CSET Mathematics (211/212/213) flashcards

32 cards. Read the term and the definition here. The deck at the top of this page is optional.

Rational number characterization

A real number is rational exactly when its decimal expansion terminates or eventually repeats. Irrational numbers have non-terminating, non-repeating expansions, so √2 and π are irrational.

Closure property

A set is closed under an operation when applying it to any two members always yields a member of the set. Integers are closed under addition and multiplication but not division.

Triangle inequality for absolute value

For real numbers a and b, |a + b| ≤ |a| + |b|. Equality holds exactly when a and b have the same sign or at least one is zero.

Complex number modulus

For z = a + bi, the modulus is |z| = √(a² + b²), the distance from the origin in the complex plane. Division uses the conjugate a − bi to rationalize.

Quadratic formula

The solutions of ax² + bx + c = 0 with a ≠ 0 are x = [−b ± √(b² − 4ac)]/(2a). The discriminant b² − 4ac gives two, one, or zero real roots.

Sum and product of roots

For ax² + bx + c = 0, the two roots satisfy sum = −b/a and product = c/a. These follow from factoring a(x − r₁)(x − r₂) and comparing coefficients.

Logarithm change of base

log_b x = log_c x / log_c b for any valid base c. For example, log₂ 7 = ln 7 / ln 2, allowing evaluation with natural logarithms.

Geometric sequence sum

For a geometric sequence with first term a₁ and ratio r ≠ 1, the nth partial sum is a₁(1 − rⁿ)/(1 − r). The infinite sum converges to a₁/(1 − r) when |r| < 1.

Horizontal asymptote rule

For a rational function, compare degrees of numerator and denominator. Denominator larger gives y = 0; equal degrees give the ratio of leading coefficients; numerator larger gives none.

Inverse function condition

A function has an inverse exactly when it is one-to-one. The inverse swaps domain and range, and its graph reflects the original across the line y = x.

Pythagorean theorem

In a right triangle with legs a and b and hypotenuse c, a² + b² = c². The converse also holds: if the sides satisfy this equation, the triangle is right.

Inscribed angle theorem

An angle inscribed in a circle measures half its intercepted arc. An angle inscribed in a semicircle is therefore a right angle.

Similarity scaling rules

If two similar figures have linear scale factor k, corresponding lengths scale by k, areas by k², and volumes by k³. Doubling dimensions multiplies volume by eight.

Regular polygon interior angle

A regular polygon with n sides has each interior angle equal to (n − 2)180°/n. The sum of all exterior angles is always 360°.

Perpendicular slope relationship

Two nonvertical lines are perpendicular exactly when their slopes are negative reciprocals, so the product of the slopes equals −1. Parallel lines have equal slopes.

Sphere volume and surface area

A sphere of radius r has volume (4/3)πr³ and surface area 4πr². Both formulas involve only the radius, unlike cylinders and cones.

Conditional probability formula

P(A|B) = P(A and B)/P(B), provided P(B) > 0. If A and B are independent, this reduces to P(A), since P(A and B) = P(A)P(B).

Permutation versus combination

Permutations count ordered arrangements: nPr = n!/(n − r)!. Combinations count unordered selections: nCr = n!/[r!(n − r)!]. Choosing a committee uses combinations.

z-score meaning

A z-score equals (x − μ)/σ and states how many standard deviations a value lies above or below the mean. Positive z means above the mean.

Expected value

For a discrete random variable, expected value is Σ x·P(x), a probability-weighted average. A game is fair when expected net gain equals zero.

Correlation coefficient range

Correlation r lies between −1 and 1 and measures only the strength of linear association. A value near zero means no linear trend, not necessarily no relationship.

Standard error of the mean

The sampling distribution of the sample mean has standard deviation σ/√n, called the standard error. Larger samples produce a narrower distribution of sample means.

Limit definition of derivative

f′(x) = lim(h→0) [f(x + h) − f(x)]/h. This limit gives the slope of the tangent line and the instantaneous rate of change.

Chain rule

d/dx f(g(x)) = f′(g(x))·g′(x). Differentiate the outer function evaluated at the inner function, then multiply by the inner derivative.

Fundamental Theorem of Calculus

If F is an antiderivative of continuous f, then ∫_a^b f(x) dx = F(b) − F(a). Differentiation and integration are inverse processes.

Second derivative concavity test

Where f″ > 0 the graph is concave up and critical points are local minima; where f″ < 0 it is concave down and critical points are local maxima.

Mean Value Theorem

If f is continuous on [a, b] and differentiable on (a, b), some c in (a, b) satisfies f′(c) = [f(b) − f(a)]/(b − a), matching instantaneous to average slope.

Al-Khwarizmi and algebra

Al-Khwarizmi, writing around 820 CE, systematized solving quadratic equations. His name yields the word algorithm, and the title al-jabr yields algebra.

Euclid's Elements

Compiled around 300 BCE, the Elements built geometry deductively from five postulates and common notions, making it a model of axiomatic proof for centuries.

Newton and Leibniz calculus

Newton and Leibniz independently developed calculus in the late 1600s. Newton used fluxions and dot notation; Leibniz introduced the dy/dx and integral notation still used.

Proof by contrapositive

To prove P → Q, prove the equivalent ¬Q → ¬P. This differs from contradiction, which assumes P and ¬Q together and derives any impossibility.

Mathematical induction structure

Induction proves statements indexed by integers: establish the base case, assume the statement holds for k, then prove it for k + 1. Both steps are required.