How letx checks its maths

Correctness is the whole point of a formula reference. Every calculator on this site is tested at every build, and nothing is published unless every test passes.

Validation summary (build of 2026-10-31)

Calculators: 100
Known test cases passed: 822 / 822
Round-trip checks passed: 1226
Worked examples recomputed: 100

Topic	Formulas	Test cases
Algebra	14	110
Geometry	22	155
Coordinate geometry	8	66
Trigonometry	9	108
Sequences and series	7	66
Probability and statistics	13	114
Finance maths	9	72
Calculus basics	12	81
Motion (kinematics)	6	50

Test cases for each formula

What is tested

Known cases. Every calculator is run against at least five results whose answers are known independently: textbook values (a 3-4-5 triangle, $C(52, 5) = 2 598 960$ , $Φ(1.96) = 0.97500$ , a 200,000 loan over 30 years at 6%) or the formula re-evaluated separately in plain code. The expected answers never come from the calculator being tested.
Edge cases. Each set includes the cases that go wrong in practice: a negative discriminant (the roots must come out as a complex pair, $−1 ± 2i$ , not as an error), a zero denominator, impossible triangles, $tan 90°$ , inverse sines of numbers outside −1 to 1, whole-number inputs that are not whole, probabilities above 1, and the same angle in degrees and in radians.
Round trips. After each successful case, the answer is fed back in and the calculator solves for each of the original inputs in turn. Solving $A = πr²$ for A and then for r must give back the radius. This catches a wrongly rearranged formula even where no textbook value exists.
Worked examples. The steps in each page's worked example are produced by the same code as the live calculator. When the site is built, every example is solved again; the build fails if the question no longer contains the numbers used, or if any number in the written answer differs from the computed one at the precision written.

How the calculators work

Each formula is stored once, as the rearrangements a student would write: $r = √(A/π)$ , $t = ln (N/N₀)/k$ . The calculator substitutes your numbers, then evaluates the expression one layer at a time so you see the intermediate results. Inputs can be expressions such as 2pi, sqrt(2)/2, 3/8 or 5%.

Precision. Arithmetic uses double-precision floating point (about 15–16 significant figures). Results are shown to 6 significant figures, money to 2 decimal places, and "≈" appears whenever a displayed number is rounded.
No algebraic rearrangement. A few unknowns cannot be isolated by algebra (the interest rate in a loan payment, for example). Those are found numerically by bisection to about 12 significant figures, the page says so, and the answer is checked by substituting it back.
Special functions. The normal distribution uses the complementary error function, computed from its Taylor series for small arguments and a continued fraction for large ones; checked against standard tables to better than $10^( −14)$ . The inverse uses Acklam's rational approximation refined with Newton steps.
Angles. Trigonometry calculators work in degrees or radians (your choice is remembered on your device). Exact values are kept exact: $sin 30° = 0.5$ , $cos 90° = 0$ , and $tan 90°$ is reported as undefined rather than as a huge number. Calculus uses radians, as the derivative rules require.
Domains. Square roots of negative numbers, logarithms of non-positive numbers, division by zero and similar cases produce an explanation, never a silent NaN.

Graphs and illustrations

Every graph and shape is drawn as SVG by the site's own code: once when the page is built, so it works without JavaScript, and again in your browser as you move the sliders. Nothing is fetched from anywhere else.

Sources and licences

The formulas themselves are standard mathematics and belong to no one. All text, calculators, graphs, illustrations and icons on letx are original work written for this site. No third-party datasets, photographs, fonts or scripts are used. Values used in the tests (normal-distribution table values, binomial coefficients, amortisation figures) are mathematical facts that were recomputed independently, not copied from a source.

Found a mistake?

If an answer looks wrong, check the inputs are in the units the calculator asks for (angles in degrees or radians, rates as 5% or 0.05). The step-by-step working shows exactly what was computed.