Complex Number Is Really Complex.

[kushwaha]

Complex number .........have you ever imagine whats it signifies ?In mathematices any number in the form (a+ib) is called a complex number.Or a number which have no existence ..........in physics complex number is assigned to any thing whose most of the properties are unknown .Like in the case of wave function ,now could anybody solve my confusion that how in this world two complex quantity on being operated with each other may give a real quantity?

Like under root of -1,equal to what,cant be predicted or we can say it is undefined .When we are multiplying iXi=-1,

so (undefined X undefined)=defined ,....how is that possible.

[JMJones0424]

Just as we can say that

[math]\sqrt n \cdot \sqrt n = n[/math]

 

We can say

[math]\sqrt -1 \cdot \sqrt -1 = -1[/math]

 

Because [math]i = \sqrt -1[/math], we can substitute [math]i[/math] for [math]\sqrt -1[/math] and end up with

 

[math] i \cdot i = -1[/math]

[maddog]

Complex number .........have you ever imagine whats it signifies ?In mathematices any number in the form (a+ib) is called a complex number.

... (undefined X undefined)=defined ,....how is that possible.

The square root of -1 (or any negative number) is Not undefined. It is just NOT Real! Once I learned about Complex numbers in high school by reading in the back of my algebra book, I got really fascinated. I went up to ask my (2nd Yr) Algebra teacher then "what is sqrt(i) or 4th root of -1". She said that number didn't exist. Now from what I read, I knew something was fishy though I didn't know why. So when I took trigonometry in summer session before I was a freshman to prepare myself for Calculus, I asked my instructor this question and she drew on a virtual blackboard in the stairwell a unit circle and showed that this is "just e^i*pi/4". I know another complex number (expressed in polar coordinates). Graphically it makes sense in that this is also the point on the circle where a line drawn from the center to a point on this unit circle of 45 deg or sin (pi/4) for y or cos (pi/4) for x when using Cartesian coordinates (x, y).

 

I remember three places where complex numbers comes in directly in physics.

1) when you have this integral that is only solvable in complex numbers to describe the light intensity from a Fresnel lens (Optics).

2) Calculating the E field potential at an arbitrary point in space in vector coordinates (x, y, z) where x, y, z could be Complex values of the type (a + ib).

3) Solving the Schroedinger equation using wave functions that were Complex Valued.

 

See the important point is Reals are contained within the Complex numbers. Complex as a domain is Algebraicly Complete which means that all equation have solutions that also are Complex. For Reals this is not the case. Some equations for Reals are unsolvable.

 

maddog

[kushwaha]

Just as we can say that

[math]\sqrt n \cdot \sqrt n = n[/math]

 

We can say

[math]\sqrt -1 \cdot \sqrt -1 = -1[/math]

 

Because [math]i = \sqrt -1[/math], we can substitute [math]i[/math] for [math]\sqrt -1[/math] and end up with

 

[math] i \cdot i = -1[/math]

 

 

Thanks for replying.

But dear friend my question still remains there.As mathematically we can not represent the square root of -1 on number line,or it is somewhere in the space we dont know.so how we can perform operation on this particular "dont know" to make it known. The question posted here is more of theoretical approach rather than calculative. i will be highly gr8full

[JMJones0424]

The question I tried to answer above was-

 

When we are multiplying iXi=-1,

so (undefined X undefined)=defined ,....how is that possible.

 

The fact that i as an imaginary number cannot be placed on a number line of real numbers is irrelevant. Of course, i can be placed on a number line of imaginary numbers, and that axis is one of the two axes that make up the complex plane.

 

The restriction that a number must be real in order to perform mathematical operations on it is entirely in your own mind.

[Qfwfq]

so (undefined X undefined)=defined ,....how is that possible.

Actually, i times i equals -1 is the very definition of i, it is a relation between what is being defined and things already defined.

 

The only fly in the ointment is that this definition does not distinguish i from -i but this isn't a problem, it entails that the complex field has a reflexive endomorphism (which is complex conjugation).

 

Math is all a matter of defining and constructing. So long as there isn't self-inconsistence, everything is fine. Numbers don't exist, they are but figments of our imagination. Very useful ones, however.

[maddog]

Thanks for replying.

But dear friend my question still remains there.As mathematically we can not represent the square root of -1 on number line,or it is somewhere in the space we dont know.so how we can perform operation on this particular "dont know" to make it known. The question posted here is more of theoretical approach rather than calculative. i will be highly gr8full

You are right that you cannot represent "square root of -1" on a number line. However, you can represent it nonetheless. Instead you represent it as a unit circle about 0. The representation of e^ix, where x is in interval [0, 2PI]. In this way you can even represent sqrt (i) => x = PI/4.

 

maddog

[maddog]

The restriction that a number must be real in order to perform mathematical operations on it is entirely in your own mind.

I realize I must have stepped over this. Yes, there is NO Requirement that for a number to exist - it must be Real. Aside from the clear and concise comments of Qfwfq that "numbers" are an invention of our minds and have no inherent/independent/intrinsic reality. They do not.

 

Thus I can create abstract algebras that can be anticommunative such as the quaternions. Quaternions are like complex numbers where i^2 = -1. With quaternions though there are two more j & k where j^2 = -1 and k^2 = -1. However with quaternions j * k != k * j --> instead j * k == - k * j. You can see this when you realize that there is an Isomorphism between the quaternion algebra and the Real 3-space vector algebra with X, Y, Z. Vectors likewise do not commute under the operation of a "cross-product".

 

maddog