Postfix%20notation

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Postfix notation
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About postfix notation

• Postfix, or Reverse Polish Notation (RPN) is an
  alternative to the way we usually write
  arithmetic expressions (which is called infix, or
  algebraic notation
• Postfix refers to the fact that the operator is
  at the end
• Infix refers to the fact that the operator is
  in between
• For example, 2 2 postfix is the same as 2 2
  infix
• There is also a seldom-used prefix notation,
  similar to postfix
• Advantages of postfix
• You dont need rules of precedence
• You dont need rules for right and left
  associativity
• You dont need parentheses to override the above
  rules
• Advantages of infix
• You grew up with it
• Its easier to see visually what is done first

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How to evaluate postfix

• Going from left to right, if you see an operator,
  apply it to the previous two operands (numbers)
• Example

2 10 4 5 / 9 3
- -

• Equivalent infix 2 10 4 / 5 (9 3)

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Computer evaluation of postfix

• Going left to right,
• If you see a number, put it on the stack
• If you see an operator, pop two numbers from the
  stack, do the operation, and put the result back
  on the stack
• (The top number on the stack is the operand on
  the right)

2 10 4 5 / 9 3
- -

• The result is the only thing on the stack when
  done
• If theres more than one thing, there was an
  error in the expression

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A helpful observation

• When you convert between infix and postfix, the
  operands (numbers) are always in the same order

• The operators are probably in a different order
• Were going to talk about how to convert manually
  between the two systems
• Your textbook has more information on how to do
  the conversion by computer

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From infix to postfix

• Figure out, using the infix rules, which
  operation to perform next
• Write the new operand or operands in their
  correct places
• Write the operator at the end
• Postfix does not use parentheses, but well put
  them in temporarily to help show the way things
  are grouped
• Example 2 10 4 / 5 (9 3)
• The multiply is done first 10 4
• Next, the divide (10 4 ) 5 /
• The addition 2 (10 4
  5 /)
• The rightmost subtraction (2 10 4 5 / )
  9 3 -
• The leftmost subtraction (2 10 4 5 / )
  (9 3 -) -
• The final result 2 10 4 5 /
  9 3 - -

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From postfix to infix

• Why would you want to?
• OK--working from left to right, for each
  operator, move it between the two preceding
  operands, and put parenthesis around the whole
  group
• Example
• 2 10 4 5 / 9 3 - -
• 2 (10 4) 5 / 9 3 - -
• 2 ((10 4) / 5) 9 3 - -
• (2 ((10 4) / 5) 9 3 - -
• (2 ((10 4) / 5) (9 - 3) -
• ((2 ((10 4) / 5) - (9 - 3))
• Now you can remove unnecessary parentheses (if
  you want to)

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Unary operators

• In infix, - means both negation (as a unary
  operator) and subtraction (as a binary
  operator)
• Likewise, means both addition and identity
• In infix notation, an operator is unary if
• It is the very first thing in the expression, or
• It immediately follows a left (opening)
  parenthesis
• We have no such clues in postfix notation
• The usual solution is simply to use a different
  symbol, such as , for unary negation

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Hewlett-Packard calculators

• Some of the finest calculators, such as expensive
  ones made by HP, use RPN (Reverse Polish
  Notation)
• Its actually quite easy to get used to using RPN
• RPN more closely mirrors the way you think about
  doing a calculation
• With an algebraic calculator, its easier to just
  copy an equation from a textbook into a
  calculator
• With just a little practice, entering an
  algebraic formula into an RPN calculator becomes
  very easy (and takes fewer keystrokes)
• Despite the advantages, RPN has a difficult time
  in the marketplace
• People in general dont like to learn new things
  if they dont have to
• ...and this isnt necessarily a bad thing

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The End