Natural Numbers, Positive IntegersDate: 04/07/97 at 10:25:22 From: Eric Manalastas Subject: Natural Numbers, Positive Integers My Algebra book says that there is a difference between the set of natural (counting) numbers, i.e. {1,2,3,...} and the set of positive integers, which should be {1,2,3,...}. The book doesn't elucidate, saying that that discussion would be too difficult for the reader. I can detect no obvious difference. I know zero is neither positive nor negative so it's not an element of the latter. What gives? Date: 04/07/97 at 15:36:06 From: Doctor Wilkinson Subject: Re: Natural Numbers, Positive Integers The distinction is one that isn't going to matter even to mathematicians most of the time. It matters only when we're concerned with what's called "the foundations of mathematics." A number of mathematicians in the late nineteenth century started to wonder about how mathematics could be built up from the simplest possible foundations. The starting point was what we call the "natural numbers." They're called "natural" because they're so universal and apparently so obvious that they seem to be given to us by nature. Everybody in every culture has always known how to count. But not every culture has always had negative numbers, fractions, etc. Now if you start with just the natural numbers, you don't have any zero or any negative numbers. Zero and negative numbers were really invented by mathematicians so they could do subtraction without having to think about whether or not the number they were subtracting was smaller than the number they were subtracting from. All this happened a very long time ago, of course. But these nineteenth-century mathematicians started worrying about where these new numbers came from and they figured out a way to manufacture the new numbers out of the natural numbers. The new set of numbers they constructed were the integers (positive, negative, and zero). And lo and behold, the positive integers, that is those new numbers which happened to be greater than zero, turned out to form a system that looked exactly like the old system of natural numbers. It had exactly the same properties, and its smallest member behaved exactly like the natural number 1, even though it was built by a somewhat complicated construction and was by no means identical to the natural number 1. Similar methods were used to build rational numbers out of integers and real numbers out of rational numbers. At each stage of the process the numbers of the previous stage acquire images in the new set of numbers which behave exactly like their predecessors but are technically not identical to them. Once this whole structure has been built up, we can pretty much forget about it and go on with our lives, treating the natural numbers, the integers, and the rational numbers as all just parts of the bigger structure of real numbers. -Doctor Wilkinson, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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