Numbers in a FractionDate: 05/20/2001 at 11:55:09 From: M. Lawrence Subject: Is a fraction 1 number or is it 2 numbers? Hi, I'm a bit stuck on some schoolwork to do with fractions. Is a fraction one number or is it two numbers? What I've figured out is that a fraction is made up of two numbers, the denominator and the numerator. For example 1/7. The denominator is 7 and the numerator is 1 - representing a fraction. So would this 'fraction' be considered one number representing one quantity but made up of two quantities? On the other hand, it is representing the process of 1 divided by 7. So it would then be two numbers wouldn't it? Can it be both 1 and 2 numbers? Very confused. Thank you for your time. M. Date: 05/20/2001 at 19:53:25 From: Doctor Ian Subject: Re: Is a fraction 1 number or is it 2 numbers? Hi M, If you say that 'Jack is the first child of Bob and Sally', you're talking about one person, even though you have to mention two other people to talk about him, right? When you divide 3 by 4, you get a particular number. It's the same number you get when you divide 6 by 8, or 9 by 12, or 75 by 100. Those are all different pairs of numbers, but the number '3/4' is a single number, which happens to have a lot of names. Does this help? Write back if you'd like to talk about this some more, or if you have any other questions. - Doctor Ian, The Math Forum http://mathforum.org/dr.math/ Date: 05/20/2001 at 22:52:43 From: Russell Lawrence Subject: Re: Is a fraction 1 number or is it 2 numbers? So the fraction 3/4 is one number representing one quantity but has many different names (3/4 is 6/8 is 9/12 etc.) So because 6/8 is another name for 3/4 it is a process of dividing 6 by 8 to equal 3/4, or a process of simplifying it to equal 3/4. So is 6/8 a process? So 3/4 then, is one number representing one quantity, and there is no process to derive it because it already has a given quantity. So 3/4 is a single number and not a process? Am I on the right track? Are there any fractions that have only one name? Lastly, is there a fraction for every number on the number line? Thanks, M. Date: 05/21/2001 at 00:18:16 From: Doctor Ian Subject: Re: Is a fraction 1 number or is it 2 numbers? Hi M, Your last two questions are the easiest to answer. There are no fractions with only one name, since you can always multiply a fraction by n/n to get a new name, e.g., 3 2 6 - * - = - 4 2 8 3 3 9 - * - = -- 4 3 12 3 4 12 - * - = -- 4 4 16 And there are many numbers on the number line that can't be expressed as fractions. For example, pi can't be expressed as a fraction. Nor can e (the base of the natural logarithms). Nor can the square root of 2, the square root of 3, or the square root of any prime number. These are all called 'irrational' numbers, because they aren't rational, i.e., they can't be expressed as fractions. Numbers are very subtle things, as you'll discover if you continue to ask the kinds of questions you're asking. We tend to think of integers as the sizes of sets - i.e., the number '7' is a label we use to signify the size of every set that contains 7 elements. It sounds circular, but it's not, since we can construct the sets without naming them. But if numbers are supposed to be set sizes, then what sets have 'sizes' like 3/4, or the square root of 2? That doesn't make sense, so we start to think of rational numbers in terms of subsets. But that doesn't work for irrational numbers, so we visualize those as distances along an imaginary 'number line'. (Then we see that this will also work for rational numbers, so we combine the rationals and irrationals to get the 'real' numbers.) And what about negative numbers? Well, we add a sign, and visualize it as a direction along the line. Then what about complex numbers? Okay, we add another dimension. You seem to be assuming that there is some distinction to be made between a 'number' and a 'process'. But is this really the case? We determine set sizes by counting, or more accurately, by matching corresponding elements in sets. Counting and matching are certainly processes, so is a set size a number, or a process? We define certain classes of numbers in terms of the kinds of equations that they satisfy. Solving an equation is certainly a process, so are those solutions numbers, or processes? When Warren McCulloch, one of the great minds in the theory of computation, was asked upon entering Haverford College what he wanted to do, he replied, "I have no idea, but there is one question I would like to answer: What is a number, that a man may know it, and a man, that he may know a number?" No one has answered that question yet, but a lot of people have had a lot of fun trying! If you're going to join the party (and I hope you will), one thing you'll have to learn to do is distinguish between a thing and its representations. Whatever this thing '3/4' is, it's not any of its representations - '3/4', '3 divided by 4', '4 into 3', '0.75', 'xxx_', a point on a number line, and so on - any more than you're the same thing as your name or a photograph or description of you. And it certainly can't be the process that gives rise to it, since there are lots of processes that can give rise to any number, so if number is the same as process, then all of the processes that give rise to the same number must be the same, which is only possible if we dilute the meaning of 'same' until the word is meaningless. People don't think much about what numbers really _are_, because you don't need to know what they are in order to use them - in much the same way that you don't need to understand what fire is in order to cook with it. The short answer to the question 'is a fraction one number or two?' is really just: A fraction is a representation of a single number, which happens to be constructed from two other numbers. If you want to go deeper than that, take a look at Was Mathematics Invented or Discovered? http://mathforum.org/dr.math/problems/angela.1.1.01.html to see where that kind of discussion can lead. Does this help? Write back if you'd like to talk about this some more, or if you have any other questions. - Doctor Ian, The Math Forum http://mathforum.org/dr.math/ Date: 05/21/2001 at 01:00:31 From: Russell Lawrence Subject: Re: Is a fraction 1 number or is it 2 numbers? Thank you very much for your time and detailed responses, it has helped a great deal and is very much appreciated. M. |
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