What is an Integer?Date: 10/03/2002 at 11:42:19 From: Patty Anderson Subject: Integer definition Why isn't -1/2 considered an integer? You can certainly have negative fraction situations in real life. My class would like this explained. Thank you. Date: 10/03/2002 at 12:27:13 From: Doctor Peterson Subject: Re: Integer definition Hi, Patty. How has your text defined integers? They must have given a very faulty definition, if any, in order for the class to ask this question, but I suspect that many students have this confusion. I have observed that integers are often introduced in a way that gives an entirely wrong impression of what they are: that any negative number is an integer. (They may not even understand that natural numbers are integers too; the set of integers extends the set of natural numbers by including negative numbers and zero.) To a mathematician, the fact that an integer can be negative is unimportant; what makes an integer an integer is that it is WHOLE. That's the root of the word "integer," which in Latin means whole (as in "integrity," and the related words "intact" and "entire"). Here's what's going on: in introducing the various types of numbers to students, schools typically start with the natural numbers, then bring in zero, then add fractions, and only then introduce the idea of negative numbers. (I believe they should be introduced long before that, because the concept is really very simple.) Both whole numbers and fractions can have a negative sign put on them, but integers are specifically the positive and negative whole numbers. But since, for some reason, negative numbers are usually introduced using the concept of integers (as if students had not yet learned about fractions), students imagine that any number with a sign ought to be an integer. That's not true. A mathematician sees numbers in a different way. We may start with the natural numbers in an axiomatic development, but then immediately move to the integers, by introducing the concept of sign. (This is done so that any equation of the form x+a=b can be solved.) Then we introduce the rational numbers (fractions), so that equations of the form ax+b=c can be solved. Finally we introduce the irrational numbers (which together with the rational numbers constitute the real numbers), and then the complex numbers, so that any equation can be solved. Looking backward, once we know about all these kinds of numbers, the integers are simply the "whole" numbers among the reals, as the rationals are those real numbers which can be written as a fraction; the property of having a sign is nothing special. The natural numbers are the only set that does not have signs. So here are the main sets of numbers your students should be aware of: natural numbers: 1, 2, 3, ... integers: ..., -3, -2, -1, 0, 1, 2, 3, ... rational numbers: any ratio of an integer to a natural number, such as -3/5 real numbers: the whole number line If you have any further questions, feel free to write back. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ |
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