NOTE: if you haven't heard of imaginary numbers or complex numbers, I would suggest doing a quick Google or YouTube search and familiarizing yourself
with this topic at least a little bit bit before you attempt to read this thread. A great starting point for the beginner is
This is a topic which, although extremely interesting, hasn't received much attention from ATS members and researchers. One reason for this is
probably because the topic is so complex and abstract, not to mention most mathematicians will admit we know barely anything about imaginary numbers
or whether they really even exist. Some people argue we shouldn't even use them because they are made up nonsense with no basis in reality.
However one might also argue that negative numbers have no basis in reality because you can't have "negative something". Yet clearly mathematicians
agree that negative numbers are "real" because they can easily be visualized, where as it's supposedly impossible to visualize an imaginary unit. Now,
I think it's important to consider what "negative something" really means. When you think about it, the phrase "something" can be reduced to
According to Einstein we can really just think of matter as one possible form of energy. In fact modern cosmology is quickly adopting the concept of
"negative energy". There are theories which predict the total amount of energy in the universe is 0 because all the negative energy cancels out all
the positive energy. In essence the theory is an attempt to answer the question of why does something exist rather than nothing.
So there may in fact be a physical manifestation of "negative energy" which only further confirms the mathematical foundations of reality. When you
realize that all particles are really particle-waves and can be expressed with a wave function you realize just how math-based our reality is. So
getting back to the main topic at hand... what does any of this have to do with imaginary numbers you ask? Well it should be obvious.
Perhaps these imaginary numbers aren't so imaginary after all and they really do have some basis in reality. After all, imaginary numbers are crucial
to electronic engineering and quantum mechanics. These so called imaginary numbers also appear to have a basis in the real world, which is quite
extraordinary if you ask me. Consider the reasons for why we started to use imaginary numbers in the first place.
It was to fill a gap, a discrepancy between what we could express algebraically and what we could solve algebraically. Put simply: we had equations
which were impossible to solve without introducing imaginary numbers. For example the square root of -1 or any negative number for that matter cannot
be solved without imaginary numbers. Now it's obvious why the square root of -1 cannot be solved if you think about it.
The square root is the opposite of the square of a number, so to find the square root you must find a number which when multiplied by itself equals
the number you are looking for the square root of. However, any positive number multiplied by its self equals a positive number and any negative
number multiplied by its self equals a positive number. There is absolutely no number on the real number line which is the square root of -1.
This is why many people argue against the use of imaginary numbers, their use violates the known rules of mathematics and introduces units which
shouldn't even exist. However there was also a time when any solution to an equation which resulted in a negative number was considered to be the
"wrong" answer. It's important to keep in mind that imaginary numbers are merely an algebraic tool to solve equations, and the rules for using them
However, the algebraic tool required is an "extra-dimensional" number system which exists outside of the real number line and contains all the
imaginary numbers. Now what is the real number line? Well imagine 0-dimensions as a point with no width, height, or depth. Now if we give our point
width it becomes a 1-dimensional line. The real number line is simply a 1-dimension line which extends infinitely in both directions.
We pick an arbitrary point on the line and call it 0. To the left of 0 we have the negative range of numbers and to the right of 0 we have the
positive range of numbers, extending to infinity of course. So we can clearly visualize negative numbers in this way. On the other hand we cannot
visualize imaginary numbers in a similar way. Now the next step would be to stretch the line out vertically and give it height, creating a
At this point you might be thinking "well what about the complex plane, that's a 2D plane with an imaginary axis and a real axis and it gives us a way
to visualize imaginary numbers". My response to that thinking would be that you are confused, the complex plane plots complex numbers
complex numbers have both a real part and an imaginary part. That is not the same thing as plotting out the structure of an imaginary number.
The spacings on the imaginary line are completely arbitrary and mean nothing because we have no way of visualizing an imaginary quantity. The
expression (2i < 3i) is not even necessary true. If imaginary numbers really are some type of multi-dimensional numbers with multiple components, like
a vector with 2 components, we should be able to apply simple vector math to them and also visualize them in a nice way.
You might say all real numbers would exist on the "event horizon" of the imaginary number space, meaning that all real numbers would have a y
component equal to 0. However there is no vector which when multiplied by itself results in a vector where x=-1 and y=0 (aka the real number -1). We
run into the same problem, a negative vector multiplied with a negative vector is a positive vector. I wish it were that easy, but if it were it would
have been figured out already.
There is a deeper secret to the mystery of the imaginary numbers. Personally I don't believe imaginary numbers represent a multi-dimensional number.
The math just doesn't work out any way you try it, these numbers don't appear to operate like normal numbers. In fact the answer may be that our
mathematical operators apply differently to imaginary numbers. It's the only way to actually reach the square root of -1 when you think about it.
i^2 = i x i = -1
i^3 = -1 x i = -i
i^4 = -i x i = 1
i^5 = 1 x i = i
The pattern above will repeat on, so i^6 is -1 and i^7 is -i, and so on. Now think about how absurd that is. You can take the real number 1 to the
power of anything and the answer will always be one, but with the imaginary unit that is not the case. Each time we multiply i by itself we shift
through negative and positive, real and imaginary values. It's as if each exponential operation on the imaginary number has some sort of repeating
recursive behavior causing it to shift between real and imaginary.
The only way we can explain this and possibly hope to visualize imaginary numbers is to understand and define the strange operational rules that apply
to imaginary numbers. Perhaps a good analogy might be to picture another universe where the laws of physics behave completely differently. It appears
that we have inadvertently stumbled across a new universe of numbers where our old rules don't apply the same way.
And yet the most mysterious thing is that these imaginary numbers seem to have a real world manifestation and basis in reality. One must question
whether our mathematical rules are incomplete or built on a faulty premise. If we have missed something important the mysterious world of imaginary
numbers may just be the road to completing our understanding of mathematics, and thusly understanding the true nature of reality.
11/6/2013 by ChaoticOrder because: (no reason given)