imaginary numbers

smslca

I googled, wikied etc., but I cant understand what it is because, may be i cant understand clearly what they said, or I have these questions in my mind because of little understanding.

What does these complex numbers represent in the real life. Where do they fit in the real 3 spatial (xyz) coordinate system.

Let us consider the real xyz coordinate system. If a person is at the origin. As he moves on x-axis forward(i.e the way he can see) it is +1*x, when he moves back(ie the way he cannot see it is -1*x. Here +1,-1 represents the direction on the line. ""Then what direction does i represent????"".

Even 3/4 (ie fractional) distance exists, and we can approximate the irrational values such as pi,sqr root 2, and show the distance from the origin(ie between 0 and 1, or 0 and -1 etc). But where should we show this 'i' distance from origin.


Answer below if my assumptions are true:

---

We know if +1 is forward and +1*-1 represent backward direction, and so on it iterates the direction. By applying similar way i must be a 90 degree direction.
Here i got an another confusion. If the person is moving in perpendicular to his facing side, then what is Y-axis in a 3 dimensional system. If Y-axis is direction of i
why shouldnt we represent every eqn like x+y=0 as x+iy=0.

If above paragraph is true, place a person on y-axis ie iy and x-axis must be real taken from above para. If we move him 90 degrees from y to z-axis then z must be real because i*i=-1 and i*-i=+1. But since we took x as real, if we move him towards z then z must be imaginary. But what is the z- as real axis really n imaginary mean.

---

can any one explain the graph in
http://en.wikipedia.org/wiki/Complex_number

LeixlipRed

There is no i distance from the origin. The clue is in the word imaginary! They don't exist in the cartesian plane. But they do have applications in the real world. Counter intuitive I know.

And the graph there is a representation of the Complex Plane. The real numbers on the x-axis and the imaginary part on the y-axis. It's not a representation of R^2.

smslca

LeixlipRed wrote: »

There is no i distance from the origin. The clue is in the word imaginary! They don't exist in the cartesian plane. But they do have applications in the real world. Counter intuitive I know.

I think numbers define life,time,and universe.
so u r saying there exists numbers which behave like just numbers and nothing else.

Actually I got this problem when im working on the kinematic(parabolic) equations.
Everyone know, sometimes we get negative solutions for time. But we will discard it as negative time doesnt make any commonsense at that equation. I think we mean it does not exists in real life
Suddenly imaginary numbers appeared in my mind, and asked myself why didnt mathematicians discarded them as they doesnt make any commonsense.

Iderown

smslca,

One of the uses of complex numbers is in the steady state analysis of ac circuits which have inductance, resistance and capacitance.
Using complex numbers allows these circuits to be analysed without using calculus.

Would you care to look at http://en.wikipedia.org/wiki/RC_circuit
But, the use of complex numbers there is mixed up with calculus.

bnt

I think you need to be investigating Phasors: have a look at this and this. Very handy when analysing 3-phase AC circuits.

LeixlipRed

smslca wrote: »

I think numbers define life,time,and universe.
so u r saying there exists numbers which behave like just numbers and nothing else.

Actually I got this problem when im working on the kinematic(parabolic) equations.
Everyone know, sometimes we get negative solutions for time. But we will discard it as negative time doesnt make any commonsense at that equation. I think we mean it does not exists in real life
Suddenly imaginary numbers appeared in my mind, and asked myself why didnt mathematicians discarded them as they doesnt make any commonsense.

Not every problem in mathematics has to have a basis in real life. There are several streams of mathematics that are completely abstract. Euclidean geometry for example

ray giraffe

All numbers are imaginary in a sense: it can be argued that they exist only in your imagination. Show me a real-life example of -2 or pi?

One way to construct new numbers is to try to solve equations. Start with the natural numbers (1,2,3,4,...), i.e. imagine these are the ONLY numbers that you know about. (alternatively, imagine you lived some thousands of years ago! :pac:)

Trying to imagine a solution to 3+x=3 produces a new number, 0.

Trying to imagine a solution to 3+x=0 produces -3, similarly we can imagine the other negative whole numbers.

Imagine a solution to 2x=3 and you get 2/3, similarly we can imagine all the other positive and negative fractions.

Imagine a solution to x^2= 2 and you are faced getting your head around an irrational number, sqrt(2). Similarly we can find many other irrational numbers (but not all! See my note later on sequences).

(But how do we know sqrt(2) is irrational (i.e. not a fraction) you ask?
There is a short, old and beautiful proof! :eek:
http://www.homeschoolmath.net/teaching/proof_square_root_2_irrational.php)

Imagine a solution to x^2=-1 and now you are up against sqrt(-1) or i, the so called 'imaginary' number. :eek:

The difference this time is that we can no longer imagine the number i on the usual number line (unlike the fractions and irrationals and negatives), so some people like to imagine it sticking out, in the direction of the usual y-axis and 1 unit long. So i is not a conventional 'length'.

If you want to learn more about imaginary numbers, take a look at:

http://www.math.toronto.edu/mathnet/answers/imaginary.html

Note on Irrationals: Most irrationals (e.g. pi and e) can't be discovered as solutions of equations (if you are only allowed to use a number in the equation that has already been discovered! For example, pi is a solution of x-pi = 0 :eek: but we are cheating because we are 'using pi to discover pi')

To discover all the irrationals, one way is to look at limits of sequences of numbers (where terms in the sequence are allowed to be only the positive and negative fractions, and 0)

Think about all possible sequences of fractions, but only the sequences where the terms get closer and closer together, the further along the sequence you go. (The technical name is 'Cauchy sequence' :eek:) What are the terms getting 'closer and closer to'?

Now we can try to imagine pi. Think about a special sequence of fractions:
(3/1),(31/10),(314/100),(3141/1000),(31415/10000),...
or equivalently,
3, 3.1, 3.14, 3.141, 3.1415,...
The terms are definitely getting closer and closer together, but what are they getting closer and closer to? The answer is pi !! :pac: It turns out that all the other irrational numbers can be imagined as answers for other sequences of fractions :cool:

Fremen

LeixlipRed wrote: »

There are several streams of mathematics that are completely abstract. Euclidean geometry for example

Eh? That's one of the most concrete bits!

LeixlipRed

Fremen wrote: »

Eh? That's one of the most concrete bits!

Euclidean geometry? Concrete in what sense now? It's no longer generally accepted that it's a good representation of the physical world. No concept of curvature for example.

equivariant

smslca wrote: »

I googled, wikied etc., but I cant understand what it is because, may be i cant understand clearly what they said, or I have these questions in my mind because of little understanding.

What does these complex numbers represent in the real life. Where do they fit in the real 3 spatial (xyz) coordinate system.

Let us consider the real xyz coordinate system. If a person is at the origin. As he moves on x-axis forward(i.e the way he can see) it is +1*x, when he moves back(ie the way he cannot see it is -1*x. Here +1,-1 represents the direction on the line. ""Then what direction does i represent????"".

Even 3/4 (ie fractional) distance exists, and we can approximate the irrational values such as pi,sqr root 2, and show the distance from the origin(ie between 0 and 1, or 0 and -1 etc). But where should we show this 'i' distance from origin.


Answer below if my assumptions are true:

---

We know if +1 is forward and +1*-1 represent backward direction, and so on it iterates the direction. By applying similar way i must be a 90 degree direction.
Here i got an another confusion. If the person is moving in perpendicular to his facing side, then what is Y-axis in a 3 dimensional system. If Y-axis is direction of i
why shouldnt we represent every eqn like x+y=0 as x+iy=0.

If above paragraph is true, place a person on y-axis ie iy and x-axis must be real taken from above para. If we move him 90 degrees from y to z-axis then z must be real because i*i=-1 and i*-i=+1. But since we took x as real, if we move him towards z then z must be imaginary. But what is the z- as real axis really n imaginary mean.

---

can any one explain the graph in
http://en.wikipedia.org/wiki/Complex_number

Imaginary numbers such as i, are no more or less abstract than negative numbers such as -1. Ask yourself this. Can you have -1 apples? No. Does that mean that -1 is not a useful concept? No.

Both -1 and i arise in mathematics from the need to find solutions to certain equations. Humans devloped the concept of numbers originally as a a way of counting things. However, the concept of number has become more and more complicated as human history and culture progressed. After a while people start to solve simple equations. But then they realised that equations like

x+1 = 0

x^2 = 2

and x^2+1=0

did not have solutions if you only allowed yourself to work with counting numbers. So people started to expand their concept of number to include solutions to these equations. When you look at the equation x^2 +1 = 0, you automatically end up thinking about imaginary numbers.

The nice thing is that these so called imaginary numbers turned out to have all sorts of applications that no one suspected initially.

Capt'n Midnight

Eleventeen

For many simple equations ( X squared + b X + c ) the solutions include imaginary numbers. And as others have pointed out they can have practical applications. If you only have a hammer then every problem looks like a nail. Equations are tools, when mathematicians are comfortable with the tools they have they can try them on new problems, or more likely they'll have them to hand when someone comes up to them with a real world problem.



Another way to look at them is what happens when you calculate roots of numbers. With the i axis at right angles to the number line the root has two components , a magnitude and an angle. The magnitude is how far from 0 it is. Let's reduce the number to 1, there are two square roots of 1 , 1 and -1 they are 180 apart (360 /2) 1 has four fourth roots ,1 -1 i -i which are 90 degrees apart ( 360/4) the same for other powers too for the third roots of 1 the three are 1, and the other two are represented by 120 degree angles




XYZ axis is used to represent the physical world.
But each of those axes can also have imaginary components too.

Off topic , there are of course Quaternions where you have not just i, but j and k too


I can't remember fully but didn't some of Hawking's stuff relating to time at the start of the universe curved back on itself in imaginary time ?

bnt

Just in case anyone needs their mind blown a little more, I have only this to add - from the Wikipedia page on Euler's Formula:



The form [latex]e^{i\,{\phi}}[/latex] is another way of expressing a Phasor. You have to use Radians, of course - not degrees. :cool:

Eliot Rosewater

LeixlipRed wrote: »

Not every problem in mathematics has to have a basis in real life.

I did a course in Abstract Algebra last semester, and our infamous lecturer Des McHale commented that Mathematics is like poetry - it deserves to be studied for its own sake.

I would definitely agree. The Abstract Algebra was probably the best learning experience I've ever had.

bnt wrote: »

Just in case anyone needs their mind blown a little more, I have only this to add - from the Wikipedia page on Euler's Formula

I suppose the most famous part of that is when you sub in pi for thetha to get the identity...

fergalr

http://xkcd.com/179/

LeixlipRed

The equation in it's one sided form contains probably the 5 most important numbers in maths too. Gorgeous it is

Fremen

LeixlipRed wrote: »

Euclidean geometry? Concrete in what sense now? It's no longer generally accepted that it's a good representation of the physical world. No concept of curvature for example.

Sure there is: x^2 + y^2 + z^2 = 1 is curved. The thing about Riemannian geometry is that Nash's embedding theorem tells us that any manifold can be isometrically embedded into Euclidean space if we pick the dimension of the Euclidean space to be large enough. From that viewpoint, "non-Euclidean" geometry is no more general than ordinary geometry.

Sorry for dragging this offtopic.

equivariant

Fremen wrote: »

Sure there is: x^2 + y^2 + z^2 = 1 is curved. The thing about Riemannian geometry is that Nash's embedding theorem tells us that any manifold can be isometrically embedded into Euclidean space if we pick the dimension of the Euclidean space to be large enough. From that viewpoint, "non-Euclidean" geometry is no more general than ordinary geometry.

Sorry for dragging this offtopic.

I will make two points to further drag this off topic. First of all, embedding manfolds into high dimesional space does not really make them more concrete. In 4 dimensions, there are already many weird things that happen that do not happen in 3 dimensions. Most (but not all) people have terrible intuition for 4 dimensional geometry.

Secondly, curvature is not the only way in which the geometry of the universe deviates from Euclidean. Modern physics tells us that certain aspects of the universe are best described in terms of noncommutative geometries, which are not typically modelled by Riemannian manifolds.

Fremen

Nonetheless, saying that Euclidean geometry is completely abstract because of the curvature of the universe is going a bit far.
In most day-to-day applications, the Euclidean viewpoint is the best one to take.

LeixlipRed

I was only kidding in the first place. Look what I've started

Capt'n Midnight

Fremen wrote: »

Nonetheless, saying that Euclidean geometry is completely abstract because of the curvature of the universe is going a bit far.
In most day-to-day applications, the Euclidean viewpoint is the best one to take.

Unless you are trying to navigate on a globe.