Scientific Calculator: How It Works
A scientific calculator adds trigonometry, logarithms, exponents and constants to basic arithmetic. Most errors people make with one come from two settings rather than from the mathematics: the angle mode, and the order in which the machine evaluates an expression.
Check the angle mode first
This causes more wrong answers than any other single thing. Trigonometric functions interpret their input as either degrees or radians, and the results are entirely different.
| Input | In degrees | In radians |
|---|---|---|
| sin(90) | 1 | 0.894 |
| cos(180) | −1 | −0.598 |
| tan(45) | 1 | 1.619 |
The quick test: if sin(90) returns 1, you are in degrees. Geometry, surveying and navigation use degrees; calculus, physics and every programming language's standard library use radians. A conversion is π ÷ 180 to go from degrees to radians.
Order of operations
Scientific calculators evaluate a full expression using standard precedence — brackets, then exponents, then multiplication and division, then addition and subtraction. Basic calculators evaluate sequentially as you type. The same keystrokes give different answers:
2 + 3 × 4 is 14 on a scientific calculator and 20 on a basic one.
When in doubt, add brackets. They cost nothing and remove all ambiguity.
Logarithms
| Notation | Base | Used in |
|---|---|---|
| log | 10 | pH, decibels, Richter scale |
| ln | e ≈ 2.71828 | Growth, decay, calculus |
| log₂ | 2 | Information theory, algorithm analysis |
For a base your calculator lacks, use the change of base rule: logb(x) = ln(x) ÷ ln(b).
Scientific notation
Large and small numbers are shown as a coefficient and a power of ten — 6.022e23 means 6.022 × 10²³. Use the calculator's EE or EXP key to enter these rather than typing × 10 ^, which introduces an extra multiplication and is a common source of error in chemistry and physics calculations.
Floating point precision
Calculators work to about 15 significant digits, which is not infinite precision. You may occasionally see results like 0.30000000000000004 from 0.1 + 0.2, because neither value has an exact binary representation. This is a property of how computers store numbers rather than a fault, and it is why financial software uses decimal or integer arithmetic rather than floating point.
Common function reference
| Function | Does |
|---|---|
| x², √ | Square and square root |
| x^y | Arbitrary power |
| n! | Factorial |
| 1/x | Reciprocal |
| sin⁻¹, cos⁻¹, tan⁻¹ | Inverse trig — returns an angle |
| π, e | Constants to full precision |
| mod | Remainder after division |
Inverse trigonometric functions return an angle in the calculator's current mode, so the same setting that affects sin also affects sin⁻¹ — a result of 1.5708 rather than 90 means you are in radians.