Prefix/Infix/Postfix Notation

ACSL tests three ways of writing mathematical expressions. You need to convert between them and evaluate prefix/postfix expressions. This topic is very mechanical once you learn the methods - which means it’s free points if you practice.

The three notations

The advantage of prefix and postfix: no parentheses needed and no ambiguity. The order of operations is built into the notation.

Operators you’ll see

Precedence (highest to lowest): ^ (exponents), then * / , then + -

Method 1: Fully parenthesize (infix to prefix/postfix)

This is the most reliable conversion method.

Example: Convert $3\times 4+8/2$ to postfix.

Step 1 - Add all implicit parentheses based on precedence:

$((3\times 4)+(8/2))$

Step 2 - For postfix, move each operator to after its closing parenthesis:

$((34\ast )(82/)+)$

Step 3 - Remove parentheses:

$34\ast 82/+$

For prefix, move each operator to before its opening parenthesis:

$(+(\ast 34)(/82))$ $=+\ast 34/82$

Method 2: Stack evaluation (evaluating postfix)

To evaluate a postfix expression, scan left to right:

• Number? Push it onto the stack.
• Operator? Pop two numbers, apply operator, push result.

Example: Evaluate $5831-/+$

Answer: 9

Try it: evaluate a postfix expression

Step through $5831-/+$ and watch how the stack builds and collapses.

1Expression: 5 8 3 1 - / +2Read 5: push 53Read 8: push 84Read 3: push 35Read 1: push 16Read -: pop 3 and 1, compute 3-1=2, push 27Read /: pop 8 and 2, compute 8/2=4, push 48Read +: pop 5 and 4, compute 5+4=9, push 9

Critical detail: When you pop two values for an operator, the first popped is the right operand and the second popped is the left operand. This matters for - and /:

• Stack has […, A, B] and you see -: the result is A - B (not B - A)
• Pop B first (right), then A (left), giving A - B

Method 3: Stack evaluation (evaluating prefix)

For prefix, scan right to left and use the same stack logic.

Example: Evaluate $+5/8-31$

Scan right to left:

Answer: 9

For prefix, operand order is reversed: the first popped is the left operand. So when scanning right-to-left with stack […, A, B] and you see -: pop B (left), then A (right), giving B - A.

Converting postfix/prefix to infix

Work through the expression using the stack method, but instead of computing, build sub-expressions.

Example: Convert $34\ast 82/+75-\hat{}$ to infix.

Answer: ((3*4)+(8/2))^(7-5)

Quick mental tricks

Spotting the operator for the root of the expression:

• In postfix, the last symbol is the main operator
• In prefix, the first symbol is the main operator

This tells you the overall structure immediately.

For simple expressions (2 operators), you can often convert mentally:

• Postfix $AB+C\ast$ means $(A+B)\times C$
• Prefix $\ast +ABC$ means $(A+B)\times C$

Common mistakes

1. Operand order with - and /. In $83-$ (postfix), the answer is $8-3=5$, not $3-8$. The first number written is the left operand.
2. Missing the precedence when parenthesizing. $3+4\times 2$ must become $(3+(4\times 2))$, not $((3+4)\times 2)$.
3. Exponentiation is right-associative. $2\hat{}3\hat{}2$ = $2\hat{}(3\hat{}2)$ = $2^9=512$, not $(2^3{)}^2=64$.
4. Spaces matter. 12 is the number twelve, not 1 and 2.

Contest strategy

• Evaluation problems: use the stack method, ~30 seconds
• Conversion problems: fully parenthesize first, ~45 seconds
• Always verify by evaluating your converted expression - it should give the same result as the original