{"id":47,"date":"2010-02-02T02:23:03","date_gmt":"2010-02-02T07:23:03","guid":{"rendered":"http:\/\/random-musings.org\/?p=47"},"modified":"2024-09-21T22:31:32","modified_gmt":"2024-09-22T03:31:32","slug":"the-number-zero","status":"publish","type":"post","link":"https:\/\/random-musings.org\/index.php\/2010\/02\/02\/the-number-zero\/","title":{"rendered":"The Number Zero"},"content":{"rendered":"\n

The number zero is an annoying and dangerous beast. It should be abolished.<\/span><\/p>\n\n\n

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Copyright Married to the Sea. Reproduced here by permission.<\/span><\/p>\n\n\n\n\n\n\n\n

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The number zero hasn\u2019t always been with us. For thousands of years \u2013 millions \u2013 we got along without it. It\u2019s true that some calculations are more difficult without it. Try multiplying Roman numerals, for example. V x VI = XXX? There wasn\u2019t much you could do about it back then.<\/span><\/p>\n\n\n\n

The first use of zero was apparently by an Indian mathematician named Brahmagupta, who was born about 600 C.E. He conceived the notion that some quantity, minus itself, must equal a number. That number was zero. It doesn\u2019t sound like a big deal, but it was innovative in its time \u2013 to treat \u201cnothing\u201d as just another number. Note that zero isn\u2019t actually \u201cnothing.\u201d If you have ten each of apples and oranges, and take 10 apples away, you have zero apples \u2013 but you still have oranges. You\u2019re not left with nothing.<\/span><\/p>\n\n\n\n

Zero has its uses. In our way or representing numbers, zero is a placeholder for empty numbers. We know that 1000 is bigger than 100, because of the zeroes. The more zeroes after a number, the bigger it is. So on a paycheck, zeroes are OK.<\/span><\/p>\n\n\n\n

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The Malignity of the Number Zero Rears Its Offensive Head<\/span><\/strong><\/p>\n\n\n\n

Ah, but there is a fly in the ointment, one little property that zero has, that no other number does. This unique property should make us suspicious. You can\u2019t divide with it. This fact was glossed over when they taught us how to calculate. We learned how to multiply, divide, add, and subtract any numbers with any others \u2013 except, we cannot divide by zero.<\/span><\/p>\n\n\n\n

Division by zero is considered \u201cundefined.\u201d It\u2019s not infinity, though it\u2019s often said to be. It\u2019s undefined. Division by zero has no meaning in mathematics, neither number nor infinity nor anything else. Undefined. It\u2019s the only number that behaves this way. Out of the infinitudes of numbers, countable and uncountable, natural, integer, real, complex, transcendental, and whatever other kinds there are, only one number can\u2019t divide into another. Zero.<\/span><\/p>\n\n\n\n

Does anyone have a problem with this? Apparently not. Mathematicians seem OK with it, accountants, everyone else. Well, almost\u2026<\/span><\/p>\n\n\n\n

I said mathematicians are OK with zero, but that\u2019s not quite true. Throughout all of mathematics, zero lurks around and bites mathematicians right in the asymptote if they\u2019re not careful. You can\u2019t use the logarithm of zero, because that involves a division by zero. You can\u2019t take the tangent of 90\u00b0 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 have an equation like 1\/(X-1), where X can have any value \u2013 oh, except for 1. If X = 1, then X-1 = 0, and you can\u2019t divide by zero. So if you get an equation like this, you have to qualify it by saying, \u201cX \u2260 1.\u201d That is, X is not equal to one.<\/span><\/p>\n\n\n\n

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The Theory of Limits<\/span><\/strong><\/p>\n\n\n\n

There is another place where zero gives trouble, but this also appears to have been glossed over somewhat. It\u2019s the concept of limits. Take the simple series of fractions: 1\/2 + 1\/4 + 1\/8 + \u2026 (The three dots indicate that this series goes on forever). You keep doubling the bottom number (the denominator). You can take this out as far as you want. Each new term (fraction) is half the previous one. As you can imagine, the numbers get small very quickly. After a few dozen terms the numbers are almost zero \u2013 but not quite. They\u2019ll never be zero, no matter how many times you repeat this process.<\/span><\/p>\n\n\n\n

Since we\u2019re adding all these terms, and we\u2019re summing an infinite number of terms, you\u2019d think that the sum would be infinity. It\u2019s not. Believe it or not, the sum never exceeds 1. In fact, the sum never reaches 1 in any finite number of steps. The number 1 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 up all the infinite number of terms, the sum would add up to exactly one. But you can\u2019t add up an infinite number of terms. And for any finite number of terms, that last term never equals zero.<\/span><\/p>\n\n\n\n

The way mathematicians get around this is to say that the series approaches 1 as closely as you want. Pick a number less than 1 \u2013 say, 0.99. That\u2019s pretty close to 1. If you add up enough of those terms, you\u2019ll get a number that\u2019s closer to 1 than that. No matter how close a number you pick, the series can come even closer. It\u2019s almost as if the last term really becomes zero \u2013 but of course, it doesn\u2019t. The last term is never zero.<\/span><\/p>\n\n\n\n

In this series, the limit is said to be 1. It will never be greater than 1. Personally, I don\u2019t think mathematicians have shown that it ever reaches 1. Common sense says it must be so \u2013 in some ways, it\u2019s \u201cobvious.\u201d But is it mathematically true?<\/span><\/p>\n\n\n\n

The concept of limits is vital to mathematics. It\u2019s the basis of much of the calculus, for example. For hundreds of years the concept of limits has been used successfully to prove many important theorems in mathematics \u2013 which in turn has helped advance science \u2013 in particular, theoretical physics. Calculating orbits, the trajectories of satellites, how stars and molecules move, how to build bridges and skyscrapers, all rely on concepts based on limits. Our rockets fly (more or less), eclipses and planets move according to our predictions, and everyone\u2019s happy. Except me.<\/span><\/p>\n\n\n

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Niels H Abels<\/span><\/span><\/a><\/p>\n\n\n\n

Actually a few others have had their doubts about limits. Niels H. Abels said:<\/span><\/p>\n\n\n\n

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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.<\/span><\/p>\n<\/blockquote>\n\n\n

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Bishop George Berkeley<\/span><\/span><\/a><\/p>\n\n\n\n

Another mathematician, Bishop George Berkeley wrote:<\/span><\/p>\n\n\n\n

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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?<\/span><\/p>\n<\/blockquote>\n\n\n\n

I like that guy. Too bad he died in 1783. I can\u2019t really count on him for backup.<\/span><\/p>\n\n\n\n

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Yet Another Problem<\/span><\/strong><\/p>\n\n\n\n

The notion of limits might almost be acceptable, except for two concepts called supremum and infimum. The mathematical definitions of these words are tiresome, so I\u2019ll skip all that and present an example, which would concern the supremum:<\/span><\/p>\n\n\n\n

Consider what\u2019s the biggest number that\u2019s still less than a given number, say 1. That\u2019s the supremum. If we\u2019re talking about integers (whole numbers), then that number is 0. There is no integer greater than 0 that is also less than 1. If we\u2019re talking about numbers with decimal places, like 0.99, it\u2019s not possible to give an answer. No matter what number you choose, I can name another number that\u2019s bigger, but still less than 1. If you say 0.9, I\u2019ll say 0.99. If you say 0.99, I\u2019ll say 0.999. And so on. I can always tack on another 9 at the end and get a larger number that\u2019s still less than 1.<\/span><\/p>\n\n\n\n

Does this look familiar? It\u2019s the same idea as limits. You can approach 1 as closely as you like, so the limit of this procedure is 1. But we\u2019re looking for a number that\u2019s less than 1. Now we\u2019ve got a problem.<\/span><\/p>\n\n\n\n

Mathematicians avoid this whole mess by simply saying that this number doesn\u2019t exist. Problem solved. Personally I think that\u2019s cheating.<\/span><\/p>\n\n\n\n

Still, there\u2019s a glimmer of hope for me in physics. Here is where things actually get interesting. Physicists don\u2019t believe in zero, at least as far as its having a physical meaning. It\u2019s something of a long story\u2026<\/span><\/p>\n\n\n\n

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The Ultraviolet Catastrophe<\/span><\/strong><\/p>\n\n\n\n

About 100 years ago, physicists were faced with a serious problem concerning black body radiation. A \u201cblack body\u201d is a theoretical object that perfectly absorbs or emits energy (called radiation). Practical (imperfect) black bodies were heated until they glowed. Measurements were taken of the different colors of light emitted at different temperatures. The data was plotted onto graphs, which looked like this:<\/span><\/p>\n\n\n

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Graph of Black Body Data As Measured<\/p>\n\n\n\n

This doesn\u2019t hold any surprises. You perform an experiment, get the results, plot them, and have a nice graph. You can see how the lines start out low, rise, reach a maximum, and go back down again. This is the graph they got from their labs.<\/span><\/p>\n\n\n\n

There was a classic physics equation that predicted what the curve should look like for different temperatures. It worked reasonably well for the lower temperatures, giving about the same curve as they got in the labs. The equation failed miserably when applied to the higher temperatures. The equation should have produced the same graph as the lab results did. Unfortunately, its graph looked like this:<\/span><\/p>\n\n\n

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Graph of Black Body Radiation As Calculated<\/span><\/span><\/span><\/p>\n\n\n\n

You can see that some of the curves just run right off the graph. Those were the ones for the higher temperatures. At high temperatures \u2013 towards the ultraviolet \u2013 the calculated energy was enormous, approaching infinity. This was a serious problem because the equation was fundamental.<\/span><\/p>\n\n\n\n

The crux of the problem is that the equation was a fraction, and the denominator could approach zero. As the temperature increases, the denominator gets smaller, and the calculated energy becomes ridiculously high \u2013 far beyond what was found in the labs. According to the equation, a body should radiate an infinite amount of energy. This made a lot of people very unhappy.<\/span><\/span><\/a><\/span><\/p>\n\n\n\n

The 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 the idea that the wavelengths of light could never reach zero, could never fall below some number. Planck considered this a convenient fiction, but when he used it in his equations, many problems disappeared and the equations started giving correct results. He called this minimum wavelength a quantum, which means a quantity or amount.<\/span><\/p>\n\n\n

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Max Planck<\/figcaption><\/figure>\n<\/div>\n\n\n

There was no mathematical justification for this move. He didn\u2019t arrive at it through careful calculations. He simply made it a requirement. Despite its lack of foundation, the idea worked beautifully. Eventually the notion of quanta was applied to the Ultraviolet Catastrophe problem. This led to equations that yielded curves in agreement with experiment.<\/span><\/p>\n\n\n\n

There was much furor over this and other quantum ideas. There really was no mathematical or theoretical justification for requiring energy or matter to exist in \u2018quanta\u2019. It just worked. This is fine in engineering, but it\u2019s not so great in theoretical physics. Einstein, whose work helped to start the quantum revolution, spent the rest of his life trying to disprove it. Many classical physicists were outraged over this seemingly ad hoc solution. No one was very happy about it, but it kept yielding stunning, accurate results. Most physicists decided that you can\u2019t argue with success, and accepted the quantum theory uneasily.<\/span><\/p>\n\n\n

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Werner Heisenberg<\/span><\/span><\/strong><\/a><\/p>\n\n\n\n

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\u2019t know the precise location of a particle, and at the same time know its precise momentum (or motion). The more you know about one, the less you know about the other, and you can\u2019t know either one exactly. There\u2019s always a minimum error.<\/span><\/p>\n\n\n\n

One consequence of the Uncertainty Principle is that you can\u2019t say that any region of space is empty. In fact, there is a phenomenon known as \u201cquantum space\u201d or \u201cquantum foam\u201d in which particles continually pop into existence and wink out almost immediately. Empty space isn\u2019t empty after all \u2013 it\u2019s boiling with \u201cvirtual particles.\u201d Meaning, there\u2019s nowhere that there are zero particles. No truly empty space. No zero.<\/span><\/p>\n\n\n\n

The consequences of this notion of quanta are far-reaching. There are certain physical units called \u201cPlanck Units\u201d that represent the minimum meaningful value for certain physical properties. For instance, there is the Planck Length, which is about 1.6 x 10-35<\/sup> meters; Planck Time, which is 5.4 x 10-44<\/sup> seconds; and so on. These are incredibly tiny values, irrelevant for almost all purposes. They\u2019re the smallest possible values for these properties. You can\u2019t say that something is shorter than the Planck Length. Speaking of zero length or zero time is physically meaningless.<\/span><\/p>\n\n\n\n

This means that we don\u2019t have situations in which zero occurs to mess up our equations. We can\u2019t divide by zero because zero has no physical meaning. This is a Good Thing\u2122.<\/span><\/p>\n\n\n\n

So quantum theory has spared me from the ravages of the number zero.<\/span><\/p>\n","protected":false},"excerpt":{"rendered":"

The number zero is an annoying and dangerous beast. It should be abolished. Copyright Married to the Sea. Reproduced here by permission.<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[30],"tags":[],"class_list":["post-47","post","type-post","status-publish","format-standard","hentry","category-rants"],"_links":{"self":[{"href":"https:\/\/random-musings.org\/index.php\/wp-json\/wp\/v2\/posts\/47","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/random-musings.org\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/random-musings.org\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/random-musings.org\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/random-musings.org\/index.php\/wp-json\/wp\/v2\/comments?post=47"}],"version-history":[{"count":21,"href":"https:\/\/random-musings.org\/index.php\/wp-json\/wp\/v2\/posts\/47\/revisions"}],"predecessor-version":[{"id":1068,"href":"https:\/\/random-musings.org\/index.php\/wp-json\/wp\/v2\/posts\/47\/revisions\/1068"}],"wp:attachment":[{"href":"https:\/\/random-musings.org\/index.php\/wp-json\/wp\/v2\/media?parent=47"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/random-musings.org\/index.php\/wp-json\/wp\/v2\/categories?post=47"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/random-musings.org\/index.php\/wp-json\/wp\/v2\/tags?post=47"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}