Pre-Calculus for Programmers

The numbers calculus works with

One endless line, the four families of numbers sitting on it, and what a decimal expansion gives away about which family a number belongs to.

Writing down a set of numbers

A thermostat, a run of accepted values, and the three notations that describe that run.

Absolute value is a distance, not a sign remover

A tolerance check on a machined part, and the one reading of $|a|$ that makes every rule about it easy to see.

Exponents, including the strange ones

Walk the powers of 2 downwards past zero and the meaning of $2^{0}$ and $2^{-3}$ is forced on you.

Fractional exponents are roots

If the exponent laws are to keep working, there is only one thing $9^{1/2}$ can possibly mean.

Simplest form, and getting a root off the bottom

$\sqrt{72}$ and $6\sqrt{2}$ are the same number; one of them is an answer and the other is not.

Rationalising a numerator, the move calculus actually needs

Watch a perfectly sensible expression turn to rubbish as a number shrinks, then repair it with one multiplication.

Multiplying out, and the products worth memorising

Every term times every term, why the middle term exists, and the three patterns that save you from doing it longhand.

Pascal's triangle and the binomial theorem

The coefficients of an expansion can be written down directly, including the one term you actually want.

Factoring: expanding, read backwards

Common factors, trinomials, grouping, and the three patterns that come up again and again.

Factoring when the exponents are negative or fractional

The tidy-up step every derivative of a root demands, and the one rule that makes it work.

Fractions with letters in them

Cancelling is legal for factors and never for terms, and everything else follows from ordinary fraction arithmetic.

Fractions inside fractions

One multiplication clears every small denominator at once, and calculus produces these constantly.

Completing the square

Rewriting a quadratic as a perfect square plus a leftover, which is the form that shows where its lowest point is.

Rearranging a formula, and the errors that ruin calculus

Solving for whichever letter you want, and a tester that will tell you when an algebra step was a lie.

Quality control report

Coding problem · Absolute value, Distance, Tolerance

When are we both free?

Coding problem · Intervals, Intersection

Expansion coefficients

Coding problem · Binomial theorem, Pascal's triangle

A factoring assistant

Coding problem · Factoring, Trinomials

Distance, powers and roots

Coding problem · Absolute value, Exponents, Rational exponents, Radicals

Expanding, the binomial theorem and factoring

Coding problem · Special products, Binomial theorem, Factoring

Fractions, conjugates and rearranging

Coding problem · Rational expressions, Complex fractions, Rationalising numerators, Completing the square, Literal equations