the science down in his books until it's actually wrong.

And the axes intersect eachother at 0, as will all axes, and are at right angles to eachother.

(Real x, imaginary x, real y, imaginary y)

[NSX]

Originally posted by MrL_JaKiri


Well, hawking isn't really all that respected because he dumbs all the science down in his books until it's actually wrong.


Really?
My Physics teacher said it would be in layman's language; but I didn't know it went that far...

[EvilMind]

Originally posted by eric
Is it just me, or is this whole imaginary number stuff just a bunch of crap? I learned this in Algebra II class the other week. Now, my teacher is trying to apply it. What meaning will this have in my life? Don't get me wrong, I understand the concept...but, why do I need to know this?

that is not such a good way to look at it. complex analysis will play an intregral part to finding a theory of the universe, if it exists. the joyce manifolds theorized in string theory are partially complex. complex variables also play a huge role in engineering.


and to the person who said that imaginary numbers are baloney basically, i dont think that is so. Gauss, one of the greats, proved that imaginary numbers must exist and have a logical backbone to it.

[EvilMind]

Originally posted by MrL_JaKiri
Well, hawking isn't really all that respected because he dumbs all the science down in his books until it's actually wrong.

And the axes intersect eachother at 0, as will all axes, and are at right angles to eachother.

(Real x, imaginary x, real y, imaginary y)

that does not even deserve a reply

[blike]

can someone present a practical application for imaginary numbers?

[EvilMind]

method of steepest descent is one, off the top of my head.

also fourier integrals, riemann-hilbert spaces, geez the list goes on dude

[dave]

Originally posted by blike
can someone present a practical application for imaginary numbers?

electrical engineering is the main one that i can think of, but there's quite a lot of others. it's also pretty important for various basic things like damped/forced harmonic motion (you can end up with a second order differential equation that requires complex numbers to provide a real solution).

[Radical Edward]

anything to do with harmonic motion is good, because you can represent sin(blah) with e(blah) saving wodges of time when integrating differentiating and so on.

[Lynn]

MrL_JaKiri said in post #17 (http://www.scienceforums.net/forums/showthread.php?s=&postid=7421#post7421):


And the axes intersect eachother at 0, as will all axes, and are at right angles to eachother.

(Real x, imaginary x, real y, imaginary y)

Sorry, I don't quite get it. 4 axes at right angles to each other, how could it happen in 3-dimentions? Would it be 4-dimentions then?

[YT2095]

Hmmm... I can only picture 3 as well? 2 to make a cross +
and one right through the middle to make like a "jack" type shape.
also, Imaginary numbers is that what the RND key on a calculator is for, it makes random numbers (I always thought it was for statistics and stuff)?

[JaKiri]

Lynn said in post #25 (http://www.scienceforums.net/forums/showthread.php?s=&postid=24059#post24059):


Sorry, I don't quite get it. 4 axes at right angles to each other, how could it happen in 3-dimentions?

It couldn't.
Lynn said in post #25 (http://www.scienceforums.net/forums/showthread.php?s=&postid=24059#post24059):


Would it be 4-dimentions then?

By definition.

[JaKiri]

YT2095 said in post #26 (http://www.scienceforums.net/forums/showthread.php?s=&postid=24066#post24066):
Hmmm... I can only picture 3 as well? 2 to make a cross +
and one right through the middle to make like a "jack" type shape.
also, Imaginary numbers is that what the RND key on a calculator is for, it makes random numbers (I always thought it was for statistics and stuff)?

Imaginary numbers are real numbers multiplied by i, the square root of minus 1 (bi, where b is real).

Complex numbers are of the form a + bi, where a and b are real.

Random numbers can be but usually AREN'T imaginary.

[YT2095]

you mention `i` being the square root of -1
so is that a "constant"? as in every time you type that into a calc, it`ll give the same result?

I`m curios now what makes that such a special number? why not sqr rt of -2 or 3 etc..?

and by REAL numbers, do you mean things that you know are fact "the fence is 3.5m long"
or real as in Whole numbers?
(I`m kinda new to this).

[JaKiri]

Real numbers = any non imaginary number (ie from negative infinity to positive infinity. examples being, say, 312, 2314.1 and pi)

What do you mean, is SQRT(-1) a constant? Does SQRT(2) change?

It's SQRT(-1) because that's the simplest way of doing it.

Why have the sqrt(-2) as i, and then have to worry about the fact that it's irrational?

Why have sqrt(-4) as i, when you might just as well as have sqrt(-1), because all that you're doing is making i = 2i (regular).

SQRT(-2) = SQRT(2)i (squaring both sides: -2 = 2 * i*i = -2

Oh, and calculators don't 'do' imaginary numbers. If you try to square root -1, you'll get an error.

[neo_maya]

MrL_JaKiri said in post #12 (http://www.scienceforums.net/forums/showthread.php?s=&postid=7125#post7125):


It does if you have a 3 (or 4) dimensional graph with each set of axes being represented by an Argand Plane.

Sorry for asking this question and I'm pretty sure it's a dumb one.
But I am gonna ask this anyway-

How can u have a four dimensional graph?

And if u replace the 3 axis (in a 3d graph) with an Argand Plane then I can only imagine two planes. I was just curious if u draw a 4d graph (I am not sure if it is possible practically - so please don't laugh), will we get 3 planes? If we do, what's the function of it? And is there function (or existance) of a third axis in an Argand Plane. I thought it had only two axis intersecting each other - one representing the real numbers and one representing imaginary ones.

And though it is not relevant - I was just curious - do u know what is a " Treserect " - I am not sure if the spelling is right - and not sure if something of this name really exists - but if it does - what is it? I have heard that this is four dimensional object which cannot be drawn on papers but u can draw the projection of its shadow. [ This may also be from a science-fiction book, so please don't blame me if it is. ]

( A side question - Can u define a point in the Argand Plane with the value of Pi - I mean a teacher of mine told us that u can never get Pi or e (exp) as a root of any equations - So I just had this crazy thought )

Please don't laugh at these questions - I'm sure these are dumb ones - but just had to ask them.

[JaKiri]

neo_maya said in post #31 (http://www.scienceforums.net/forums/showthread.php?s=&postid=24074#post24074):
( A side question - Can u define a point in the Argand Plane with the value of Pi - I mean a teacher of mine told us that u can never get Pi or e (exp) as a root of any equations - So I just had this crazy thought )

That's incorrect.

Example.

If the radius of a circle is 1/2(unit), what's the circumference?

Reply to rest following.

[JaKiri]

neo_maya said in post #31 (http://www.scienceforums.net/forums/showthread.php?s=&postid=24074#post24074):
How can u have a four dimensional graph?

And if u replace the 3 axis (in a 3d graph) with an Argand Plane then I can only imagine two planes. I was just curious if u draw a 4d graph (I am not sure if it is possible practically - so please don't laugh), will we get 3 planes? If we do, what's the function of it? And is there function (or existance) of a third axis in an Argand Plane. I thought it had only two axis intersecting each other - one representing the real numbers and one representing imaginary ones.


Well, you're mistaken about the number of planes in a 3d object. I can think of 3; xy, xz and yz. Similarly, in a 4d situation there will be 6 (xy, xz, xa, yz, ya, za).

As for the function, it's to have a complex graph with 2 variables. (As I stated above, one direction for each of real x, imaginary x, real y, imaginary y).

And as to the tesseract...

If a square is the 2d version, and the cube is the 3d version, then the tesseract is the 4d version.

We can view it in 3d (sort of) by perspective, similar to implying a 3d object in 2d. (I've seen one by this method.)

A tesseract is more commonly known as a hypercube.

[neo_maya]

MrL_JaKiri said in post #33 (http://www.scienceforums.net/forums/showthread.php?s=&postid=24080#post24080):
[B]

Well, you're mistaken about the number of planes in a 3d object. I can think of 3; xy, xz and yz. Similarly, in a 4d situation there will be 6 (xy, xz, xa, yz, ya, za).



Oooooppppppppsss. Told ya I was dumb. :embarass:

But still can't think of the fourth axis (in a 4d situation). Three axis intersects each other (in 3d) perpendicularly, but 4d??

But what is the fourth dimension of a tesseract?( I always thought the fourth dimension was time )

I have just said something silly - haven't I?

:scratch:

[JaKiri]

http://www.scienceforums.net/forums/showthread.php?s=&threadid=1068

[neo_maya]

R u still there? I had some more questions? If u r free and not annoyed with all my dumb questions, may I please ask them?

[JaKiri]

Fire away.

Even if I'm not here now, I can answer them later.

And even if I don't, someone else will.

[neo_maya]

Thanks. First of all -

http://dogfeathers.com/java/hyprcube.html

What is this? (This is not the main question)

[neo_maya]

I am still reading the thread that u posted. But the problem is these stuff are way over my head. Trying....

Sorry for keeping u waiting - If u want I will write the questions and u can always answer them later.

Thanks a lot. Did u go to the website?

[neo_maya]

This whole conception of dimensions is a bit confusing. U said we can have xy, xz, xa, yz, ya, za planes in a 4d situation, but what does the axis a stand for?

Someone once told me that the fourth dimension is arbitrary - it can be time or something else (according to his will or according to the equation he is solving).

[JaKiri]

neo_maya said in post #38 (http://www.scienceforums.net/forums/showthread.php?s=&postid=24093#post24093):
Thanks. First of all -

http://dogfeathers.com/java/hyprcube.html

What is this? (This is not the main question)

It's an attempt to show a 4 dimensional object in 2 dimensions.

As you can see, it doesn't work.

[YT2095]

mr_l so `i` is a fixed number then?
it`s the result of sqr rt -1
and no I did`nt know that a calculator wouldn`t do it (can you show how it`s worked out then? I wouldn`t know where to start)
is `i` classed as a Function? ( I read about those in calculus threads) or is it a constant?

neo-mayo, NOT knowing something here is fine, that`s partly why I`m here too, but PLEASE stop wearing it like a badge, if you want to know ask! :)

[neo_maya]

This Entropy thing is the most confusing of all. It's the operative of the thermal co-efficient of a closed system - what's that supposed to mean?

dS = dQ/T (had some integration signs)

I read somewhere that in a adiabatic process the change of entropy is 0 and that in a closed system it will keep increasing until the temperature of the whole universe is equal and since it is irreversible, we can't go back in time (though it is irrevlent here). My problem is that - I don't unnderstand the concept of entropy. (Actually I don't understand anything) and why is it irreversible?



[ PS : There r a whole lot of things that I need to ask and all of these are surely some dumb questions. ]

[JaKiri]

YT2095 said in post #42 (http://www.scienceforums.net/forums/showthread.php?s=&postid=24101#post24101):

and no I did`nt know that a calculator wouldn`t do it (can you show how it`s worked out then? I wouldn`t know where to start)


You don't work it out, it's just.... i. The square root of -1. That's how its written.

It's not going to be equal to 35 or something.

[JaKiri]

neo_maya said in post #43 (http://www.scienceforums.net/forums/showthread.php?s=&postid=24102#post24102):
My problem is that - I don't unnderstand the concept of entropy. (Actually I don't understand anything) and why is it irreversible?


The second law of thermodynamics states that within any closed system, entropy always increases.

Entropy is basically a measure of how much energy in the system is not 'ordered' or useful; like when running a car, there's heat given off.

As for the universe thing, that is correct. It is called the heat death of the universe.

It is irreversable because no matter what action we take to increase order, there will be a subsequent larger increase in disorder (because entropy [disorder] cannot decrease in any closed system).

(A closed system is basically something which is cut off from everything else. This part is essential, and is one of the bits of science ignored by creationists, but that's going off on a tangent)

[neo_maya]

I think it's like infinity - u can't define it - yet there it is - an infinite amount of infinity inside infinity [ At least that's how this crappy head sees it ]

[YT2095]

that makes no sense at all whatsoever to me?

[neo_maya]

My last post was for YT.

[YT2095]

neo-mayo, ok so it would fit into the "function" catagory then , yes?
a bit like the reciprocal of 0 = infinity (or at least any number you choose to mention)

[neo_maya]

MrL_JaKiri said in post #45 (http://www.scienceforums.net/forums/showthread.php?s=&postid=24104#post24104):


The second law of thermodynamics states that within any closed system, entropy always increases.

Entropy is basically a measure of how much energy in the system is not 'ordered' or useful; like when running a car, there's heat given off.




But if a certain amount of energy is not ordered or useful - then wouldn't that energy transform to some other form of energy? Then why is it irreversible?

[Kedas]

Just want to add that in electronics we use the letter 'j' to represent 'i'.
This to avoid confusion with the symbol of current.

An ideal capacitor and coil only have an imaginary value because the voltage is exactely 90° out of phase.
Capacitor: Current 90° before voltage or -xj
Coil: current 90° behind voltage or +xj

The current of a resistor is in phase and therefore has no imaginary value, you can also say it has one but it is always zero like 0j.

Yes, you can multiply/divide imaginary numbers with a calculator.
(you have to convert to an other representation though, real & real -->real & angle, a little calculator programming helps a lot)

[neo_maya]

YT2095 said in post #49 (http://www.scienceforums.net/forums/showthread.php?s=&postid=24108#post24108):
neo-mayo, ok so it would fit into the "function" catagory then , yes?
a bit like the reciprocal of 0 = infinity (or at least any number you choose to mention)


Hey don't attack me - I'm just a crappy head.

But here is how I see this -

No - it wouldn't fit into a function catagory either. Like the illuminated one said - it's just ........i. And what I understand is - +ve infinity means a number greater than the largest number u can think of and -ve infinity means a number smaller than the smallest number u can think of.

Though this brings the question - I heard there were 7 special forms like 0/0, infinity/infinity, 0^0 and several others that can't be defined. And they don't fit even into the catagory of infinity. They are like i, just imaginary.

[ Someone please book a sit for me in the mental hospital , I am mad ]

[neo_maya]

neo_maya said in post #50 (http://www.scienceforums.net/forums/showthread.php?s=&postid=24109#post24109):



But if a certain amount of energy is not ordered or useful - then wouldn't that energy transform to some other form of energy? Then why is it irreversible?


Oooooppsss sorry - got the point of the ever increament of entropy now. So, basically entropy refers to the disorder in the universe - but what is the heat death then?

[JaKiri]

YT2095 said in post #49 (http://www.scienceforums.net/forums/showthread.php?s=&postid=24108#post24108):
neo-mayo, ok so it would fit into the "function" catagory then , yes?
a bit like the reciprocal of 0 = infinity (or at least any number you choose to mention)

It's not a function.

It's a number.

You might as well say 'How do you work out 1? Is it the same every time?'

[Kedas]

if everything in the universe would have the same temperature then there wouldn't be any energy transfer anymore.

[YT2095]

and so, if you calculate the sqr rt of -1, what is the answer? in one post you say it`s not a number like "35" and in another you say is IS a number? Kinda WELL LOST here!

[neo_maya]

Ok - I read this in a scinece-fiction book - if u take a right handed golves outside of the universe and then bring it back - it will be left-handed. - Is there any scientific basis of this theory.

[YT2095]

why should the gloves change? definately sounds like sci-fi to me!? :)

[neo_maya]

YT2095 said in post #58 (http://www.scienceforums.net/forums/showthread.php?s=&postid=24119#post24119):
why should the gloves change? definately sounds like sci-fi to me!? :)


First, u tell me - what's outside of this universe?

:-p

[JaKiri]

YT2095 said in post #56 (http://www.scienceforums.net/forums/showthread.php?s=&postid=24117#post24117):
and so, if you calculate the sqr rt of -1, what is the answer? in one post you say it`s not a number like "35" and in another you say is IS a number? Kinda WELL LOST here!

It's not a real number (ie on the number line from negative infinity to positive infinity).

It's just... a number.

[JaKiri]

neo_maya said in post #59 (http://www.scienceforums.net/forums/showthread.php?s=&postid=24120#post24120):



First, u tell me - what's outside of this universe?

:-p

I can categorically say that whatever it is won't be a mirror.

[Kedas]

i^2=-1 not the same as i=sqrt(-1)

There is a funny 'prove' about it that says that -1=1

i.i=-1
sqrt(-1).sqrt(-1)=-1
sqrt(-1.-1)=-1
sqrt(1)=-1
1=-1

[JaKiri]

No.

That's equivilent to saying that -4 = 4, because SQRT(16) = 4

SQRT(16) = -4

Hence -4 = 4.

[YT2095]

"It's not a real number (ie on the number line from negative infinity to positive infinity).

It's just... a number. "

HUH? now I`m totaly lost :)
the only thing I can think of then is that it must be a variable like (n) or X like in a computer program

for X = 1 to 10
next x

X isn`t a definate number exactly, but at any one time it`ll have a definate value between 0 and 10.
is it that sort of thing?

[JaKiri]

No.

It's a number.

i is a number.

It's not on the number line from - infinity to + infinity, but it's a number just like 1 or 462.

i is the symbol that represents it.

[YT2095]

but that sounds SO contradictory, so I`m sure I must be missing something here?

[JaKiri]

i doesn't have a value in the real numbers. Otherwise it would just be that number.

As I said, it doesn't feature on a number line; it exists on an Argand plane, which has the usual number line as the x axis and a similar one for the imaginary numbers as the y axis.

[JaKiri]

You may as well say 'Why is 15 not 3?'

[YT2095]

I kinda regret asking now :-p
Thnx for trying anyway Mr-L, appreciated! :)

[neo_maya]

IMAGINARY NUMBER. There are two modern meanings of the term imaginary number. In Merriam-Webster's Collegiate Dictionary, 10th ed., an imaginary number is a number of the form a + bi where b is not equal to 0. In Calculus and Analytic Geometry (1992) by Stein and Barcellos, "a complex number that lies on the y axis is called imaginary."

Chech out these sites. There are a number of them out there. If u want I can search them for u.

http://www.friesian.com/imagine.htm
http://www.jimloy.com/algebra/imaginar.htm

[These sites have long articles, but by the time u have finished reading them, I think that will do ]

_______________________________________________


An Argand Plane is where u have to axis X and Y both intersecting each other at (0,0) point. X represents all the real numbers from -infinity to +infinity. And Y axis represents the imaginary numbers.

ai + b = each and every number that u see. But in the case of real number a = o and in the case of imaginary number a has a value. So, every real number is a kinda imaginary number (sort of , where the imaginary part doesn't exist) only where a = 0.


Cube roots of unity and their properties :

:lcomega: ^3 = 1
1+ :lcomega: + :lcomega: ^2 = 0

where, :lcomega: = 1/2 {-1+ sqrt(-3)}
and :lcomega: ^2 = 1/2 {-1- sqrt(-3)}


There r a whole lot other stuff regarding i, but basically these are the basic ones. An imaginary number has some other properties like - if u multiply two conjugate imaginary numbers - u will get a real number or if u add them u will get a real number. And there is the modulus and argument of imaginary number. You can even work out the sqrt of [ 7-30sqrt(-2) ].


_______________________________________________

The problem with imaginary numbers are that - yet we can't define them. So, no matter how much we can imagine - we will never imagine an imaginary number until someone defines them or describes their characteristics or properties. But it is indeed a number - we just can't understand it - that's why it's imaginary (u have to imagine it).
________________________________________________

I don't know if each and every line of what I have written is right. But that's basically what can think of an imaginary number.

PS : There r other forms of undefineables (I think) like - 0/0 , infinity/infinity. 0^0, infinty^infinity etc.

[neo_maya]

Hey, I think there should be a sqrt sign in the smilies list - don't you think?

[JaKiri]

x^1/2?

[neo_maya]

MrL_JaKiri said in post #72 (http://www.scienceforums.net/forums/showthread.php?s=&postid=24209#post24209):
x^1/2?


Ooooppppss. Right.


:embarass:

[neo_maya]

Hey, I was having some problems with differentiation and integration. Can I post them now?

[JaKiri]

Make a new thread :p

[Sayonara³]

neo_maya said in post #71 (http://www.scienceforums.net/forums/showthread.php?s=&postid=24202#post24202):
Hey, I think there should be a sqrt sign in the smilies list - don't you think?
Use the superscript tags.

[JaKiri]

Sayonara³ said in post #76 (http://www.scienceforums.net/forums/showthread.php?s=&postid=24223#post24223):

Use the superscript tags.

I was going to find an image of a square root sign to demonstrate what she meant, but when searching google for square root, the first thing up was This (http://www.annettemoen.com/images/square-root.jpg), which is much more appealing as a smiley.

[neo_maya]

MrL_JaKiri said in post #75 (http://www.scienceforums.net/forums/showthread.php?s=&postid=24221#post24221):
Make a new thread :p



I already have. :D

Sayonara is a she !!!!!!!!!!!!!!??????????!!..........

[neo_maya]

No offence.

[JaKiri]

My fingers, they type of their own accord.

[Kedas]

MrL_JaKiri said in post #63 (http://www.scienceforums.net/forums/showthread.php?s=&postid=24126#post24126):
No.

That's equivilent to saying that -4 = 4, because SQRT(16) = 4

SQRT(16) = -4

Hence -4 = 4.

what ??

I start with -1=-1 and end with 1=-1 without making any obvious mathematical error.

BTW I know that 1=-1 is wrong but that's the point of the message.

with numbers:
2.2=4
sqrt(-4).sqrt(-4)=4
sqrt(-4.-4)=4
sqrt(16)=4

The difference is that no one claims that sqrt(-4)=2 but they do claim sqrt(-1)=i.

[Tom Mattson]

YT2095 said in post #56 (http://www.scienceforums.net/forums/showthread.php?s=&postid=24117#post24117):
and so, if you calculate the sqr rt of -1, what is the answer?


The answer is 'i'.


in one post you say it`s not a number like "35" and in another you say is IS a number? Kinda WELL LOST here!

OK, how about this:

'i' is not an element of Z, as 35 is.
'i' is an element of C, as 35 also is.

[Tom Mattson]

Kedas said in post #62 (http://www.scienceforums.net/forums/showthread.php?s=&postid=24124#post24124):
i^2=-1 not the same as i=sqrt(-1)


You're right: i2=-1 contains more information than does i=sqrt(-1). -1 (like all numbers) actually has two square roots: +i and -i.


There is a funny 'prove' about it that says that -1=1

i.i=-1
sqrt(-1).sqrt(-1)=-1
sqrt(-1.-1)=-1
sqrt(1)=-1
1=-1


The part in red is where the mistake was made. MrL Jakiri's comment was right. I can do the exact same thing with real numbers.

2*(-2)=-4
sqrt(4)*sqrt(4)=-4
sqrt(16)=-4
4=-4

Ta-Daaaa!

[Kedas]

yeah, it's a funny thing to keep people busy :)

I only wanted to point out the difference between the two with some proof.
Not to start a debate about the comparison with real numbers.