F2:    Unique Sum and Product of Several Real Numbers (2)

Rosenlicht states this consequence as follows:  "In a sum or product of several real numbers the order of the terms is immaterial."  (see ita-2-1-6-pg17)  He does not give a proof either of this or of the preceding one, and in my first reading of the material I did not attempt to do the proof because it seemed too difficult.  However, in this second reading I found the demonstration quite easy, and this was largely due to the fact that I was thinking about transferring the material to an oral tradition.  Consequently, I immediately disassociated my thinking from the written expressions, such as  (a+b+c+d)  and  (a·b·c·d).  Therefore, it was easier to see that we must prove that the numbers have a unique sum and product.  Once this is done, we can use order of terms and parentheses if we wish to control how the sum or product is obtained.  Two examples follow.  Note that the assignment of a value to the variable  s  returns that value to the calculation.  The first example indicates that the innermost parenthetical expression is evaluated first because the final value of  s  is  14.  The second example shows that the parentheses are evaluated from left to right because the final value of  s  is  13.

Example 1:   Use of Parenthesis

Example 2:   Use of Parenthesis


Converted [ in part ] by Mathematica      December 4, 2007