Reverse Polish notation (RPN), also called postfix notation, writes each operator after the value or values it applies to. For example, 1 2 + means 1 + 2. In a complete, well-formed RPN expression, the order of the tokens specifies how operations are grouped, so ordinary arithmetic parentheses are unnecessary.
How RPN differs from infix and prefix notation
The names describe where the operator appears in relation to its operands. In ordinary infix notation, the operator sits between operands; in prefix notation, it comes first; in RPN, it comes last.
| Notation | Example for adding 1 and 2 | Operator position |
|---|---|---|
| Infix | 1 + 2 |
Between the operands |
| Prefix (Polish) | + 1 2 |
Before the operands |
| RPN (postfix) | 1 2 + |
After the operands |
In infix notation, readers use parentheses and precedence conventions to interpret groupings. RPN puts the grouping into the token order itself; it does not make arithmetic precedence disappear so much as avoid needing to infer the intended grouping from conventional infix rules.
How to evaluate an RPN expression
A common way to evaluate RPN is with a stack, a last-in, first-out structure. Read tokens from left to right: push each number onto the stack; when an operator appears, take the required operands from the top, apply the operation, and push the result back. For a complete, well-formed expression, one value remains at the end.
The Tool Desk
Outbyte PC Repair FREERepair Windows errors before they cause bigger problemsFix Now →Outbyte Driver Updater FREEFix the driver behind crashes, sound loss and screen glitchesFind Drivers →#1 Best Overall
- Robust and versatile scientific calculator
- Efficient data entry using RPN
- Over 120 built-in functions for business, finance, mathematics, and statistics, including date calculations
- Portable Design
- With WYNGS protective case black
Worked example: 1 2 3 + 4 5 * * +
This sequence is equivalent to 1 + ((2 + 3) * (4 * 5)). The table shows the stack after each token is processed; the leftmost value is at the bottom and the rightmost is at the top.
| Token | Stack after processing |
|---|---|
1 |
1 |
2 |
1, 2 |
3 |
1, 2, 3 |
+ |
1, 5 |
4 |
1, 5, 4 |
5 |
1, 5, 4, 5 |
* |
1, 5, 20 |
* |
1, 100 |
+ |
101 |
The first plus combines 2 and 3; the next multiplication combines 4 and 5; the following multiplication combines those results; and the final plus combines that product with 1.
Rank #2
- ADVANCED GRAPHING & CAS – Explore complex math with computer algebra system, dynamic geometry, advanced graphing, and spreadsheet applications plus RPN. This color graphing calculator lets you switch between symbolic, graphical, and numerical views with dedicated keys.
- 3.5-INCH MULTI-TOUCH COLOR DISPLAY – Graphing scientific calculator with a large high-resolution color screen, pinch-to-zoom, interactive graph manipulation, and customizable backgrounds. Add background images and use your finger to sketch and adjust functions for blended learning.
- EXAM APPROVED & CLASSROOM READY – Approved for PSAT/NMSQT, SAT, IB, and select AP exams. Suitable as an SAT calculator, statistics calculator, calculus calculator, precalculus calculator, engineering calculator for college, algebra calculator, or geometry calculator.
- WIRELESS & APP INTEGRATION – Use HP Prime Wireless + Connectivity Kits to poll students, share data, and project screens. HP Prime Mobile App (Windows, iOS, Android) mirrors full calculator functionality. Brushed metal design includes a slide-on cover and rechargeable lithium-ion battery.
- EXPANDABLE MEMORY & STEM ECOSYSTEM – 256 MB flash memory stores programs, exam configurations, and images. Integrates with optional accessories for advanced classroom control, real-time collaboration, and interactive engagement across the complete HP STEM environment.
Why operand order matters
For a binary operator, the top stack value is the right operand and the value immediately beneath it is the left operand. Thus, an evaluator processing 8 3 - computes 8 - 3, not 3 - 8. Likewise, 8 2 / means 8 ÷ 2. Reversing the operands would change subtraction or division, even though it would not change addition or multiplication.
How token order replaces parentheses
Compare 2 3 + 5 * with 2 3 5 * +. The first says to add 2 and 3, then multiply that result by 5: (2 + 3) * 5. The second says to multiply 3 by 5, then add 2: 2 + (3 * 5). Because the operators arrive only after their operands, the sequence shows which values each operation consumes without written parentheses.
What’s actually slowing this PC down?
Pick the symptom - the matching free tool is one click away.
Rank #3
- View multiple calculations at the same time: Compare results and explore patterns on-screen with the MultiView display that supports up to four lines
- See math exactly as it appears in textbooks: Display math expressions, symbols and stacked fractions exactly the way they appear in textbooks — no need to adapt to a technical syntax; provides quick access to frequently used functions
- Scientific notation output: View scientific notation with the proper superscripted exponents and see the output in scientific notation
- Explore (x,y) table of values: Students can easily explore an (x,y) table of values for a given function automatically or by entering specific x values
- The TI-30XS MultiView scientific calculator is ideal for general math, Pre-Algebra, Algebra 1 and 2, Geometry, Statistics, general science, Biology and Chemistry
Where RPN is used
RPN is useful for teaching stack-based expression evaluation: a straightforward evaluator can process the sequence from left to right. It is also associated with stack machines and calculator input. Princeton’s instructional material discusses the notation in a calculator context and identifies the HP-35 as a handheld calculator; that historical example does not establish which RPN calculators are currently sold or available.
For more on the stack procedure, see Princeton’s Stacks and Queues material and Emory’s explanation of postfix expressions. Princeton’s COS 126 lecture notes show the nested arithmetic example; Carnegie Mellon’s stack-computer notes connect postfix operations with stack machines, and University of Alaska Fairbanks course notes compare infix and RPN examples.
Quick Recap
Best Value
- RPN Entry: Reverse Polish Notation (RPN): you enter the numbers first, then the operation, with no parentheses or equals key. Runs the open source Free42 software.
- Math Functions: 34-digit precision, an equation solver, numerical integration, matrix operations and complex numbers.
- Programs: Write programs in a multi-line editor and assign any program to a single key.
- Case: Matte black stainless steel with a PVD coating and laser-engraved markings. 77 x 144 x 12 mm (3.0 x 5.7 x 0.5 in), 205 g (7.2 oz).
- Display And Power: 400 x 240 pixel memory LCD. One CR2032 coin cell, and USB-C for backups to a computer. Designed in Switzerland.
Rank #4
- Fundamental, two-line calculator that combines statistics and advanced scientific functions for high school math and science
- Two-line display shows the entry and calculated result at the same time for easy understanding of the calculation
- Fraction features, conversions, and basic scientific and trigonometric functions
- Solar and battery powered
- Approved for use on SAT, ACT and AP exams
Product prices and availability are accurate as of the date/time indicated and are subject to change. Any price and availability information displayed on Amazon at the time of purchase will apply.
