Plus or Minus Sign

Date: 03/08/2002 at 17:59:07
From: Kim
Subject: Algebra 1 (equations)

What does this equation mean: y = + k 
                                  -

The - sign is directly under the + sign.

Date: 03/09/2002 at 12:54:31
From: Doctor Ian
Subject: Re: Algebra 1 (equations)

Hi Kim,

This is really shorthand for two separate equations:

  y = k
  y = -k

It's a little like when you say 'Pat is at either McDonalds or Burger 
King'. You know he's at one of those places, but you don't know which.

Why would we need to express this with numbers?  Here is an example.  
The equation 

  y = x^2

describes a parabola. If we choose a value of x, we can compute the 
corresponding value of y:

  x=2   y=2^2 =  4
  x=3   y=3^3 =  9
  x=4   y=4^2 = 16

But note that for any given value of y, there are _two_ values of x 
that give rise to it:

  x=2   y=2^2    = 4
  x=-2  y=(-2)^2 = 4

  x=3   y=3^2    = 9
  x=-3  y=(-3)^2 = 9
 
So what if we want to go the other way? What if we have a value for y, 
and we want to know what the corresponding value of x would be?  Well, 
there are two values:

  y=4   x=sqrt(y) = either 2 or -2
  y=9   x=sqrt(y) = either 3 or -3

and so on. In this case, we use the shorthand '+/-' ('plus or minus') 
to indicate the ambiguity:

  y=4   x=sqrt(y) = +/- 2
  y=9   x=sqrt(y) = +/- 3

In lots of cases, we can simply ignore the negative value. But 
ignoring it is different from not realizing that it exists. Forgetting 
about it completely is the basis of many so-called 'false proofs', 
like this one:

  [1]           -2 = -2

  [2]        4 - 6 = 1 - 3

  [3]  4 - 6 + 9/4 = 1 - 3 + 9/4

  [4]  (2 - 3/2)^2 = (1 - 3/2)^2                

  [5]      2 - 3/2 = 1 - 3/2            
                                   
  [6]            2 = 1

Do you see the problem? There are really four possible versions of 
step [5]:

  [5a]     2 - 3/2 = 1 - 3/2            False.

  [5b]  -(2 - 3/2) = 1 - 3/2            True.

  [5c]     2 - 3/2 = -(1 - 3/2)         True.   

  [5d]  -(2 - 3/2) = -(1 - 3/2)         False.  

This could be written more simply as

  [5d]  +/-(2 - 3/2) = +/-(1 - 3/2)         

But this would rule out the final step that makes the proof 'work'.

I hope this helps. Write back if you'd like to talk more about this, 
or anything else. 

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