Foundations of Mathematics 11

Inductive reasoning generalizes from cases: many examples suggest a rule but never prove it.

One counter-example kills a conjecture — a single failing case outweighs a thousand confirmations.

Good conjecture-testing hunts the weird cases: zero, negatives, fractions, giant numbers.

Deduction runs from accepted truths through logic to a certain conclusion — it covers every case at once.

Algebra is the proof engine: writing three consecutive integers as n, n+1, n+2 makes their sum 3n + 3 = 3(n + 1), visibly divisible by 3.

Corresponding angles are equal, alternate angles are equal, co-interior angles sum to 180° — all three need genuinely PARALLEL lines.

A scale factor k multiplies every LENGTH by k — but area picks up k² and volume k³.

Vertically opposite angles are equal at any crossing, parallel or not.

Similar objects share shape, not size: all corresponding lengths hold the same ratio.

One known angle unlocks all eight: chase equal pairs and supplements around the figure, naming the rule at each step.

The boundary line comes from the matching equation; ≤ or ≥ draws it solid (included), < or > dashed (excluded).

The vertex is the max or min; the axis of symmetry passes through it and mirrors the intercepts.

A test point settles the shading: substitute (0,0) — if true, shade its side.

End behaviour follows the leading coefficient: positive opens up (both ends rise), negative opens down.

Where two boundary lines intersect marks a corner of a combined region — technology reads those points precisely.

Standard deviation measures typical distance from the mean — small σ means clustered, large σ means scattered.

Equal means with different σ tell different stories: consistency is its own quality.

Technology computes σ; the human work is interpreting it in the data's units.

z = (x - mean)/σ counts how many standard deviations a value sits from the mean — comparable across different scales.

In a normal distribution, about 68% of data lies within 1σ, 95% within 2σ, 99.7% within 3σ.

A = P(1 + i)^n with i = rate per period and n = number of periods — the pair must match the compounding frequency.

Time dominates: doubling the years does far more than doubling the rate suggests, because growth compounds on growth.

Comparing frequencies at the same annual rate: more frequent compounding nudges the total up.

Regular deposits each compound for their own duration — spreadsheets or finance solvers sum the stack.

A loan's true cost = total of all payments minus the amount borrowed; longer terms lower payments but raise total interest.