Scientific Calculator with Step Breakdown
Evaluate expressions with trigonometry, logarithms, roots, powers, constants, parentheses, and factorial using a full scientific keyboard in degree or radian mode.
Scientific Calculator — interactive calculator, computed instantly in your browser.
How a Scientific Calculator Thinks: Parsing, Precedence & Angle Modes
From Keystrokes to Value — the Shunting-Yard Path
Key Takeaway: Expressions are tokenized, reordered to postfix (RPN), then evaluated on a stack — deterministic precedence with zero code-execution risk.
A scientific calculator extends plain arithmetic with trigonometry, logarithms, roots, powers, and constants — but its real intelligence is parsing. Before a single number is computed, the expression is split into tokens, precedence is resolved with the shunting-yard algorithm, and evaluation runs on a stack in postfix (RPN) order. That is why (5+3)×2 = 16 while 5+3×2 = 11 — multiplication binds tighter than addition, and parentheses override both.
1. Operator Precedence & Associativity
- Right-associativity of ^: 2^3^2 groups as 2^(3²) = 512, not (2³)² = 64 — powers chain right-to-left
- Unary minus vs power: the sign attaches to its literal, so −2² = (−2)² = 4; write −(2²) for −4
- Factorial binds first: 2^3! = 2^(3!) = 64, because 3! resolves before the exponent
- Modulo % returns the remainder (7 % 3 = 1) and follows multiplication's precedence
2. Degrees vs Radians
Trigonometric functions expect radians in calculus and physics, but degrees feel natural in geometry — one wrong mode silently produces answers like sin(30) = −0.988 (30 rad ≈ 1,719°) instead of 0.5. The DEG/RAD toggle converts the argument before evaluation and converts inverse results (asin, acos, atan) back afterward, so tan(45) reads exactly 1 in degree mode and tan(π/4) reads 1 in radian mode.
3. Logarithms, Roots & Factorials
- log vs ln: log is base-10 (log(1000) = 3), ln is base-e (ln(e) = 1) — inverse pairs of 10ˣ and eˣ
- Roots as powers: √x = x^0.5 and ∛x = x^(1/3) — sqrt() rejects negatives, while cbrt() handles them
- Factorial limits: n! grows super-exponentially; beyond 170! ≈ 7.26 × 10³⁰⁸ a 64-bit float overflows to Infinity
- Constants: π ≈ 3.141592653589793 and e ≈ 2.718281828459045 carry full double precision
🔗 Advanced Numeric Evaluation Tools
Scientific expressions combine arithmetic with transcendental functions and physical constants — the same building blocks used in physics, engineering coursework, and data analysis. Visit our ➗ Math & Statistics Hub for more step-by-step solvers.
Powers, notation formats, and statistical spreads pair naturally with function evaluation — explore the related tools below.
Frequently Asked Questions (Scientific Calculator)
Because the calculator is in DEG mode — sin of 30 degrees is exactly 0.5. In RAD mode, 30 is interpreted as 30 radians (≈ 1,719°) and sin(30) ≈ −0.988. Match the mode to the units of your problem: geometry uses degrees, calculus and physics usually use radians.
log (log₁₀) answers '10 to what power gives x?' — log(1000) = 3. ln (natural log) answers 'e to what power gives x?' — ln(e) = 1. They invert 10ˣ and eˣ respectively; convert with log(x) = ln(x) ÷ ln(10).
Use the ^ key: 2^3^2 evaluates right-to-left as 2^(3²) = 512 because exponentiation is right-associative. For (2³)² = 64, add parentheses: (2^3)^2. When unsure, parentheses always win.
Real-valued square roots require a non-negative argument — √−1 has no real solution. The calculator flags it rather than returning NaN; complex results (i = √−1) belong to complex-number tools.
170! ≈ 7.26 × 10³⁰⁸ is the last value inside double-precision floating point; 171! overflows to Infinity. Beyond that use logarithms: ln(n!) = Σ ln(k) or Stirling's approximation n! ≈ √(2πn)(n/e)ⁿ.
Yes — this calculator tokenizes and validates every character, then evaluates on an operand stack. No code strings are executed (no eval), so malformed input like '2+' simply returns a parse error instead of running anything.