It is extremely difficult to believe that Descartes did not anticipate the consequences of his choices for a geometrical and hence, numerical system. The man was a consummate intellectual. A top dog among the big dogs of his age. And yet, there it is: the square root of negative one, which we designate as ‘i’ (to keep it from looking quite so ridiculous, I guess).
Like so many profound and dangerous concepts, it is deceptively small and unassuming. All numbers must have square roots, so negative one must have one also. However, due to the multiplication rules expressed in the previous chapter, it would appear that more special rules would be needed to create one for it.
Since a negative value multiplied by a negative value, by definition, yields a positive value, then because of those rules, no negative number can have a true square root consisting of two equal values within the Real Numbers.
It is worth noting at this point, how logically difficult the concept of less than zero actually is. How can you have less than none? Let’s look at some physical examples.
Take temperature, for example. It’s true that the commonly used temperature scales are (at least roughly) calibrated to the properties of water, and that both the Fahrenheit and Celsius scales have a zero value that is well within the range of human experience. But if you talk to a scientist or physicist, you’ll quickly be informed that there is an absolute zero. Temperature is a function of atomic and subatomic movement. Absolute zero is the point at which all atomic motion stops and there is no movement at all. None. Zero.
We have temperature scales that start at that zero point, one is called the Kelvin scale (after the famous Lord Kelvin) and it relates to Celsius in that its units are the same size. The other, that relates to the Fahrenheit scale, is called Rankin. Those scales have no negative values. They start with zero being absolute zero, and go up in value using positive numbers. Since the zero point refers to a physical state where there is no atomic movement, any number associated with temperature must imply movement, which is conceived to be a positive value. There simply cannot be a value that is defined as ‘less than no movement’. (Note: recent developments in cryogenic molecular physics have stated that temperatures less than absolute zero have been created. However, utilizing the definition that absolute zero is a point where no subatomic movement is present, this new state of matter does not qualify as being less than zero. Subatomic movement still exists in these states, it is just different, and the author would argue, inverse.)
Pressure? Unh-unh. Zero pressure means no pressure at all. Zero is at the bottom.
Energy? If anyone can discern and prove the existence of negative energy there’s a Nobel Prize in it for them, claims of ‘Dark Energy’ notwithstanding.
Gravity? We wish.
Time? Theories abound, and this will be discussed at more length later, but there exists no proof that there is, or can be a backward time vector.
Well, there is accounting….
So why is the concept of numbers that are less than zero so ubiquitous and pervasive? Unfortunately, this is one of many questions that this author is not prepared to answer. Numbers that are less than zero, except when presented on a number line, and only on a number line, just do not make sense in a physical way. Maybe it’s the power of that one classical example. Maybe it’s mass hysteria. Who can say? But the concept has led our most advanced thinkers, and really, all of the civilized world down a dangerous and misleading path, that has created more problems and ignorance than was ever anticipated.
The language constrains the discussion.
This statement relates strongly to the old adage that says “If all you have is a hammer, every problem begins to look like a nail”. If you accept negative numbers, and the flawed logic behind them, your solutions that use them will all be suspect, and it colors your overall thinking.
The language of negative numbers leads to thinking in ‘less than zero’ terms, and those terms are unable to communicate a full understanding of the subject matter, slanting the discussion and leading, many times, to inappropriate conclusions.
An outstanding example of this comes from the realm of physicists who are working on solutions to the theory of Relativity and the implications of those solutions. Without going into much detail, there is a small group working on novel constructs, specifically ‘worm holes’, that has found a solution that allows them to exist. It seems that if one had enough negative mass, one could, theoretically speaking of course, open up the end of a wormhole big enough to get a spaceship in. They even calculated the energy that it would take to create that negative mass, which was, as you would expect, a very large number.
There’s just so much to love in this example. How big is the spaceship? Where would the worm hole go? Who has ever seen one? But the best is the part about the negative mass. What is negative mass? How could you make it? How could you keep it? Does it exhibit negative gravity?
This is just the kind of conceptual problem that was being referenced. The very existence of negative numbers permits, even encourages this sort of wild speculation. And it’s not the speculation itself that is worrisome. All in all, speculation is a good thing as it stretches the boundaries of the imagination and can lead to surprising insights. But negative mass is just a goofy concept, like flying cars and turning lead into gold. This wasn’t published in a comic book, but rather from the halls and journals of esteemed academics. It is the mathematical language that allows this aberrant train of thought, and unwittingly, constrains and eliminates the consideration of other, more accurate and succinct ones.
But the concept of the square root of negative one has had an even bigger impact. Once it was realized that the number system demanded that there be square roots for negative numbers, the mathematicians finally gave in and allowed that ‘number’ to be included within the numerical system. However, since it is not a logical (or even provable) number, it was assigned the term ‘Imaginary Number’.
It works like this; suppose that you have a negative ‘real’ number like -4. If it were a positive number, it could easily be factored into two components, 2 * 2, giving 2 as the obvious square root. Similarly, -4 can be factored to be: -4 = 2 * 2 * -1. Taking the square root of this then gives a real component; ‘2’, and an imaginary component, the square root of negative one, ‘i’. This strategy allowed all negative numbers to have calculable square roots.
The square root of negative one, since it is difficult to write (and even more difficult to conceive) was given the designation ‘i’ and would be used as follows:
(-4) SQRT = 2i. Pretty handy, eh? No more messy writing or logic. The problem is that that, theoretically, it still can’t be integrated into the Real Numbers (at least they weren’t that crazy) and so a complete alternate universe of numbers was created: the Imaginary Numbers.
You just have to love the name. The Imaginary Numbers. The phrase brings all sorts of possibilities to mind. But in this context it was used to designate those values having a factor of the square root of negative one. The best part is that, similar to the way we are required to treat negative numbers, an imaginary number multiplied by an imaginary number yields a (negative) ‘real’ number, and the product completely loses its imaginary component. How convenient.
In and of itself, this wouldn’t have been so bad. If the Imaginary Numbers were only relegated to the realm of mathematical oddities discussed in topical journals, it might have been alright. But unfortunately, uses were found for this number in real physics. It turns out that when calculating the phase shifts in electrical design, if one uses the square root of negative one in your equations, you get answers that work in the physical world.
And it gets even worse than that. Unfortunately, as we shall see in the discussion of Quantum Mechanics, it has found its way into the foundation of what has become one of the most successful theoretical physics postulates ever proposed.
Modern mathematics, by adopting the language developed to support Cartesian geometry has boxed itself into a logical trap by ensconcing imaginary, illogical concepts into its vocabulary and in so doing, has obscured the true nature of the phenomena that it purports to describe.