Equation without a SolutionDate: 11/14/2001 at 18:25:37 From: Steven and Richard Subject: Sqrt(x) = -2, order of operations, and other problems with the square root What is the solution to the equation sqrt(x) = -2 ? Some people say that there is no solution, since the square root is defined to be the side of a square with a given area. We allow for negative area by allowing these imaginary numbers. If I were to have a negative side the area would most certainly be imaginary, or would it? We thought of looking at sqrt(4i^4). Using the laws of exponents first, the answer is most definitely -2. Using the laws of imaginary numbers first, the answer is 2. Is there an order of operations for which law to use first? Where is the breakdown in the square root? I have ceased calling the square root a function because 4 = 4i^4 and y = sqrt(x) seems to be providing me with two real solutions. If we allow negative area, don't we have to allow negative sides? Date: 11/14/2001 at 22:40:10 From: Doctor Peterson Subject: Re: Sqrt(x) = -2, order of operations, and other problems with the square root Hi, Steven. When you start thinking about negative or imaginary numbers, it's best to think algebraically rather than geometrically. A square root of a number is defined as any number whose square is that number. A positive number has two square roots, one positive and the other negative. When we write the radical sign (which we're representing as "sqrt"), that represents only the POSITIVE square root of the number; so the two square roots of 4 are called sqrt(4) and -sqrt(4). We do this so that the square root is a function with one value, not something that somehow takes two values at once. Therefore, your equation sqrt(x) = -2 has no solutions, simply because by definition sqrt(x) is always positive. This has nothing to do with imaginary numbers, which are the solutions of equations like x^2 = -2 Although these look similar, they have no real solutions for entirely different reasons. Going back to geometry, if a square could have sides with negative lengths (which it can't), then the area would still be positive, since the product of two negative numbers is positive. Only with imaginary sides (?) could a square have a negative area. Now, when you move into complex numbers, there is NO square root FUNCTION. You're exactly right: any definition that chooses a single square root for every complex number will be inconsistent, in the sense that sqrt(ab) = sqrt(a) sqrt(b) will not always be true. It's only a fortunate convenience that a square root function over the reals can be defined. This means that you can't look to the complex numbers for a solution to your equation, because any equation involving square roots must allow only reals. You will be interested in these pages that deal with various issues you have raised: Square Root of 100 - Dr. Math archives http://mathforum.org/dr.math/problems/cry.04.15.99.html What is i? - Dr. Math archives http://mathforum.org/dr.math/problems/toeti24.html False Proofs, Classic Fallacies - Dr. Math FAQ http://mathforum.org/dr.math/faq/faq.false.proof.html (see the link near the bottom to "1 = 2: A Proof using Complex Numbers"). - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ |
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