Functionality of the Square Root
Date: 4/27/96 at 15:26:23 From: Gunnar Jakob Briem Subject: Re: Disputing the "functionality" of the square root... Hello. I am a student of Philosophy at the University of Iceland, and I have been quarrelling with my logic teacher on a rather simple matter. On an introductory course in logic we recently discussed the "formal qualities of relations". I don't know if my translation into English makes much sense, but what I mean by it is simply a discussion of the various relations between subjects, i.e. transitive relations, symmetric relations, reflexive relations etc. An example of a relation between A and B is that of B being a function of A. By a function I (and presumably everyone else) mean a relation whereby A is connected exclusively to B (I hope my clumsy phrasing doesn't preclude any chance of making sense to you). Now, to get to the point, I maintain that the relation indicated by the statement: SQRT(x) = 3D y (limited to real numbers) is a function, whereas my teacher claims it is not. In an attempt to put an end to his folly, I quoted the math book I used in high school. There, it is stated that SQRT(x) = 3D |y| i.e. the absolute value of y which means that it is necessarily a positive number. Also, it shows a graph of the function f(x) = 3D SQRT(x) with both x and f(x) only taking positive values (as seems obvious to me - otherwise it wouldn't be a function). Now, that didn't convince him, far from it. He quoted 'A Dictionary of Science' (Penguin 1964) as saying: "...the square root is one of two equal factors; e.g. 9 =3D 3 * 3 =3D -3 * -3; hence SQRT(9) is +/- 3..." He also quoted 'Hutchinsons Popular Encyclopaedia' (Random 1990) as saying: "...square root: in mathematics, a number, which when squared equals another given number. For example, the square root of 25 is +/- 5 because 5 * 5 =3D 25 and -5 * -5 =3D 25." Now, I have no problem with those definitions of the square root. However, as I see it, the statement: SQRT(x) = 3D y = 20 cannot yield a negative number on the right side, as the left side cannot be negative. Therefore, this statement indicates that relation between x and y, which is called a function. Whose folly is it, mine or my teacher's? (I hope this isn't too nonsensical for you.) Kind regards, Gunnar J. Briem
From: Doctor Steven I wouldn't worry too much about your English, you have quite a good grasp of it. In answer to your question, you're both right. In mathematics the Sqrt symbol is used to mean the positive real root of a number. So if I said Sqrt(9), I would mean 3. Unfortunately, this can get confusing, since Sqrt(9) actually can take on two values. Say we have the equation x = 3D Sqrt(9) and we want to find x. We square both sides to get x^2 = 3D 9. Move the nine to the opposite side to get x^2 - 9 = 3D 0. Factor this and we have (x-3)(x+3) = 3D 0, which tells us that both x = 3D 3, and x = 3D -3 solve the problem. Another way to look at this is by graphing the function f(x) =3D Sqrt(x=). If we mean all roots we get a parabola like this. | / | / |/ ------------C--------------- |\=20 | \ | \ Well... by using the equation f(x) = 3D Sqrt(x), haven't I limited myself to the set of positive real numbers? (Since the right side of the equation can yield a negative number iff the square root symbol is defined as including the +/- symbol, which to my knowledge it doesn't.) I checked a mathematics dictionary (James & James) at the library and there the square root is defined thus: A number which, when multiplied by itself, produces the given number. There are always two of these. The sign before an indicated square root of a positive number indicates which root is meant: Sqrt(4) = 3D 2, -Sqrt(4) = 3D -2 and +/-Sqrt(4) = 3D +/-2. So according to this definition, the equation which was the source of our debate, namely Sqrt(x) = 3D y is a function. I would think that the scientific community is in agreement on how to define the symbol Sqrt, that is, for my purpose, whether it includes the +/- symbol, and if it doesn't, then I am right. Right? And so we can tell its not a function by the vertical line test (if a vertical lines intersects the graph at more than one point anywhere on the graph it's not a function). But if we mean only the positive values we get half a parabola, in fact only the part above the x-axis, so a vertical line test will show this to be a function. So it all depends on what you take Sqrt to mean. I hope I have cleared things up somewhat, or at least added some fuel to the fire :). -Doctor Steven
From: Briem Thank you for tolerating my nagging. Gunnar Jakob Briem Netfang: email@example.com
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