Purpose of Zero
Date: 17 Nov 94 18:20:48 EST From: Mike Romanoff Subject: Question Dr. Math, What is the actual purpose of the number zero? Thank you, Regan Romanoff
Date: Sat, 19 Nov 1994 13:52:39 -0500 (EST) From: Dr. Ken Subject: Re: Question Hello there! What an excellent question! I'm glad to see you're really thinking about the fundamental concepts behind mathematics. The invention of Zero was one of the most important breakthroughs in the history of civilization. More important, in my opinion, than the invention of the wheel. I think that it's a fairly deep concept. One crucial purpose that Zero holds is as a placeholder in our system of notation. When we write the number 408, we're really using a shorthand notation. What we really mean by 408 is "4 times 100, plus 0 times 10, plus 8 times 1." Without the number zero, we wouldn't be able to have a placeholder in the tens place, and we wouldn't be able to tell the numbers 408, 48, 480, 408000, and 4800 apart. So yes, zero is important. Another crucial role that zero plays in mathematics is that of an "additive identity element." What this means is that when you add zero to any number, you get the number that you started with. For instance, 5 + 0 = 5. That may seem obvious and trivial, but it's actually quite important to have such a number. For instance, when you're manipulating some numerical quantity and you want to change its form but not its value, you might add some fancy version of zero to it, like this: x^2 + y^2 = x^2 + y^2 + 2xy - 2xy = x^2 + 2xy + y^2 - 2xy = (x + y)^2 - 2xy Now if we wanted to, we could use this as a proof that (x + y)^2 is always greater than 2xy; the expression we started with was positive, so the one we ended up with must be positive, too. Therefore, subtracting 2xy from (x + y)^2 must leave us with a positive number. Neat stuff. Anyway, if you're interested in finding out more about the number zero, the field of study you should look into is called "Group Theory", a substudy of Modern Abstract Algebra. Thanks for the question! -Ken "Dr." Math
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