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### Square root of a negative number

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Date: Wed, 23 Nov 1994 12:58:34 -0800
From: (Paul Pragin)

Is it theoretically possible to calculate the square root of a negative
number?
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```
Date: Mon, 28 Nov 1994 16:51:23 -0500 (EST)
From: Dr. Sydney

Hi there!  Thanks for writing to Dr. Math.  Taking the square root of a
negative number is a problem that mathematicians struggled with for a
very long time.  It was not until the 1600's and 1700's that the great

mathematicians really got a firm grasp on what the square root of a
negative number is.  Why were there so many problems?  From the basic
axioms that describe the real numbers we see that a positive real times a
positive real yields a positive real and a negative real times a negative
real yields a positive real.  So, in the real numbers it is impossible to
square a number and get a negative number.

So, a new field had to be introduced: the complex field.  Do you know what
complex numbers are?  The square root of -1 was given the name i (i is for
imaginary).  All numbers that are square roots of negative numbers can be
written as a real number times i.  Do you see how this works?  How would
you write the square root of -5 in terms of i?

I hope this helps answer your question.  If you have any more

--Sydney, a math doctor
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Associated Topics:
High School Negative Numbers

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