Functionality of the Square Root

Date: 4/27/96 at 15:26:23
From: Gunnar Jakob Briem
Subject: Re: Disputing the "functionality" of the square root...

Hello.

I am a student of Philosophy at the University of Iceland, and I
have been quarrelling with my logic teacher on a rather simple matter.

On an introductory course in logic we recently discussed the
"formal qualities of relations".  I don't know if my translation
into English makes much sense, but what I mean by it is simply
a discussion of the various relations between subjects, i.e.
transitive relations, symmetric relations, reflexive relations etc.

An example of a relation between A and B is that of B being a
function of A.  By a function I (and presumably everyone else) mean
a relation whereby A is connected exclusively to B (I hope my clumsy
phrasing doesn't preclude any chance of making sense to you).

Now, to get to the point, I maintain that the relation indicated by
the statement:

   SQRT(x) = 3D y   (limited to real numbers)

is a function, whereas my teacher claims it is not.
In an attempt to put an end to his folly, I quoted the math book
I used in high school.  There, it is stated that

       SQRT(x) = 3D |y|

i.e. the absolute value of y which means that it is necessarily
a positive number.  Also, it shows a graph of the function

       f(x) = 3D SQRT(x)

with both x and f(x) only taking positive values (as seems obvious
to me - otherwise it wouldn't be a function).

Now, that didn't convince him, far from it.  He quoted 'A Dictionary
of Science' (Penguin 1964) as saying:

   "...the square root is one of two equal factors; 
   e.g. 9 =3D 3 * 3 =3D -3 * -3; hence SQRT(9) is +/- 3..."

He also quoted 'Hutchinsons Popular Encyclopaedia' (Random 1990)
as saying:

   "...square root: in mathematics, a number, which when squared
   equals another given number.  For example, the square root of
   25 is +/- 5 because 5 * 5 =3D 25 and -5 * -5 =3D 25."

Now, I have no problem with those definitions of the square root.

However, as I see it, the statement:

       SQRT(x) = 3D y = 20

cannot yield a negative number on the right side, as the left side
cannot be negative.  Therefore, this statement indicates that relation
between x and y, which is called a function.

Whose folly is it, mine or my teacher's?

(I hope this isn't too nonsensical for you.)

Kind regards,

Gunnar J. Briem

From: Doctor Steven

I wouldn't worry too much about your English, you have quite a good 
grasp of it.  

In answer to your question, you're both right.

In mathematics the Sqrt symbol is used to mean the positive real root 
of a number.  So if I said Sqrt(9), I would mean 3.

Unfortunately, this can get confusing, since Sqrt(9) actually can take 
on two values.  Say we have the equation x = 3D Sqrt(9) and we want to 
find x. We square both sides to get x^2 = 3D 9.  Move the nine to the 
opposite side to get x^2 - 9 = 3D 0.  Factor this and we have 
(x-3)(x+3) = 3D 0, which tells us that both x = 3D 3, and x = 3D -3 
solve the problem.

Another way to look at this is by graphing the function 
f(x) =3D Sqrt(x=). If we mean all roots we get a parabola like this.

                                  |    /
                                  |  /
                                  |/
                      ------------C---------------
                                  |\=20
                                  |  \
                                  |    \

Well... by using the equation f(x) = 3D Sqrt(x), haven't I limited 
myself to the set of positive real numbers? (Since the right side of 
the equation can yield a negative number iff the square root symbol is 
defined as including the +/- symbol, which to my knowledge it 
doesn't.)

I checked a mathematics dictionary (James & James) at the library 
and there the square root is defined thus:

	A number which, when multiplied by itself, produces the given
	number.  There are always two of these.  The sign before an
	indicated square root of a positive number indicates which root
	is meant:  
     Sqrt(4) = 3D 2,  -Sqrt(4) = 3D -2  and  +/-Sqrt(4) = 3D +/-2.

So according to this definition, the equation which was the source of 
our debate, namely

	Sqrt(x) = 3D y

is a function.  I would think that the scientific community is in
agreement on how to define the symbol Sqrt, that is, for my
purpose, whether it includes the +/- symbol, and if it doesn't, then
I am right.  Right?

And so we can tell its not a function by the vertical line test (if a
vertical lines intersects the graph at more than one point anywhere on 
the graph it's not a function).

But if we mean only the positive values we get half a parabola, in 
fact only the part above the x-axis, so a vertical line test will show 
this to be a function.

So it all depends on what you take Sqrt to mean. I hope I have cleared
things up somewhat, or at least added some fuel to the fire :).

 -Doctor Steven

From: Briem

Thank you for tolerating my nagging.

Gunnar Jakob Briem	
Netfang: gjb@rhi.hi.is