The Error Families

The course's twenty-one error twins fall into three families — distribution, custody, and misread notation — and knowing the family diagnoses errors the course never had time to film.

The distribution family wishes an operation spread over a sum. The radical split takes √(9 + 16) to 7 when it is 5; the denominator split takes 12/(2 + 4) to 9 when it is 2; the missing middle squares a binomial term by term and loses the 2ab; assumed linearity does it to functions, and the split-sum log error does it to logarithms — five costumes, one wish, and the diagnosis is always the same question: does this operation actually distribute over addition? Multiplication does. Almost nothing else does. The custody family loses or fabricates solutions. The lost root keeps 3 and abandons −3; the divided-away solution erases x = 0 by dividing where factoring was due; unchecked squaring reports impostors, and the log equations of Unit 8 manufactured a candidate at −2 the same way. Custody errors share one cure, and the course repeats it like law: candidates are cheap, and the original equation is the only judge. The misread-notation family is smaller and subtler — symbols meaning less than they seem to say: multiplying logs where the exponent-log mirror wrote a sum, dividing logs and calling it a quotient rule when the division is honest change-of-base, and reading the plus inside a quadratic's standard form as a positive h when the form subtracts. Three families, one habit that beats them all: test the claim on numbers. Every error twin in this course dies by its own example — twenty-one errors, twenty-one executions by arithmetic — and that was the design from the intake forward.