{"id":84,"date":"2006-07-21T14:57:37","date_gmt":"2006-07-21T14:57:37","guid":{"rendered":"http:\/\/scientopia.org\/blogs\/goodmath\/2006\/07\/21\/zero\/"},"modified":"2006-07-21T14:57:37","modified_gmt":"2006-07-21T14:57:37","slug":"zero","status":"publish","type":"post","link":"http:\/\/www.goodmath.org\/blog\/2006\/07\/21\/zero\/","title":{"rendered":"Zero"},"content":{"rendered":"

Back during the DonorsChoose fundraiser, I promised a donor that I’d write an article about the math of zero. I haven’t done it yet, because zero is actually a suprisingly deep subject, and I haven’t really had time to do the research to do it justice. But in light of the comment thread that got started around [this post][fspog] yesterday, I think it’s a good time to do it with whatever I’ve accumulated now.
\nHistory
\n———
\nWe’ll start with a bit of history. Yes, there’s an actual history to zero!
\nIn general, most early number systems didn’t have any concept of “zero”. Numbers, in early mathematical systems, were measurements of quantity. They were used to ask questions like “How much grain do we have stored away? If we eat this much now, will we have enough to plant crops next season?” A measurement of zero doesn’t really mean much; even when math is applied to measurements in modern math, leading zeros in a number – even if they’re *measured* – don’t count as significant digits in the measurement. (So if I’m measuring some rocks, and one weighs 99 grams, then that measurement has only two significant digits. If I use the same scale to weigh a very slightly larger rock, and it weighs 101 grams, then my measurement of the second rock has *three* significant digits. The leading zeros don’t count!) *(In the original version of this post, I managed to stupidly blow my explanation of significant digits, which several alert commenters pointed out. As usual, my thanks for the correction.)*
\nAristotle is pretty typical of the reasoning behind why zero wasn’t part of most early number systems: he believed that zero was like infinity: an *idea* related to numbers, but not an actual number itself. After all, you can’t *have* 0 of anything; zero of something isn’t *any* something: you *don’t have* anything. And you can’t really *get* to zero as he understood it. Take any whole number, and divide into parts, you’ll eventually get a part of size “1”. You can get to any number by dividing something bigger. But not zero: zero, you can never get to by dividing things. You can spend eternity cutting numbers in half, and you’ll still never get to zero.
\nThe first number system that we know of to have any notion of zero is the babylonians; but they still didn’t really quite treat it as a genuine number. They had a base-60 number system, and for digit-places that didn’t have a number, they left a space: the space was the zero. (They later adopted a placeholder that looked something like “\/\/”.) It was never used *by itself*; it just kept the space open to show that there was nothing there. And if the last digit was zero, there was no indication. So, for example, 2 and 120 looked exactly the same – you needed to look at the context to see which it was.
\nThe first real zero came from an Indian mathematician named Brahmagupta in the 7th century. He was quite a fascinating guy: he didn’t just invent zero, but arguably he also invented the idea of negative numbers and algebra! He was the first to use zero as a real number, and work out a set of algebraic rules about how zero, positive, and negative numbers worked. The formulation he worked out is very interesting; he allowed zero as a numerator or a denominator in a fraction.
\nFrom Brahmagupta, zero spread both east (to the Arabs) and west (to the Chinese and Vietnamese.) Europeans were just about the last to get it; they were so attached to their wonderful roman numerals that it took quite a while to penetrate: zero didn’t make the grade in Europe until about the 13th century, when Fibonacci (he of the series) translated the works of a Persian mathematican named al-Khwarizmi (from whose name sprung the word “algorithm” for a mathematical procedure). As a result, Europeans called the new number system “arabic”, and credited it to the arabs; but as I said above, the arabs didn’t create it; it originally came from India. (But the Arabic scholars, including the famous poet Omar Khayyam, are the ones who adopted Brahmagupta’s notions *and extended them* to include complex numbers.)
\nWhy is zero strange?
\n———————-
\nEven now, when we recognize zero as a number, it’s an annoyingly difficult one. It’s neither positive nor negative; it’s neither prime nor compound. If you include it in the set of real numbers, then they’re not a group – even though the concept of group is built on multiplication! It’s not a unit; and it breaks the closure of real numbers in algebra. It’s a real obnoxious bugger in a lot of ways. One thing Aristotle was right about: zero is a kind of counterpart to infinity: a concept, not a quantity. But infinity, we can generally ignore in our daily lives. Zero, we’re stuck with.
\nStill, it’s there, and it’s a real, inescapable part of our entire concept of numbers. It’s just an oddball – the dividing line that breaks a lot of rules. But without it, a lot of rules fall apart. Addition isn’t a group without 0. Addition and subtraction aren’t closed without zero.
\nOur notation for numbers is also totally dependent on zero; and it’s hugely important to making a polynomial number system work. Try looking at the [algorithm for multiplying roman numerals][roman-mult] sometime!
\nBecause of the strangeness of zero, people make a lot of mistakes involving it.
\nFor example, based on that idea of zero and infinities as relatives, a lot of people believe that 1\/0=infinity. It doesn’t. 1\/0 doesn’t equal *anything*; it’s meaningless. You *can’t* divide by 0. The intuition behind this fact comes from the Aristotelean idea about zero: concept, not quantity. Division is a concept based on quantity: Asking “What is x divided by y” is asking “What quantity of stuff is the right size so that if I take Y of it, I’ll get X?”
\nSo: what quantity of apples can I take 0 of to get 1 apple? The question makes no sense; and that’s exactly right: it *shouldn’t* make sense, because dividing by zero *is meaningless*.
\nThere’s a cute little algebraic pun that can show that 1 = 2, which is based on hiding a division by zero.
\n1. Start with “x = y”
\n2. Multiply both sides by x: “x2<\/sup> = xy”
\n3. Subtract “y2<\/sup>” from both sides: “”x2<\/sup> – y2<\/sup> = xy – y2<\/sup>”
\n4. Factor: “(x+y)(x-y) = y(x-y)”
\n5. Divide both sides by the common factor “x-y”: “x + y = y”
\n6. Since x=y, we can substitute y for x: “y + y = y”
\n7. Simplify: “2y=y”
\n8. Divide both sides by y: “2 = 1”
\nThe problem, of course, is step 5: x-y = 0, so step five is dividing by zero. Since that’s a meaningless thing to do, everything based on getting a meaningful result from that step is wrong – and so we get to “prove” false facts.
\nAnyway, if you’re interested in reading more, the best source of information that I’ve found is an online article called [“The Zero Saga”][saga]. It covers not just a bit of history and random chit-chat like this article, but a detailed presentation of everything you could ever want to know, from the linguistics of words meaning zero or nothing to cultural impacts of the concept, to detailed mathematical explanation of how zero fits into algebras and topologies.
\n[fspog]: http:\/\/scienceblogs.com\/goodmath\/2006\/07\/restudying_math_in_light_of_th.php
\n[saga]: http:\/\/home.ubalt.edu\/ntsbarsh\/zero\/ZERO.HTM
\n[roman-mult]: http:\/\/www.phy6.org\/outreach\/edu\/roman.htm<\/p>\n","protected":false},"excerpt":{"rendered":"

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