🚀 go-pugleaf

                                 2      2          2
You need to remember that (x + y)   =  x  + 2xy + y  .
This equality can be illustrated by a diagram:

	    x          y
      +------------+--------+
      |            |    2   |
    y |     xy     |   y    | y 
      |            |        |      The area of the large
      +------------+--------+
      |            |        |                       2 
      |      2     |        |      square is (x + y) .
    x |     x      |   xy   | x 
      |            |        |
      |            |        |
      +------------+--------+
            x          y

Now let us see why sqrt(a + b) is not equal to sqrt(a) + sqrt(b).
Consider the equation sqrt(a + b) = sqrt(a) + sqrt(b), and square both sides.
You get
                   2                       2
        sqrt(a + b)  =  (sqrt(a) + sqrt(b))

                   2           2                             2
	sqrt(a + b)  =  sqrt(a) + 2 sqrt(a) sqrt(b) + sqrt(b)
(Applying the above rule, with x = sqrt(a) and y = sqrt(b) )

	     a + b   =     a    + 2 sqrt(a) sqrt(b) +   b
	       0     =            2 sqrt(a) sqrt(b).

This last line is only true when a = 0 or b = 0.
In general, sqrt(a + b) < sqrt(a) + sqrt(b) unless a = 0 or b = 0.

Here are some challenging problems to ponder:

  1.  Show by a picture that (x + y)(x - y) = x^2 - y^2.
  2.  Show by a picture that the sum of the first n odd numbers is n^2. 
  3.  Show by a picture that (x + y)^3 = x^3 + 3(x^2)y + 3x(y^2) + y^3. 
  4.  Draw a picture which illustrates the method of completing the square. 
  5.  Prove the Pythagorean theorem c^2 = a^2 + b^2 as follows:
      Draw a square of side a + b.  Divide the square into a square
      of side c and four right triangles.  Then divide the same
      square into a square of side a, a square of side b, and
      four right triangles.  Conclude that c^2 = a^2 + b^2.

--
David Radcliffe                                
radcliff@csd4.csd.uwm.edu