Square Root of a Negative Number
Date: 01/25/97 at 19:04:00 From: Dustin Pedlar Subject: Square root of negative numbers Is it possible to find the square root of a negative number and if so, to what number system do these square roots belong? Could you please explain how this number system works?
Date: 01/25/97 at 20:47:34 From: Doctor Wallace Subject: Re: Square root of negative numbers Hi Dustin! As you no doubt know, a square root is a number that when multiplied by itself is equal to a given number. For example, 4 is the square root of 16, since 4 x 4 = 16. Note, however, that -4 x -4 = 16, too. We call 4 the positive square root of 16, and -4 the negative square root of 16. Now, you want to know if we can find the square root of a negative number. Let's take -16. We need to find a number, call it x, such that: x times x (x^2) = -16 Now, we know that any number times itself must be positive, not negative. Therefore, there is no such number x in the set of real numbers. A number x is defined, however, in the set of complex numbers. The complex numbers are a superset of the real numbers. That is, the complex numbers form a bigger set. The reals are a subset of the complex. A complex number has the form a + bi, where a and b are real numbers and the i is a special number. The "a" is called the real part; the "bi" is called the imaginary part. If we let a equal 0, then we have an imaginary number. The set of imaginary numbers is also a subset of the complex numbers. If we let b equal 0, then we have a regular real number. This is why the reals are a subset of the complex: the reals are just complex numbers that all have b=0, that is, no imaginary part. Now, the number i is defined to be equal to the square root of -1. This means that i^2 (i squared) is equal to -1. So now we can find the square root of -16. Since -16 = (-1) 16, we can write: sqr(-16) = sqr(-1) times sqr(16) (property of square roots) sqr(-16) = i times 4 This is usually written as 4i. We can check by squaring 4i. We get 4 x 4 = 16 times i x i = sqr(-1) times sqr(-1) = -1, giving 16 times -1 or -16. There is much more to the complex and imaginary sets of numbers than I can go into here. There are entire branches of mathematics (like complex analysis) which deal with these numbers. For more information, consult the math section of your local library or check out this page in our FAQ: http://mathforum.org/dr.math/faq/faq.imag.num.html -Doctor Wallace, The Math Forum Check out our web site! http://mathforum.org/dr.math/
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