
Complex Analysis Complex Analysis Math Forum 
 LinkBack  Thread Tools  Display Modes 
March 1st, 2017, 12:50 PM  #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, 02:38 PM  #2 
Global Moderator Joined: Dec 2006 Posts: 16,590 Thanks: 1199 
No.

March 1st, 2017, 05:21 PM  #3 
Global Moderator Joined: May 2007 Posts: 6,203 Thanks: 486  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, 07:21 PM  #4  
Senior Member Joined: May 2016 From: USA Posts: 515 Thanks: 230  Quote:
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, 06:30 PM  #5 
Math Team Joined: Jan 2015 From: Alabama Posts: 2,279 Thanks: 570 
"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. 

Tags 
gain, integers, intergers, negative, rooting, square 
Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
Manipulation of negative square root of a negative term/#  daigo  Algebra  3  June 30th, 2012 07:06 AM 
Non negative integers  Keroro  Algebra  3  June 9th, 2012 06:30 PM 
why the product of two negative integers is positive  fantom.1040  Algebra  7  June 28th, 2011 05:22 PM 
Integers: Subtracting Negative Numbers  Mighty Mouse Jr  Algebra  7  May 11th, 2010 04:29 AM 
Cardinality of the set of sequences of non negative integers  Artur  Real Analysis  1  November 8th, 2007 11:52 AM 