Foundations

Fraction arithmetic preserves value by using common denominators or reciprocals.

Absolute value measures distance from zero on the real line.

A percentage states a ratio per hundred and needs an explicit base.

Scaling an inequality depends on the factor's sign.

Negative exponents use reciprocals; rational exponents use principal roots.

A logarithm is the inverse of exponentiation for an admissible positive base.

The discriminant classifies the real solutions of a quadratic equation.

Composition feeds one output into another function; inversion requires unique preimages.

Specified matching sides and angles can establish the same triangle shape and size.

Triangle similarity keeps corresponding angles and one common positive side scale.

Opposite and adjacent legs depend on the chosen acute angle.

Radian measure is arc length divided by radius.

Area measures covered surface; perimeter measures boundary length.

Volume and surface area measure different aspects of a solid.

Mean, median, mode and sample variance summarize observed values.

Grouping and equal-priority arithmetic evaluation.

A normalized decimal coefficient with an integer power of ten.

Constant-ratio and constant-product relationships.

Euclidean line-angle relationships.

Interior and exterior angle sums for convex polygons.

Side-angle laws beyond right triangles.

A finite assignment forces a shared output when inputs outnumber outputs.

Finite-partition probabilities and reversal of event conditioning.

The smallest positive common multiple of positive integers.

Finite sums of nonnegative integer powers with fixed coefficients.

Decimal places, retained significant digits and rounding ties.

Euclidean cone and sphere mensuration.

Inscribed angles and radius-tangent perpendicularity.

Slope-intercept form of a nonvertical line.

Boundary length and enclosed area measure different properties of a plane shape.