two dimensional equality

[AntonioLao]

Although the set of complex numbers constitute a field, unlike the fields of the rational numbers or the field of the real numbers, it is not an ordered field. That is to say there is no defined mathematical measurability of ‘greater than’ or ‘lesser than’ between any two complex numbers. By definition, complex numbers are found only on the complex plane except on the two axes which equally divide it into 4 quadrants. The horizontal real axis is also known as the real number line which extends to negative infinity on the left and to positive infinity on the right. At the midpoint of this line is the whole number zero. The vertical imaginary axis is also divided by zero which extends to positive imaginary infinity at the top and negative imaginary infinity at the bottom.

Nonetheless, two-dimensional equality can be defined as the equality of complex moduli. Since an arbitrary complex number Z is given by Z=A+Bⅈ where A and B are both real numbers and ⅈ is called the imaginary unit defined as the square root of negative unity, the complex modulus, |Z| is defined as the square root of the sum of A² and B² or equivalently: |Z|²=A²+B². In this sense, it is mathematically analogous to the Pythagorean Theorem for positive integers. This theorem was discovered by Pythagoras, between 585-500 BCE, more than 2500 years ago. In 300 BCE, Euclid gave a geometric proof in proposition 47 of Book I of his 13 books of the Elements, which is based on the defined area of triangles. The algebraic proof is given for 2 integers M and N such that M>N and M²-N², 2MN, M²+N² comprise a unique set of Pythagorean triples. However, in analytic geometry (combining algebra and geometry) this theorem is equivalent to the equation of a circle whose radius is equal the complex modulus. Therefore, two-dimensional equality is defined as all complex numbers that give the same complex modulus. Not surprisingly, there are infinitely many complex circles with infinitely many complex numbers that satisfy this equality condition.

[Profpat]

So can we have a 2 dimensional reality or is the 2nd dimension imaginary only?

I gave lloyd a thought experiment where I posed the question: "What if you had a little tiny particle which through movement is creating a great big field" Is that field 2 dimensional?

Best,

Pat

[labelwench]

Okay.

I can measure length x width, that is two dimensions, is it not?

Length, width and height become three dimensions?

Can't quite decide what one dimension would be.

What about something like 'thought'?

Would that be one dimensional?

Would a shared thought become two dimensional?

Time to recharge the dilithium crystal for night shift. My thoughts get pretty strange just before I 'Timex'.

Antonio, as the Master of strange math, should be able to make heads or tails of my query. Later, gentlemen.....

[Profpat]

1 dimension is length only. As far as thought being 1 or 2 dimensional, I don't know, but I usually think in 3D.

[AntonioLao]

can we have a 2 dimensional reality

We get 2D reality each time we look ourselves in the mirror but the image is reversed between left and right.

little tiny particle which through movement is creating a great big field

This is best described using Einstein's general relativity. The tiny particle is called by cosmologists as the big bang singularity. But my idea is that it cannot move because of infinite mass.

What about something like 'thought'? Would that be one dimensional?

Light travels in a 1D straight line and on curve surfaces the straight line is called the geodesic, so if thought travels as lightspeed then it also travels in 1D straight line or geodesic. I think thoughts travel faster than light.

[Profpat]

[QUOTE=AntonioLao;111192]We get 2D reality each time we look ourselves in the mirror but the image is reversed between left and right.

This is best described using Einstein's general relativity. The tiny particle is called by cosmologists as the big bang singularity. But my idea is that it cannot move because of infinite mass

Let's take the electron and it's shell that it produces, is that shell dimensional? And if so what? I view it as a 2 dimensional surface area, created by the electron moving very fast.

[AntonioLao]

Let's take the electron and it's shell that it produces

This shell is properly described as the cloud of virtual photons which clothed the nakedness of the electron such that experiments can never detect the infinite self-energy or infinite bare mass of the naked electron but experiments always detect a finite mass value.

[Profpat]

Thanks Antonio;

What about a wave 1, 2 or 3 dimensional?

Best,

Pat

[AntonioLao]

What about a wave 1, 2 or 3 dimensional

The classical electromagnetic theory describes an EM wave as the Poynting vector which moves in one dimension but the wave's energy density propagates in a 2 dimensional spherical surface of concentric spherical shells, the density decreases inversely with increasing distance away from the source; but quantum mechanically, wherever and whenever this wave interacts with matter, say an electron, it becomes a 3 dimensional photon with linear momentum given by its energy E divided by its speed c.