Conventions of Square Roots

Date: 11/24/98 at 17:00:35
From: Andrew
Subject: Square Roots

On a review problem in my Sequential 3A math class, I had the square 
root of 81 equals positive or negative 9. My math teacher said it was 
wrong, that if there is no negative before the whole square root 
problem, only the positive answer is wanted. Who is right?

-Andrew

Date: 11/24/98 at 17:56:58
From: Doctor Pat
Subject: Re: Square Roots

Andrew,

The convention in mathematics is that x^2 = 81 has two solutions, 
x = 9 or x = -9, but when we take the square root of a value, we want 
the principal square root. For real numbers, that is the positive 
value. 

It is important that you know that there are two values that may be 
squared to give every real number (even the negatives), but it is also 
important that you understand that we may want to make a distinction 
between all the particular roots of an equation, and the request to 
give a particular solution. Your teacher clearly wants you to give the 
positive square root of the value when the square root symbol is used. 

If situations come up in which there is ambiguity, we need to develop a 
mutual consent between us as to how we would use certain symbols; 
otherwise no one would know what I wanted when I wrote 3x4 because you 
would be unsure what I intended the symbol x to represent. You accept 
those conventions every day without thinking. You assume problems are 
in base ten form unless otherwise stated, for example, and that ratios 
are written with the numerator on top. But once it was common to write 
3.4 for 3/4. 

Hope this helps.  

- Doctor Pat, The Math Forum
  http://mathforum.org/dr.math/