Significance of Rational NumbersDate: 01/11/2003 at 11:24:12 From: Chirag Subject: Significance of rational numbers We know that any rational number can be constructed using integers, and that an irrational number can be constructed using a sequence of rational numbers, but isn't that entirely due to the way rational numbers are defined? Why are rational numbers defined the way they are? Date: 01/12/2003 at 12:05:01 From: Doctor Kastner Subject: Re: Significance of rational numbers Hi Chirag - Why are the rationals defined the way they are? That's an interesting question. Let me address it from a formal point of view, and to do that we need to talk about abstract algebra a little. Abstract algebra is the branch of mathematics that studies number systems and their associated operations. An example will help make this clearer. A group (G) is a set of elements, together with a binary operator (.) that satisfies the following properties: 1) Closure. If a and b are in G, a.b is in G 2) Associativity: If a,b,c are in G, then a.(b.c) = (a.b).c 3) Identity. There is an identity element (I) in G such that a.I = a for every element in G 4) Inverse. For each element a in G, there is an inverse b = a^(-1) such that a.(a^(-1)) = I I should make it clear that the a^(-1) does not necessarily mean 1/a. We are dealing with generic inverses here. Groups are all around us in math. The classic example is the set of integers together with the binary operator +. It is clear that it is closed and associative, and the identity is 0, while the inverse of a is -a. But notice that the integers with the operator * do NOT form a group. Multiplication of the integers satisfies the first three properties (the identity is 1), but given an element a, the inverse element 1/a is not always an integer. Even worse, what if the number you want to find an inverse of is 0? Other topics of study in abstract algebra include (in order of increasing number of properties) rings and fields. In addition, rings and fields require two operations (+ and * if we are talking about the rationals), not just one. Groups, rings and fields are all connected, and it is common for mathematicians to start with a group and try to expand it into a ring or a field. The group of integers under + forms a group, but suppose I want to make a field using that as my base. First, I know that I need another operation, in this case *. Both operators in a field must also satisfy the group properties (except for a multiplicative inverse for the 0 element), and we also need commutivity (a+b = b+a, a*b= b*a) and the distributive property (a*(b+c) = a*b+a*c)). Those extra ones aren't bad, but I still don't have the right set of numbers for the multiplicative inverse. If I consider the integer a, my field must also contain the inverse, 1/a. So let's put those in the set. But since this should be closed, I need to include the numbers b*1/a in my field too. Now put those in. You can see where this is going. The result is the rational numbers. It turns out that the rationals are the simplest field that can be constructed using the integers as a starting point. Finally, I should mention that the set of irrational numbers does not form a field (or even a group for that matter) since it isn't closed under +. Pi and its additive inverse -pi are both in the collection, but their sum is 0 which is not an irrational number. I hope this helps. Write back if you're still stuck, or if you have other questions. - Doctor Kastner, The Math Forum http://mathforum.org/dr.math/ Date: 01/16/2003 at 12:23:50 From: Chirag Subject: Thank you (significance of rational numbers ) Thanks so much for answering my question so promptly. It was really very satisfying to get such a comprehensive answer. |
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