# Infix to Postfix Converter with Step-By-Step Conversion Tutorial This free online converter will convert a mathematical infix expression to a postfix expression (A.K.A., Reverse Polish Notation, or RPN) using the stack method.

Plus, the converter's results also include the step-by-step, token-by-token processing used to complete the conversion. You can use these dynamic tutorials to learn how to perform the infix to postfix conversions manually, or to check your practice conversions.

If you would like to evaluate an infix expression, please use the Order of Operations Calculator. Or, to evaluate a postfix expression, use the Postfix Evaluator.

## Infix to Postfix Converter

Convert infix to postfix and see step-by-step how the conversion is made using the stack method.

#### Selected Data Record:

A Data Record is a set of calculator entries that are stored in your web browser's Local Storage. If a Data Record is currently selected in the "Data" tab, this line will list the name you gave to that data record. If no data record is selected, or you have no entries stored for this calculator, the line will display "None".

DataData recordData recordSelected data record: None
Example expressions:

#### Example infix expressions:

To see an example of how the Infix to Postfix Converter works, and what types of expressions the converter is set up to handle, select an infix expression from the drop-down menu. To clear the expression field to enter your own infix expression, select "Example Expressions" or click the "Reset" button.

Infix expression:

#### Infix expression:

Enter an infix expression that fits within the following guidelines:

• No spaces between numbers or operators.
• Numbers, parenthesis, and operators only (no letters or variables).
• Contains only numbers, decimal points, and these valid characters: ( ^ * / + - ).
• Contains the same number of left and right parenthesis.
• Numbers with a leading decimal point must be preceded by a zero (enter .5 as 0.5).
• If a number (numeric operand) immediately precedes a left parenthesis (indicating multiplication), the calculator will attempt to insert a multiplication operator between the number and the left parenthesis (convert 2(2+1) to 2*(2+1) ). However, you might get more predictable results if you insert the multiplication operator when you enter the infix expression.
 # Infix expression
Postfix expression:

#### Postfix expression:

This line will display the postfix equivalent of the entered infix expression. Note that a postfix expression is also known as a reverse polish notation, or RPN.

If you would like to save the current entries to the secure online database, tap or click on the Data tab, select "New Data Record", give the data record a name, then tap or click the Save button. To save changes to previously saved entries, simply tap the Save button. Please select and "Clear" any data records you no longer need.

## Learn

### What infix is, what postfix is, and how to convert infix to postfix using stack.

#### What is an Infix Expression?

An Infix Expression (or Infix Notation) is characterized by a math expression wherein the operators are placed between operands, (as in 2 + 3).

In the case of infix expressions, parentheses (or brackets) must be used to indicate the order in which the author wants the operations to be executed. Otherwise, the person solving the problem is left to apply an order of operation convention (PEMDAS, BODMAS, etc.) to solve the expression.

#### What is a Postfix Expression?

As the name implies, a Postfix Expression (or Postfix Notation, or Reverse Polish Notation) is characterized by a math expression wherein the operators are placed after their operands (2 + 3 infix becomes 2 3 + postfix).

Since each postfix operator is evaluated from left to right, this eliminates the need for parenthesis. This is why postfix expressions are easier for computer algorithms to evaluate than infix expressions.

#### How to Convert Infix to Postfix Using Stack

In case you're not familiar, a stack is a collection or list wherein the last element added to the stack is always the first element to be removed.

Using a stack to temporarily store operators is necessary because as we are evaluating each operator token of the infix expression from left to right, we can't instantly know an operator's right-hand operand. Therefore we need to temporarily add (push) an operator to the stack and only remove (pop) it from the stack and append it to postfix expression once we know its right operand.

So now that you know what a stack is and why it is used, the next order of business is to assign precedence and associativity to each operator type based on an order of operations convention (PEMDAS or BODMAS). The only exception is, in the stack method parentheses don't have precedence because they don't exist in a postfix expression. Instead, parentheses are used to isolate parts of the infix notation that are to be executed independently of the operators outside of them. Any nested parentheses are evaluated prior to evaluating their encompassing parentheses.

OperatorSymbolPrecedenceAssociativity
Exponent^4Right to Left
Multiplication*3Left to Right
Division/3Left to Right
Subtraction-2Left to Right

Now that you know what a stack is and have assigned precedence and associativity to each operator, the following are the steps to converting infix to postfix using stack.

Working from left to right, one token at a time, determine whether or not the current token is an operand, an operator, or an opening or closing parenthesis.

If the token is an operand, append it to the postfix expression. Operands always appear in the same order in the postfix expression as they do in the infix expression.

If the token is an opening parenthesis, push it to the stack. This marks the beginning of an expression that should be evaluated separately.

If the token is a closing parenthesis, until an opening parenthesis is encountered at the top of the stack, pop each operator from the stack and append to the postfix expression. This marks the end of the operands and operators located within the current set of parentheses.

If the token is an operator and it has greater precedence than the operator on the top of the stack, push the token to the stack.

If the token is an operator and it has lower precedence than the operator on the top of the stack, until the operator on top of the stack has lower precedence than the token, or until the stack is empty, pop each operator from the stack and append them to the postfix expression. Then push the current token to the stack.

If the token is an operator and it has the same precedence as the operator on the top of the stack, and the operator on top of the stack is left-to-right associative, until the operator on top of the stack has lower precedence than the token, or until the stack is empty, pop each operator from the stack and append them to the postfix expression. Then push the current token to the stack.

If the token is an operator and it has the same precedence as the operator on the top of the stack, but the operator on top of the stack is right-to-left associative, push the current token to the stack.

Finally, once all of the infix tokens have been evaluated, pop any remaining operators from the stack, one at a time, and append each of them to the postfix expression.

#### Infix to Postfix Conversion Examples

Since the step-by-step infix to postfix examples are quite long, I will first provide a simple example without any parentheses, and then provide a more complex example that includes parentheses and a case of right-to-left associativity. Both examples were generated by the infix to postfix calculator on this page.

##### Example #1: 6*4+2^5-3

Token #1 Evaluation

Infix: 6*4+2^5-3

Since 6 is an operand, append it to the postfix expression.

Postfix: 6

Stack:

Token #2 Evaluation

Infix: 6*4+2^5-3

Since the * is an operator and the stack is empty, push the * to the stack.

Postfix: 6

Stack:

 *

Token #3 Evaluation

Infix: 6*4+2^5-3

Since 4 is an operand, append it to the postfix expression.

Postfix: 6 4

Stack:

 *

Token #4 Evaluation

Infix: 6*4+2^5-3

Since + is an operator and has lower precedence than the * on the top of the stack, pop * from the stack and append it to the postfix expression.

Postfix: 6 4 *

Stack:

Push the + to the stack.

Postfix: 6 4 *

Stack:

 +

Token #5 Evaluation

Infix: 6*4+2^5-3

Since 2 is an operand, append it to the postfix expression.

Postfix: 6 4 * 2

Stack:

 +

Token #6 Evaluation

Infix: 6*4+2^5-3

Since ^ is an operator and has greater precedence than the + on the top of the stack, push ^ to the top of the stack.

Postfix: 6 4 * 2

Stack:

 ^+

Token #7 Evaluation

Infix: 6*4+2^5-3

Since 5 is an operand, append it to the postfix expression.

Postfix: 6 4 * 2 5

Stack:

 ^+

Token #8 Evaluation

Infix: 6*4+2^5-3

Since - is an operator and has lower precedence than the ^ on the top of the stack, pop ^ from the stack and append it to the postfix expression.

Postfix: 6 4 * 2 5 ^

Stack:

 +

Since - is an operator, is left-to-right associative, and has the same precedence as the + at the top of the stack, pop + from the stack and append it to the postfix expression.

Postfix: 6 4 * 2 5 ^ +

Stack:

Push the - to the stack.

Postfix: 6 4 * 2 5 ^ +

Stack:

 -

Token #9 Evaluation

Infix: 6*4+2^5-3

Since 3 is an operand, append it to the postfix expression.

Postfix: 6 4 * 2 5 ^ + 3

Stack:

 -

All 9 Token Evaluations Completed

Since we are done processing the infix expression, pop the remaining operator from top of the stack and append it to the postfix expression.

Pop - from the stack and append it to the postfix expression.

Postfix: 6 4 * 2 5 ^ + 3 -

Stack:

Completed Conversion:

 Infix: 6*4+2^5-3 to Postfix: 6 4 * 2 5 ^ + 3 -

##### Example #2: 4*((2+6)/3)-2^2^3

Token #1 Evaluation

Infix: 4*((2+6)/3)-2^3^4

Since 4 is an operand, append it to the postfix expression.

Postfix: 4

Stack:

Token #2 Evaluation

Infix: 4*((2+6)/3)-2^3^4

Since the * is an operator and the stack is empty, push the * to the stack.

Postfix: 4

Stack:

 *

Token #3 Evaluation

Infix: 4*((2+6)/3)-2^3^4

Since ( is an opening parenthesis, push it to the top of the stack.

Postfix: 4

Stack:

 (*

Token #4 Evaluation

Infix: 4*((2+6)/3)-2^3^4

Since ( is an opening parenthesis, push it to the top of the stack.

Postfix: 4

Stack:

 ((*

Token #5 Evaluation

Infix: 4*((2+6)/3)-2^3^4

Since 2 is an operand, append it to the postfix expression.

Postfix: 4 2

Stack:

 ((*

Token #6 Evaluation

Infix: 4*((2+6)/3)-2^3^4

Since + is an operator and has greater precedence than the ( on the top of the stack, push + to the top of the stack.

Postfix: 4 2

Stack:

 +((*

Token #7 Evaluation

Infix: 4*((2+6)/3)-2^3^4

Since 6 is an operand, append it to the postfix expression.

Postfix: 4 2 6

Stack:

 +((*

Token #8 Evaluation

Infix: 4*((2+6)/3)-2^3^4

Since the ) is a closing parenthesis, pop each operator from the stack one at a time and append to the postfix expression. Keep popping from the stack until an opening parenthesis is encountered.

Pop + from the top of the stack and append to the postfix expression.

Postfix: 4 2 6 +

Stack:

 ((*

Since we are done with the expression inside the current parenthesis, pop the opening parenthesis from the top of the stack and discard.

Postfix: 4 2 6 +

Stack:

 (*

Token #9 Evaluation

Infix: 4*((2+6)/3)-2^3^4

Since / is an operator and has greater precedence than the ( on the top of the stack, push / to the top of the stack.

Postfix: 4 2 6 +

Stack:

 /(*

Token #10 Evaluation

Infix: 4*((2+6)/3)-2^3^4

Since 3 is an operand, append it to the postfix expression.

Postfix: 4 2 6 + 3

Stack:

 /(*

Token #11 Evaluation

Infix: 4*((2+6)/3)-2^3^4

Since the ) is a closing parenthesis, pop each operator from the stack one at a time and append to the postfix expression. Keep popping from the stack until an opening parenthesis is encountered.

Pop / from the top of the stack and append to the postfix expression.

Postfix: 4 2 6 + 3 /

Stack:

 (*

Since we are done with the expression inside the current parenthesis, pop the opening parenthesis from the top of the stack and discard.

Postfix: 4 2 6 + 3 /

Stack:

 *

Token #12 Evaluation

Infix: 4*((2+6)/3)-2^3^4

Since - is an operator and has lower precedence than the * on the top of the stack, pop * from the stack and append it to the postfix expression.

Postfix: 4 2 6 + 3 / *

Stack:

Push the - to the stack.

Postfix: 4 2 6 + 3 / *

Stack:

 -

Token #13 Evaluation

Infix: 4*((2+6)/3)-2^3^4

Since 2 is an operand, append it to the postfix expression.

Postfix: 4 2 6 + 3 / * 2

Stack:

 -

Token #14 Evaluation

Infix: 4*((2+6)/3)-2^3^4

Since ^ is an operator and has greater precedence than the - on the top of the stack, push ^ to the top of the stack.

Postfix: 4 2 6 + 3 / * 2

Stack:

 ^-

Token #15 Evaluation

Infix: 4*((2+6)/3)-2^3^4

Since 3 is an operand, append it to the postfix expression.

Postfix: 4 2 6 + 3 / * 2 3

Stack:

 ^-

Token #16 Evaluation

Infix: 4*((2+6)/3)-2^3^4

Since ^ is an operator and has the same precedence as the ^ at the top of the stack, and the ^ at the top of the stack is right-to-left associative, push the ^ to the stack.

Postfix: 4 2 6 + 3 / * 2 3

Stack:

 ^^-

Token #17 Evaluation

Infix: 4*((2+6)/3)-2^3^4

Since 4 is an operand, append it to the postfix expression.

Postfix: 4 2 6 + 3 / * 2 3 4

Stack:

 ^^-

All 17 Token Evaluations Completed

Since we are done processing the infix expression, pop each of the remaining operators from top of the stack one at a time and append each to the postfix expression.

Pop ^ from the stack and append it to the postfix expression.

Postfix: 4 2 6 + 3 / * 2 3 4 ^

Stack:

 ^-

Pop ^ from the stack and append it to the postfix expression.

Postfix: 4 2 6 + 3 / * 2 3 4 ^ ^

Stack:

 -

Pop - from the stack and append it to the postfix expression.

Postfix: 4 2 6 + 3 / * 2 3 4 ^ ^ -

Stack:

Completed Conversion:

 Infix: 4*((2+6)/3)-2^3^4 to Postfix: 4 2 6 + 3 / * 2 3 4 ^ ^ -

Move the slider to left and right to adjust the calculator width. Note that the Help and Tools panel will be hidden when the calculator is too wide to fit both on the screen. Moving the slider to the left will bring the instructions and tools panel back into view.

Also note that some calculators will reformat to accommodate the screen size as you make the calculator wider or narrower. If the calculator is narrow, columns of entry rows will be converted to a vertical entry form, whereas a wider calculator will display columns of entry rows, and the entry fields will be smaller in size ... since they will not need to be "thumb friendly".