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 Complex Analysis Complex Analysis Math Forum

 March 1st, 2017, 11:50 AM #1 Newbie   Joined: Feb 2017 From: Netherlands Posts: 8 Thanks: 2 Math Focus: Trigonometry and complex numbers Square rooting to gain negative integers Are there any (complex, real etc.) numbers that satisfy the following equation? $\displaystyle \sqrt{x} < 0$ March 1st, 2017, 01:38 PM #2 Global Moderator   Joined: Dec 2006 Posts: 20,919 Thanks: 2203 No. Thanks from Xxmarijnw March 1st, 2017, 04:21 PM   #3
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 Originally Posted by Xxmarijnw Are there any (complex, real etc.) numbers that satisfy the following equation? $\displaystyle \sqrt{x} < 0$
The square root operation always has two solutions. However by custom the square root of a positive real number is the positive root. However the negative square root is there, but by custom is ignored. March 1st, 2017, 06:21 PM   #4
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 Originally Posted by mathman The square root operation always has two solutions. However by custom the square root of a positive real number is the positive root. However the negative square root is there, but by custom is ignored.
I express this differently.

A positive real number always equals the square of two distinct numbers, one positive and one negative but with equal absolute values.

By definition, $\sqrt{a} > 0\ if\ a > 0.$

So what is the negative root? $-\ \sqrt{a}.$ March 11th, 2017, 05:30 PM #5 Math Team   Joined: Jan 2015 From: Alabama Posts: 3,264 Thanks: 902 "The" square root, of a positive real number, a, is defined to be the positive number whose square is a. But this question is asking for a complex number either of whose square roots is a negative real number. Writing the complex number in polar form, where r is a positive real number and is between 0 and , the square roots are and . Either of those is a negative real number only if or . Those give and which are really the same positive real number. But that puts us back to the "positive real number" case where is, by definition, the positive root. Thanks from Xxmarijnw Tags gain, integers, intergers, negative, rooting, square Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post daigo Algebra 3 June 30th, 2012 07:06 AM Keroro Algebra 3 June 9th, 2012 06:30 PM fantom.1040 Algebra 7 June 28th, 2011 05:22 PM Mighty Mouse Jr Algebra 7 May 11th, 2010 04:29 AM Artur Real Analysis 1 November 8th, 2007 10:52 AM

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