Definition of Negative Square Roots
Date: 03/08/2004 at 14:26:46 From: Linda Subject: Square Roots I know that the square root of 49 = 7 since 7 x 7 = 49. But the negative square root of 49 is -7. Is this because (-7) x (-7) also equals 49 or because the square root of 49 is 7 and the negative stays because it is not involved with the operation? The teacher wrote -SQRT(49) = -7 because (-7) x (-7) = 49.
Date: 03/08/2004 at 16:28:48 From: Doctor Peterson Subject: Re: Square Roots Hi, Linda. Both statements in English are valid; but what your teacher wrote in symbols is not quite right. When we say "the negative square root of 49," we may mean either of two things. The first option is "of the two square roots of 49, the one that is negative," which says that the negative number -7 is a square root of 49, and therefore is the negative square root of 49: (-7)*(-7) = 49 The second option is "the negative of the (principal) square root of 49," which is what we are saying when we write __ -\/49 or -sqrt(49) which is -(7) or -7 Since the radical symbol refers specifically to the principal square root, which is the non-negative one, the expression above means "take the principal square root, which is 7, and negate it to get -7." This is not a statement about what you get when you square -7. So when your teacher wrote (presumably using symbols) -sqrt(49) = -7 because (-7)*(-7) = 49 the first equation technically means only that -7 is the negative of the square root of 49, and not that -7 itself, squared, is 49. But the two facts are so closely related that it's hard to really say the teacher is wrong! The expression "-sqrt(X)," while it properly means "the negative OF THE square root," does always refer to the same number as "the negative square root," so it becomes natural to think of it as the latter. In that sense, the expression becomes sort of a mathematical idiom, as if "-sqrt" meant "the negative root." But it can be dangerous to teach it that way. Sometimes students see the quadratic formula and think that the "+/-" before the radical just means to take either sign for the root, and think that is multiplied by -b. In fact, the "+/-" means to add or subtract the (principal) root from -b. If we make it clear that the radical represents the positive root, and anything outside of that is just an operation on that number, such misunderstandings should not arise. Here is probably the best reason not to say it as your teacher did: If I use "cbrt" to mean the cube root, then it is true that -cbrt(343) = -7 but it is NOT true that (-7)^3 = 343 Here it is absolutely wrong to think of "-cbrt" as meaning "the negative cube root"; there is only one cube root, and it is positive! 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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