Algebra Tools

Twenty-eight algebra tools, and not one floating-point number

Each of these answers a question from a maths course, and each of them shows how it got there: the rule every derivative step used, the intermediate matrices of an elimination, the table of implicants, the iterations of a root-finding method with the error at each. They run as WebAssembly in this browser tab and upload nothing.

Nothing here is a floating-point number, and in an algebra tool that is the feature rather than pedantry. Exact work is done in fractions and approximate work in decimals to a number of places you choose. An elimination in floating point produces rows with 0.0000000000000002 where a zero belongs — and a tool whose whole value is showing its working then shows working that is wrong. Here a third is 1/3, the rank of a matrix is a count rather than a judgement about how small a number has to be before it counts as nothing, and a matrix times its inverse is *exactly* the identity.

What cannot be done is named rather than approximated. The rational root theorem finds every rational root and no irrational one, and says so. A symbolic integral is found for the forms it recognises and refused by name for the rest, with the numerical answer offered instead — many perfectly ordinary functions have no antiderivative that can be written down at all, and a plausible wrong one would be worse than none. A tangent at ninety degrees has no value, and this says so rather than returning an enormous number, which is the commonest wrong answer a calculator gives.

The equations come first, then the calculus, then matrices and vectors, then counting and logic, and last the tools about numbers themselves: polynomials, complex numbers, trigonometry, primes, modular arithmetic, integers with as many digits as you type, and continued fractions.

Equations

Quadratic equation
Roots, discriminant, vertex and factors, with the working

System of equations
Two to six equations by elimination, with the type of solution

Inequality
Intervals, a sign table and a number line

Factorising
Common factors and rational roots, with the method named

Simplify an expression
Brackets out, like terms in, and the conditions it needs

Calculus

Derivative
Symbolic, with the rule named at every step

Integral
The forms it recognises, and plain honesty about the rest

Limit
From each side, with the table of approach that is the evidence

Area under a curve
Four rules compared, with the strips drawn

Roots of a function
Bisection, Newton and secant, with every iteration shown

Matrices and vectors

Matrix calculator
Inverse, product, rank and trace, all in exact fractions

Determinant
By expansion, by Sarrus or by elimination, with the working

Vector calculator
Dot, cross, length, angle and projection, drawn

Interpolation
A value between the ones in a table, with the working

Counting and logic

Combinations and permutations
Two questions decide the formula; the answer is exact

Binomial expansion
Every term, or just the one the question asks for

Pascal's triangle
The rows, the row sums, and the patterns hiding in it

Sequences and series
The nth term, the sum, and whether an infinite one converges

Set operations
Two pasted lists compared, with the counts in a Venn diagram

Truth table
A column for every sub-expression, and the normal forms

Boolean minimiser
Quine–McCluskey and a Karnaugh map, with the implicant table

Numbers

Polynomials
Add, multiply, divide with a remainder, and find the roots

Complex numbers
The three forms, every nth root, and the Argand diagram

Trigonometry
All six functions, three units, and the exact values

Prime numbers
Every prime in a range, with the gaps, the twins and the density

Modular arithmetic
Powers, inverses, orders and the Chinese remainder theorem

Big numbers
Integers with as many digits as you type, exact to the last

Continued fractions
The expansion, its convergents, and the best fraction under a limit

Questions

What does “exact” mean here?: That the answer was never rounded on the way. A system of equations is solved in fractions, so `3x = 1` gives `1/3` rather than 0.3333333333333333; a binomial coefficient is computed by multiplying and dividing one step at a time, so `52 choose 5` is 2,598,960 exactly; two hundred-digit integers multiply to every digit of their product. Where an answer genuinely cannot be exact — a square root, an integral, a limit — the number of decimal places is yours to choose and is stated on the result.
Which tools from the group are missing?: None of them: all twenty-eight in the group are here. What is *limited* is stated instead. Symbolic integration recognises the standard forms rather than implementing a general algorithm, and says which. Factorisation works over the rational numbers and says so. The discrete logarithm is a search, because there is no fast general method — several cryptosystems rest on exactly that.
Why is the working shown at such length?: Because it is usually the thing being asked for. Somebody looking up a derivative has to hand in the rule at each step; somebody solving a system has to show the elimination; somebody minimising a logical function has to produce the implicant table. A result that is only the answer has withheld the part that was wanted.
Do the tools work on a phone?: Yes. The whole interface lays out from a phone held upright to a desktop, and the drawings scale with it. The heavier tools — a sieve over ten million, a factorial of ten thousand — run on the phone's own processor, which is why they have stated ceilings rather than a queue.

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