Simplifying and Working with Imaginary NumbersDate: 04/11/2008 at 11:05:07 From: L Subject: simplifying imaginary square roots I gave a question on a test that said to simplify the square root of 50 over the square root of -5. The answer I accepted was -i times the square root of 10 but one of my students claims he is right with the answer of i times the square root of 10. I have spoken with many other math teachers and we don't all agree. Please help. Thanks. Date: 04/11/2008 at 11:37:35 From: Doctor Ian Subject: Re: simplifying imaginary square roots Hi L, There doesn't seem to be universal agreement on whether sqrt(4) is equal to just 2, or equal to both 2 and -2. This site, MathWorld - Square Root http://mathworld.wolfram.com/SquareRoot.html uses "principal square root" to refer only to the positive one; I think that makes sense, but it still leaves open the question of convention, i.e., when someone writes "sqrt(x)", does this default to the principal square root, or not? I generally go for the more inclusive definition--especially once imaginary numbers are being considered, since the actual definition of i is not i = sqrt(-1) but rather i^2 = -1 However, as you've found, not everyone agrees. Having said that, it seems to me that the student's reasoning is as important as the answer he gave. Did he explain how he got his answer? I would simplify the expression this way. If I use psr(x) to refer to the principal square root, then sqrt(50) -------- sqrt(-5) sqrt(5) * sqrt(10) = ------------------ sqrt(5) * sqrt(-1) sqrt(10) = -------- sqrt(-1) +/- psr(10) = ----------- +/- psr(-1) psr(10) -psr(10) psr(10) -psr(10) = ------- OR -------- OR ------- OR -------- i i -i -i which is to say, psr(10) +/- ------- i So personally, I'd be inclined to mark it wrong if EITHER answer was omitted. :^D Does this help at all? - Doctor Ian, The Math Forum http://mathforum.org/dr.math/ Date: 04/11/2008 at 12:43:53 From: Doctor Peterson Subject: Re: Thank you (simplifying imaginary square roots) Hi, L. I'd like to add a little to what Dr. Ian said. A lot depends on what the student has been taught--what does your text do with square roots of negative numbers? Although it is common to take sqrt(x) to mean the principal root when talking about real numbers only (roots of positive numbers), there is no such thing when working with complex numbers, so if (as usual) you want any expression to have only one value, you really have to refuse to define sqrt(-5). (The alternative is to take Dr. Ian's position and take sqrt as a multiple-valued function, which is not generally done at an elementary level.) The reason for this is that if you take the square root of a negative number as having a single value, then no matter what rule you choose to select that value, there will be some cases where it is not true that sqrt(ab) = sqrt(a)sqrt(b), which is a highly desirable rule to have. That means that problems involving such a root are technically invalid. However, it seems to be common for texts to define sqrt(-x) = i sqrt(x) when x>0, so that they are allowed to write such roots, and then tell students that when they see such an expression, they must IMMEDIATELY translate it to the form on the right, BEFORE applying any rules such as the invalid product rule above, or the equivalent quotient rule. That makes sqrt(-5) essentially no more than a shorthand for i sqrt(5), rather than an expression you can manipulate by the usual rules. So, under these rules, your problem works this way: sqrt(50) sqrt(25)*sqrt(2) 5 sqrt(2) * i sqrt(5) 5 sqrt(10) -------- = ---------------- = --------------------- = ---------- i sqrt(-5) i sqrt(5) i sqrt(5) * i sqrt(5) -5 = -i sqrt(10) You could also (with care) do it this way: sqrt(50) sqrt(50)*sqrt(-5) sqrt(2)*sqrt(25)*sqrt(5)*i -------- = ----------------- = -------------------------- = ... sqrt(-5) sqrt(-5)*sqrt(-5) -5 But you CAN'T do this: sqrt(50) sqrt(50)*sqrt(-5) sqrt(-250) sqrt(25)*sqrt(-10) -------- = ----------------- = ---------- = ------------------ = sqrt(-5) sqrt(-5)*sqrt(-5) sqrt(25) sqrt(25) = sqrt(-10) = i sqrt(10) The error is in the simplification in the denominators where an invalid rule is invoked. So only your answer is correct if you have taught that you can treat sqrt(-5) as i sqrt(5). If you haven't dealt with this issue at all, then the question itself is really invalid. The student's answer is not really valid (giving only the one answer as correct), but he can't be blamed for it if he hasn't been taught to be careful with this kind of problem. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ Date: 04/11/2008 at 15:01:33 From: L Subject: Thank you (simplifying imaginary square roots) Excellent answer. The student was taught that the sqrt(-5) is i times the sqrt(5), so thank you. |
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