The Number Zero

The Number Zero

I think the number zero should be outlawed.  Zero is not only useless, it is downright dangerous.  This number is a nuisance and a blight upon society.

Copyright by Married To the Sea. Reproduced here by permission.

The number zero wasn't always with us. For thousands of years we got along just fine without it. True, some calculations were very difficult without it. Try multiplying with Roman Numerals, and you'll see what I mean. Why does V + V = X? Doesn't really make much sense. Using zero makes calculations much easier.

The first use of zero was apparently by an Indian mathematician by the name of Brahmagupta, who was born about 600 C.E. He somehow conceived the notion that some quantity, minus itself, would equal a number. That number was zero. It doesn't sound like a big deal, but it was innovative at the time, to treat "nothing" as just another number. Note that zero is not actually "nothing". If you have 10 each of apples and oranges, and you remove 10 apples, you have zero apples, but you still have oranges. You're not left with "nothing".

Zero does have its uses. Our whole way of representing numbers relies on zeroes to hold places that are supposed to be empty. We use an imaginary arrangement of columns that, from right to left are 1, 10, 100, etc. Each column is 10 times the one preceding it. $1,000,000 is different from $1. We can tell the difference because the zeroes show us exactly where the 1 needs to be placed to show the empty columns. So in the context of receiving a check, I acknowledge that the more zeroes after the first digit, the better.

The Malignity of Zero Rears Its Offensive Head

Ah, but there is a fly in the ointment, one little property that zero has, that no other number has. That alone should make us suspicious about its validity. You can't divide with it. This fact is glossed over when they teach us how to calculate. We're told we can do anything we like - multiply, divide, add, subtract - with any numbers at all, except we cannot divide by zero. Division by zero is considered "undefined". It's not "infinity", though often it is said to be that; it is undefined. You can't divide by zero. It has no meaning in mathematics, neither number nor infinity nor anything else. Undefined. It is the only number that behaves in that way. Out of all the infinity of numbers, countable and uncountable, natural, integer, real, irrational, complex, transcendental, and whatever other kinds there are, only one single number can't divide into another number. Zero.

Does anyone have a problem with this? Apparently not. Mathematicians are at ease with zero, as are accountants and everyone else. I'm not a mathematician, so the fact that I have a problem with zero counts for very little - almost zero, in fact. What right do I have to object, when every mathematician in the world acknowledges zero's right to exist?  At least here I can complain about it, fat lot of good it does me.

I said mathematicians are OK with zero, but that's not quite true. Throughout all of math, zero lurks around and bites mathematicians in places best left to the reader's imagination. You can't use the logarithm of zero, because that results in a division by zero. You can't take the tangent of 90°, for the same reason. You have to watch out with your equations, to make sure nothing can ever equate to dividing by zero. So you get something like (x+1)/(x-1), and you can have any value for x - oh, except for 1. Because if x = 1, then (x-1) = 0, and you can't divide by zero. So if your equation comes out to look like this, you then have to qualify it by saying "x <> 1" (those funny brackets are how you say "not equal to", when you don't have the right character on your keyboard).

The Theory of Limits

There is another place where zero gives trouble, but that also appears to have been glossed over somewhat. It is with the concept of limits. Take the simple series of fractions, 1 + 1/2 + 1/4 + 1/8 + ..., The next fraction is 1 divided by double whatever that last number was. You can take this out as far as you want. Each new term is half the previous one, and as you go on the new value quickly becomes very small. You get something like 1/4,294,967,296, and that comes out to be about 0.00000000232, which is really tiny. You can see that as the series goes on, the new numbers become almost zero - but never quite zero. Not in any finite number of terms, anyway. Ending the series with three dots is shorthand for indicating that this is an infinite series.

You might think that since you're adding an infinite number of terms, the end sum would be infinite, but it's not. The number never exceeds 2. In fact, the number never reaches 2, in any finite number of steps. The number 2 is said to be the "limit" of the series, as the number of terms approaches infinity. The idea is that if you somehow could add all the infinite number of terms, it would add up to exactly 2. The way mathematicians approach this difficulty is to say that the series will approach 2 as closely as you like. Pick any number less than 2 - say, 1.99999999. Carry out the sums enough times, and it will be closer to 2, than the number I named. This applies to any number I name that is less than 2. So, if I write 1.999999999, you can always make a number that is larger than that, but still less than 2. Just add another 9, for example. And another, and another. No matter how many nines I write, you add one and cut the distance to 2 to a tenth of its former amount. Over and over and over again, without ever stopping. Still, though the difference is minute - one often says it's "negligible" - it's not zero. It will never be zero.

In this particular series, the limit is said to be 2. The series will never exceed 2. Personally, I don't think mathematicians have really shown that this series ever reaches 2. Common sense says that this must be the case - it's "obvious", in some ways. But is it mathematically true?

This concept of limits is vital to mathematics. It is the basis of much of the calculus, for example. For hundreds of years, the idea of limits has been used successfully to prove many important theorems in mathematics, which in turn has helped advance science - in particular, theoretical physics. Calculating orbits, trajectories of satellites, how stars and molecules move, how to build bridges and skyscrapers, all rely on concepts that are based on limits. And the math works. Our rockets fly (more or less), eclipses and planets move according to our predictions, and almost everyone is happy. Except me. I think something is fishy.
Actually, some others have been suspicious of the idea of limits and of infinitesimal amounts adding up to some specific number. Mathematician Niels H. Abel, for instance, said:

If you disregard the very simplest cases, there is in all of mathematics not a single infinite series whose sum has been rigorously determined. In other words,the most important parts of mathematics stand without a foundation.

Another mathematician, Bishop George Berkeley, wrote:

And what are these same evanescent Increments? They are neither finite Quantities nor Quantities infinitely small, nor yet nothing. May we not call them the Ghosts of departed Quantities?

I like that guy. Unfortunately, Berkeley died in 1783, and he may have been the last person to feel this way.

These people were certainly in the minority, though. Most mathematicians seem to be able to accept the concept of limits without choking.

Still, there is a glimmer of hope for me in physics. Here is where things actually get interesting. Physicists don't believe in zero, either. Well, they do, but not quite as something having a physical reality. It's something of a long story...

The Ultraviolet Catastrophe

About 100 or so years ago, physicists were faced with a bit of a problem concerning black body radiation. Experiments were performed to measure the heat of so-called "black bodies". A black body is a theoretical object that perfectly radiates or absorbs energy. Without getting too technical, these "black bodies" were heated until they glowed, and the intensities of the colors they emitted at various temperatures were noted. Curves of these intensities could be drawn to show where the energy peaked. The following graph shows what the curves looked like:

This image is published under the Gnu Free Documentation License

You don't have to understand the details of the graph, just note that the curve starts out low, rises to a maximum, and then goes back down again. It doesn't keep going up, but returns almost to zero. This is what physicists found when they did the measurements in their labs.

There was a classical physical equation that predicted what the curves should be for various temperatures. It works fairly well for lower temperatures - more towards the red end of the spectrum. Unfortunately, with higher temperatures the equation failed miserably.

This image is published under the Gnu Free Documentation License

Note how the graph of the equation shows the curves going off to infinity as the number of K increases.  The prediction doesn't match the results found in the experiments.  This is a serious problem, because the equation is basic.

The crux of the problem was that the equation was a fraction, and the denominator could approach zero. The denominator consisted of a single number, the fourth power of the light's wavelength. The equation was predicting enormous amounts of energy when the light was in the ultraviolet range - energies far beyond what were observed. Something was terribly wrong. This problem was referred to as the "Ultraviolet Catastrophe", because the equation tended to fall apart as the light's color moved in that direction. Many people weren't happy about this, because the math is reasonably clear, yet produces impossible results at higher frequencies.

This Ultraviolet Catastrophe was one of the problems in classical physics that led to the quantum theory. Max Planck, while working on a different problem, had conceived of the notion that the wavelengths never reach zero, can never fall below a certain number. He considered this merely a convenient fiction, but when he used this idea in his equations, many problems disappeared. He called this unit of energy a 'quantum', meaning a quantity or an amount. Note that there was no mathematical justification for this move. He didn't arrive at it through careful calculations. He simply made it a requirement. Despite this lack of foundation, the idea works beautifully. Eventually the notion of quanta was applied to the Ultraviolet Catastrophe problem. Doing so caused the equations to yield curves in agreement with experiment.

There was much furor over this and other quantum ideas. There really was no mathematical or theoretical justification for making things have to exist in 'quanta'. It just worked, which may be OK in engineering, but wasn't so great in theoretical physics. Einstein, who was actually one of the early developers of quantum theory, struggled for the rest of his life trying to disprove it somehow. Many classical physicists were outraged over this seemingly ad hoc solution. No one was very happy about it, but applying quantum ideas to classical physics continued to yield stunning, accurate results. Most physicists decided that you can't argue with success, and accepted the quantum theory uneasily.

Werner Heisenberg was an early quantum theorist who developed the Uncertainty Principle. There are several ways of expressing this non-mathematically, but the gist of it is that you can't know the precise location of a particle, and also its precise momentum. The more you know one, the less you know the other - and you can never know either of these exactly. There is always a minumum error. The best you can do is to have a fairly good idea of where it is, and a reasonable idea of its momentum.

"Big deal", you say. Who cares where a particle is, anyway? Well, the point is that, if you can't say where a particle is, you can't say where it isn't, either. So if you try to say that some region of space is empty, you can't know that. In fact, there is something known as a "quantum vacuum", in which particles continually come into existence, and wink out almost immediately. Empty space isn't empty after all - it's boiling with what are called "virtual particles". Meaning, there's nowhere that there are zero particles. No empty space. No zero.

The consequences of this notion of quanta are far-reaching. There are certain physical units called "Planck Units" that represent the minimum meaningful value for that particular quality. For instance, there is the Planck Length, which is about 1.6 x 10-35 meters; the Plank Time, which is about 5.4 x 10-44 seconds; and so on. These are incredibly tiny values. They are so minute that they are irrelevant for almost all purposes. They make even subatomic particles look bigger than solar systems by comparisons. But, and this is very important, they aren't zero. There is no meaningful distance shorter than the Planck Length - you can't say something is 1/2 that length, because that has no physical meaning. Similarly, there is no shorter time than a Planck Time. What this means is, speaking of zero length, zero seconds, etc., is physically meaningless.

Since lengths or times smaller than the Planck Units have no physical meaning, we do not have situations in which zero occurs to mess up our equations. Even if we have a length in the denominator, it's OK - the smallest it can ever reach is the Planck Length, extremely tiny, but non-zero. The equation may yield some spectacularly huge result, but it is at least defined. We get a number, not a 'divide by zero' error. This is a Good Thing.