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1

Why Some Calculators Change the Placement of Operators in Algebraic Expressions

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2

The reason some calculators change the placement of operators in algebraic expressions is to maintain a way of changing the order of operations without having to use parentheses. For example, compare the following two expressions:

         1 + 2 * 3

and

         (1 + 2) * 3

The first expression evaluates to 7 and the second expression evaluates to 9. The effect of the parentheses in the second expression is to perform the addition before the multiplication.

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3

Changing the Notation

The way some calculators avoid parentheses is to change the notation to postfix (or Polish) notation from infix notation. With infix notation, the operator is placed in the middle of the operands. For example, in the infix expression

         1 + 2 

1 and 2 are operands and + is the operator in the middle. The postfix version of the expression places the + after both operands as in

         1 2 +

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4

Using a Calculator that Expects Posfix Expressions

Enter the numbers in the exact order in which they appear. Enter the operators in their order of operation as soon as the appropriate operands have been entered. For example, for the expression

 1 + 2 * 3 - 4

The order of the numbers is 1 2 3 4
The order of the operations is * + -
The * should be entered after 3
The + should be entered after 2 but
  not before *
The - should be entered after 4

The correct sequence is:

 1 Enter  {1 is the current value}
 2 Enter  {The calculator pushes
           2 on top of 1--1 is no
           longer shown }
 3 Enter  {The calculator pushes 3
           on top of 1 and 2--Note
           that * is the first
           operation and 3 is the 
           second operand for *}
 *        {The calculator multiplies
           2 * 3 to get 6 and pushes
           the 6 on top of the 1}
 +        {The calculator adds the 
           1 and 6 to get 7}
 4 Enter  {The calculator pushes
           4 on top of 7}
 -        {The calculator subtracts
           4 from 7 to get 3}

The postfix version of the expression
is the same as the entered order:

 1 2 3 * + 4 -

A second expression

 (1 + 2) * (3 - 4)

would be entered as follows:

The order of the numbers is 1 2 3 4
The order of the operations is + - *
The + should be entered after 2
The - should be entered after 4
The * should be entered after
  3 but not before -


 1 Enter  {1 is the current value}
 2 Enter  {The calculator pushes
           2 on top of 1--1 is no
           longer shown}
 +        {Note that + is the first
           operation--The calculator
           adds 1 and 2 to get 3 and
           returns the result}
 3 Enter  {The calculator pushes
           3 on top of 3}
 4 Enter  {The calculator pushes
           4 on top of 3 and 3}
 -        {The calculator subtracts
           4 from 3 to get -1 and
           pushes -1 on top of 3}    
 *        {The calculator multiplies 
           3 * -1 to get -3}

The postfix version of the expression
is the same as the entered order:

 1 2 + 3 4 - *

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5

Exercises

For the following expressions, show the order of operations for a calculator that expects postfix notation:

 1.     5 + 9 / 3

 2.     (5 + 9) / 2

 3.     4 * 3 / 6 + 8

 4.     4 * 3 / (4 + 8)

 5.     4 * (3 / (1 + 2))

 6.     4 * 3 + 6 / 3
 
 7.     4 * (3 + 6) / 3
 
 8.     4 * 3 + (6 / 3)
 
 9.     4 * ((3 + 6) / 3)
 
 10.    4 * (3 + (6 / 3))