How this instrument works
Polish notation writes an operator before its two operands rather than between them, so an expression like + a * b c is read as add a to the result of multiplying b and c. Nothing about that reading depends on remembering that multiplication happens before addition; the fact that '* b c' sits nested inside '+ a ...' already says everything about which piece gets evaluated first.
Ordinary infix notation, the kind used in most everyday arithmetic, writes a + b × c and leaves it to a memorized rule, multiply before you add, to avoid two different readings of the same string of symbols. Prefix notation sidesteps that requirement entirely: the way operators are nested is the only information needed, so there is never a moment where an expression is genuinely unclear without outside rules.
The notation is named for Jan Łukasiewicz, a Polish logician who introduced it in the 1920s specifically to remove the need for parentheses in logical formulas. A close relative, reverse Polish notation, flips the pattern so the operator comes after its operands instead — 'a b c * +' for the same expression — and shows up in some calculators and in stack-based computing more generally; this page sticks to the original, operator-first form.
- Enter the first value into the a field — it plays the role of a in + a * b c.
- Enter the second value into b and the third into c — together they form the nested * b c piece.
- Read Result for the fully evaluated expression: a plus the product of b and c.
- Change any of the three inputs and watch Result update while the nesting itself, unchanged, keeps deciding the order.
Worked example — evaluating + 2 * 3 4
Take the prefix expression with a=2, b=3, c=4. Reading from the innermost nesting outward, '* b c' means multiply 3 and 4, which is 12; then '+ a' wraps around that, meaning add 2 to 12, which is 14. No parentheses were needed anywhere in that reading — the nesting of the operators already told the whole story, matching the ordinary infix arithmetic 2+3×4=14.
Two more sets of values confirm the same reading holds regardless of what a, b, and c happen to be. With a=0, b=5, c=5, the nested multiplication gives 5×5=25 and adding a zero leaves it unchanged, so the result is 25. With a=10, b=2, c=3, the nested piece gives 2×3=6, and adding that to 10 produces 16.
Questions
What is Polish notation?
It is a way of writing expressions where every operator comes before its operands instead of between them, so + a * b c means add a to the product of b and c. It was introduced by the Polish logician Jan Łukasiewicz in the 1920s to write logical formulas without needing any parentheses.
How do you read a nested prefix expression like + a * b c?
Start from the innermost operator and work outward: '* b c' is evaluated first, since it sits nested inside the addition, giving b times c. Only after that does the outer '+' combine a with that inner result, so the full reading is a plus the product of b and c.
Why does prefix notation avoid the need for parentheses?
Because the position of each operator relative to its operands, and the nesting of one operator inside another, already fixes exactly which pieces belong together. Infix notation needs rules like multiply before you add precisely because + and × sit between numbers with no built-in nesting to lean on; prefix notation never runs into that gap.
Is reverse Polish notation the same thing?
No, it is the mirror image: reverse Polish notation, or postfix, places each operator after its operands instead of before, writing the same idea as 'a b c * +'. It shows up in some calculators and in stack-based computing, but this page covers the original operator-first form.
Who invented Polish notation and why is it called that?
Jan Łukasiewicz, a logician working in Poland, introduced it in the 1920s — the name simply honors his nationality. His goal was a way of writing logical and arithmetic expressions that needed no parentheses at all to stay unambiguous.
Does changing the value of a affect how * b c is evaluated?
No — the nested multiplication '* b c' only ever depends on b and c; a only enters once the outer addition combines it with whatever that inner multiplication already produced. Changing a shifts the final result but never touches the 12, or 25, or 6 that the inner piece computes on its own.