Multiplying Square Roots of Negative Numbers
Date: 11/25/2003 at 04:28:09 From: Justin Subject: sqrt(-2) * sqrt(-2) =2 or -2 ? It seems to me that there are two ways to do a problem where you are multiplying square roots of negative numbers. Here's an example: Method 1: sqrt(-2) * sqrt(-2) = sqrt((-2)*(-2)) = sqrt(4) = 2 Method 2: where i = sqrt(-1) sqrt(-2) * sqrt(-2) = sqrt(2)i * sqrt(2)i = 2(i^2) = -2 Which solution is correct? What's wrong with the other approach? Did I miss any mathematics concept?
Date: 11/25/2003 at 09:36:02 From: Doctor Ian Subject: Re: sqrt(-2) * sqrt(-2) =2 or -2 ? Hi Justin, What you've missed is that the square root returns two values. To make it easier to see what's going on, let's use a perfect square. Because we can square either (2i) or (-2i) to get -4, we have to consider several cases: sqrt(-4) * sqrt(-4) = (2i) * (2i) -> -4 - OR - (-2i) * (-2i) -> -4 - OR - (2i) * (-2i) -> 4 - OR - (-2i) * (2i) -> 4 Each of your derivations considers one of these cases separately, thus losing the connection between them. Here's one way to think of it: Each person has a parent() function, which returns two possible values. Suppose the parents of Pat are Chris and Jean. Then your original question was sort of like this: Method 1: parent(Pat) + parent(Pat) = Chris + Chris Method 2: parent(Pat) + parent(Pat) = Jean + Jean Should we then conclude that Chris is the same as Jean? No. Should we conclude that there is a problem with the definition of the parent function? No. Should we conclude that we have to be very careful with multi-valued functions? Bingo! Does this help? - Doctor Ian, The Math Forum http://mathforum.org/dr.math/
Date: 11/25/2003 at 08:57:03 From: Doctor Peterson Subject: Re: sqrt(-2) * sqrt(-2) =2 or -2 ? Hi, Justin. The problem is that the question is bad! When we talk about complex numbers, there are two square roots; we can't define "the" square root, the principal root, which in the real numbers would be the positive one. So the symbol "sqrt(-2)" doesn't have the meaning you expect; it has two values, not just one. And the rule that sqrt(ab) = sqrt(a)sqrt(b) is not true any more. Take a look at these pages for more explanations and comments: Multiplying Radicals of Negative Numbers http://mathforum.org/library/drmath/view/53873.html Simplifying Complex Numbers http://mathforum.org/library/drmath/view/62973.html If you have any further questions, feel free to write back. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/
Date: 11/25/2003 at 09:52:38 From: Justin Subject: Thank you (sqrt(-2) * sqrt(-2) =2 or -2 ?) Thanks for your quick response! It helps a lot.
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