CSET Mathematics (211/212/213) cheat sheet

Rational numbers are exactly the terminating or eventually repeating decimals; irrationals have non-terminating, non-repeating expansions.

Closure: integers are closed under +, −, × but not ÷; rationals are closed under all four operations except division by zero.

Order of operations: grouping symbols, exponents/roots, multiplication and division left to right, then addition and subtraction left to right.

Field axioms for rationals and reals: commutativity, associativity, distributivity, additive and multiplicative identities and inverses.

Absolute value satisfies |ab| = |a||b| and the triangle inequality |a + b| ≤ |a| + |b|; equality holds when ab ≥ 0.

Scientific notation: a × 10^n with 1 ≤ |a| < 10; multiplying adds exponents, dividing subtracts them.

Complex numbers: i² = −1; |a + bi| = √(a² + b²); the conjugate a − bi gives quotient (a + bi)(c − di)/(c² + d²).

Percent change = (new − old)/old × 100; successive percent changes multiply rather than add.

Divisibility: a number is divisible by 3 or 9 iff its digit sum is; by 4 iff its last two digits form a multiple of 4.

Quadratic formula: x = [−b ± √(b² − 4ac)]/(2a); the discriminant b² − 4ac tells 2, 1, or 0 real roots.

Vertex of y = ax² + bx + c occurs at x = −b/(2a); in vertex form y = a(x − h)² + k the vertex is (h, k).

Sum and product of roots of ax² + bx + c = 0 are −b/a and c/a.

Exponent rules: a^m · a^n = a^(m+n); (a^m)^n = a^(mn); a^(−n) = 1/a^n; a^(m/n) = ⁿ√(a^m).

Log rules: log(xy) = log x + log y; log(x/y) = log x − log y; log(x^k) = k log x; log_b b = 1, log_b 1 = 0.

Change of base: log_b x = ln x / ln b; exponential y = a·b^x has growth rate b − 1 per period.

Arithmetic sequence: a_n = a_1 + (n − 1)d, sum = n(a_1 + a_n)/2. Geometric: a_n = a_1 r^(n−1), sum = a_1(1 − r^n)/(1 − r).

Function composition is not commutative: (f∘g)(x) = f(g(x)); inverse functions satisfy f(f⁻¹(x)) = x and swap domain and range.

Rational function f(x) = p(x)/q(x) has vertical asymptotes where q(x) = 0 (unless the factor cancels) and horizontal asymptote from degree comparison.

Triangle angle sum is 180°; exterior angle equals the sum of the two remote interior angles.

Pythagorean theorem a² + b² = c²; triples include 3-4-5, 5-12-13, 8-15-17, and their multiples.

Area formulas: triangle ½bh; trapezoid ½(b₁ + b₂)h; circle πr²; sector (θ/360)πr².

Similar figures: corresponding sides proportional; areas scale by k² and volumes by k³ for linear scale factor k.

Circle: inscribed angle is half its intercepted arc; a tangent is perpendicular to the radius at the point of tangency.

Distance formula d = √[(x₂ − x₁)² + (y₂ − y₁)²]; midpoint is the average of coordinates.

Slopes: parallel lines have equal slopes; perpendicular lines have slopes whose product is −1.

Volume: prism/cylinder Bh; pyramid/cone ⅓Bh; sphere (4/3)πr³; surface area of sphere 4πr².

Regular polygon with n sides: interior angle = (n − 2)180/n; sum of exterior angles is always 360°.

Probability of an event = favorable outcomes / total equally likely outcomes; P(A or B) = P(A) + P(B) − P(A and B).

Independent events: P(A and B) = P(A)·P(B); conditional probability P(A|B) = P(A and B)/P(B).

Permutations count order: nPr = n!/(n − r)!; combinations ignore order: nCr = n!/[r!(n − r)!].

Mean = sum/n; median is the middle value of ordered data; mode is most frequent; the mean is pulled toward skew.

Standard deviation measures typical distance from the mean; about 68% of normal data lies within 1 SD, 95% within 2 SD.

z-score = (x − μ)/σ; it states how many standard deviations a value lies from the mean.

Expected value of a discrete variable = Σ x·P(x); a fair game has expected net gain of zero.

Correlation r ranges from −1 to 1 and measures linear association only; correlation does not imply causation.

Sampling distribution of the mean has standard error σ/√n; larger samples narrow the spread.

Limit definition of derivative: f′(x) = lim(h→0) [f(x + h) − f(x)]/h; the derivative is the slope of the tangent line.

Power rule d/dx x^n = n x^(n−1); product rule (uv)′ = u′v + uv′; quotient rule (u/v)′ = (u′v − uv′)/v².

Chain rule d/dx f(g(x)) = f′(g(x))·g′(x); derivative of sin x is cos x, of cos x is −sin x, of e^x is e^x, of ln x is 1/x.

A function has a local extremum where f′ = 0 or is undefined; sign change of f′ confirms max or min.

Second derivative f″ > 0 means concave up (local minimum); f″ < 0 means concave down (local maximum).

Fundamental Theorem: ∫_a^b f(x) dx = F(b) − F(a) where F′ = f.

Area between curves y = f(x) and y = g(x) on [a, b] is ∫_a^b |f − g| dx.

Riemann sums with n rectangles approximate the definite integral; the limit as n → ∞ is exact.

Mean Value Theorem: if f is continuous on [a, b] and differentiable on (a, b), some c has f′(c) = [f(b) − f(a)]/(b − a).

Euclid's Elements (c. 300 BCE) organized geometry as an axiomatic-deductive system with five postulates.

Pythagoras and his school (6th century BCE) linked number and geometry; the discovery of irrational √2 is attributed to that circle.

Archimedes (c. 287–212 BCE) used exhaustion to approximate π and derived areas and volumes of curved figures.

Al-Khwarizmi (c. 820 CE) wrote on solving quadratic equations; his name yields the word algorithm, and al-jabr yields algebra.

Descartes (1637) joined algebra and geometry with coordinate axes; Fermat independently developed analytic geometry.

Newton and Leibniz independently developed calculus in the late 1600s; Newton used fluxions, Leibniz the dy/dx notation.

Euler (1707–1783) introduced e, the function notation f(x), and the identity e^(iπ) + 1 = 0.

Cantor (late 1800s) founded set theory and proved the reals are uncountably infinite.

Non-Euclidean geometry was developed in the 1800s by Lobachevsky, Bolyai, and Riemann, showing the parallel postulate is independent.

Direct proof assumes the hypothesis and derives the conclusion through valid inferences.

Proof by contradiction assumes the negation of the statement and derives a contradiction, establishing the original.

Proof by contrapositive proves ¬Q → ¬P, which is logically equivalent to P → Q.

Mathematical induction: base case P(1), then assume P(k) and prove P(k + 1); valid only for statements indexed by integers.

A counterexample disproves a universal statement; one example never proves a universal claim.

Converse (Q → P) and inverse (¬P → ¬Q) are equivalent to each other but not to the original conditional.

A statement is a biconditional (iff) only when both the conditional and its converse hold.

Van Hiele levels describe geometry learning: visualization, analysis, informal deduction, formal deduction, rigor.

Manipulatives and multiple representations build conceptual understanding before procedural fluency; formative assessment guides instruction.

Rational numbers are exactly the terminating or eventually repeating decimals; irrationals have non-terminating, non-repeating expansions.

Closure: integers are closed under +, −, × but not ÷; rationals are closed under all four operations except division by zero.

Order of operations: grouping symbols, exponents/roots, multiplication and division left to right, then addition and subtraction left to right.

Field axioms for rationals and reals: commutativity, associativity, distributivity, additive and multiplicative identities and inverses.

Absolute value satisfies |ab| = |a||b| and the triangle inequality |a + b| ≤ |a| + |b|; equality holds when ab ≥ 0.

Scientific notation: a × 10^n with 1 ≤ |a| < 10; multiplying adds exponents, dividing subtracts them.

Complex numbers: i² = −1; |a + bi| = √(a² + b²); the conjugate a − bi gives quotient (a + bi)(c − di)/(c² + d²).

Percent change = (new − old)/old × 100; successive percent changes multiply rather than add.

Divisibility: a number is divisible by 3 or 9 iff its digit sum is; by 4 iff its last two digits form a multiple of 4.

Quadratic formula: x = [−b ± √(b² − 4ac)]/(2a); the discriminant b² − 4ac tells 2, 1, or 0 real roots.

Vertex of y = ax² + bx + c occurs at x = −b/(2a); in vertex form y = a(x − h)² + k the vertex is (h, k).

Sum and product of roots of ax² + bx + c = 0 are −b/a and c/a.

Exponent rules: a^m · a^n = a^(m+n); (a^m)^n = a^(mn); a^(−n) = 1/a^n; a^(m/n) = ⁿ√(a^m).

Log rules: log(xy) = log x + log y; log(x/y) = log x − log y; log(x^k) = k log x; log_b b = 1, log_b 1 = 0.

Change of base: log_b x = ln x / ln b; exponential y = a·b^x has growth rate b − 1 per period.

Arithmetic sequence: a_n = a_1 + (n − 1)d, sum = n(a_1 + a_n)/2. Geometric: a_n = a_1 r^(n−1), sum = a_1(1 − r^n)/(1 − r).

Function composition is not commutative: (f∘g)(x) = f(g(x)); inverse functions satisfy f(f⁻¹(x)) = x and swap domain and range.

Rational function f(x) = p(x)/q(x) has vertical asymptotes where q(x) = 0 (unless the factor cancels) and horizontal asymptote from degree comparison.

Triangle angle sum is 180°; exterior angle equals the sum of the two remote interior angles.

Pythagorean theorem a² + b² = c²; triples include 3-4-5, 5-12-13, 8-15-17, and their multiples.

Area formulas: triangle ½bh; trapezoid ½(b₁ + b₂)h; circle πr²; sector (θ/360)πr².

Similar figures: corresponding sides proportional; areas scale by k² and volumes by k³ for linear scale factor k.

Circle: inscribed angle is half its intercepted arc; a tangent is perpendicular to the radius at the point of tangency.

Distance formula d = √[(x₂ − x₁)² + (y₂ − y₁)²]; midpoint is the average of coordinates.

Slopes: parallel lines have equal slopes; perpendicular lines have slopes whose product is −1.

Volume: prism/cylinder Bh; pyramid/cone ⅓Bh; sphere (4/3)πr³; surface area of sphere 4πr².

Regular polygon with n sides: interior angle = (n − 2)180/n; sum of exterior angles is always 360°.

Probability of an event = favorable outcomes / total equally likely outcomes; P(A or B) = P(A) + P(B) − P(A and B).

Independent events: P(A and B) = P(A)·P(B); conditional probability P(A|B) = P(A and B)/P(B).

Permutations count order: nPr = n!/(n − r)!; combinations ignore order: nCr = n!/[r!(n − r)!].

Mean = sum/n; median is the middle value of ordered data; mode is most frequent; the mean is pulled toward skew.

Standard deviation measures typical distance from the mean; about 68% of normal data lies within 1 SD, 95% within 2 SD.

z-score = (x − μ)/σ; it states how many standard deviations a value lies from the mean.

Expected value of a discrete variable = Σ x·P(x); a fair game has expected net gain of zero.

Correlation r ranges from −1 to 1 and measures linear association only; correlation does not imply causation.

Sampling distribution of the mean has standard error σ/√n; larger samples narrow the spread.

Limit definition of derivative: f′(x) = lim(h→0) [f(x + h) − f(x)]/h; the derivative is the slope of the tangent line.

Power rule d/dx x^n = n x^(n−1); product rule (uv)′ = u′v + uv′; quotient rule (u/v)′ = (u′v − uv′)/v².

Chain rule d/dx f(g(x)) = f′(g(x))·g′(x); derivative of sin x is cos x, of cos x is −sin x, of e^x is e^x, of ln x is 1/x.

A function has a local extremum where f′ = 0 or is undefined; sign change of f′ confirms max or min.

Second derivative f″ > 0 means concave up (local minimum); f″ < 0 means concave down (local maximum).

Fundamental Theorem: ∫_a^b f(x) dx = F(b) − F(a) where F′ = f.

Area between curves y = f(x) and y = g(x) on [a, b] is ∫_a^b |f − g| dx.

Riemann sums with n rectangles approximate the definite integral; the limit as n → ∞ is exact.

Mean Value Theorem: if f is continuous on [a, b] and differentiable on (a, b), some c has f′(c) = [f(b) − f(a)]/(b − a).

Euclid's Elements (c. 300 BCE) organized geometry as an axiomatic-deductive system with five postulates.

Pythagoras and his school (6th century BCE) linked number and geometry; the discovery of irrational √2 is attributed to that circle.

Archimedes (c. 287–212 BCE) used exhaustion to approximate π and derived areas and volumes of curved figures.

Al-Khwarizmi (c. 820 CE) wrote on solving quadratic equations; his name yields the word algorithm, and al-jabr yields algebra.

Descartes (1637) joined algebra and geometry with coordinate axes; Fermat independently developed analytic geometry.

Newton and Leibniz independently developed calculus in the late 1600s; Newton used fluxions, Leibniz the dy/dx notation.

Euler (1707–1783) introduced e, the function notation f(x), and the identity e^(iπ) + 1 = 0.

Cantor (late 1800s) founded set theory and proved the reals are uncountably infinite.

Non-Euclidean geometry was developed in the 1800s by Lobachevsky, Bolyai, and Riemann, showing the parallel postulate is independent.

Direct proof assumes the hypothesis and derives the conclusion through valid inferences.

Proof by contradiction assumes the negation of the statement and derives a contradiction, establishing the original.

Proof by contrapositive proves ¬Q → ¬P, which is logically equivalent to P → Q.

Mathematical induction: base case P(1), then assume P(k) and prove P(k + 1); valid only for statements indexed by integers.

A counterexample disproves a universal statement; one example never proves a universal claim.

Converse (Q → P) and inverse (¬P → ¬Q) are equivalent to each other but not to the original conditional.

A statement is a biconditional (iff) only when both the conditional and its converse hold.

Van Hiele levels describe geometry learning: visualization, analysis, informal deduction, formal deduction, rigor.

Manipulatives and multiple representations build conceptual understanding before procedural fluency; formative assessment guides instruction.

Rational numbers are exactly the terminating or eventually repeating decimals; irrationals have non-terminating, non-repeating expansions.

Closure: integers are closed under +, −, × but not ÷; rationals are closed under all four operations except division by zero.

Order of operations: grouping symbols, exponents/roots, multiplication and division left to right, then addition and subtraction left to right.

Field axioms for rationals and reals: commutativity, associativity, distributivity, additive and multiplicative identities and inverses.

Absolute value satisfies |ab| = |a||b| and the triangle inequality |a + b| ≤ |a| + |b|; equality holds when ab ≥ 0.

Scientific notation: a × 10^n with 1 ≤ |a| < 10; multiplying adds exponents, dividing subtracts them.

Complex numbers: i² = −1; |a + bi| = √(a² + b²); the conjugate a − bi gives quotient (a + bi)(c − di)/(c² + d²).

Percent change = (new − old)/old × 100; successive percent changes multiply rather than add.

Divisibility: a number is divisible by 3 or 9 iff its digit sum is; by 4 iff its last two digits form a multiple of 4.

Quadratic formula: x = [−b ± √(b² − 4ac)]/(2a); the discriminant b² − 4ac tells 2, 1, or 0 real roots.

Vertex of y = ax² + bx + c occurs at x = −b/(2a); in vertex form y = a(x − h)² + k the vertex is (h, k).

Sum and product of roots of ax² + bx + c = 0 are −b/a and c/a.

Exponent rules: a^m · a^n = a^(m+n); (a^m)^n = a^(mn); a^(−n) = 1/a^n; a^(m/n) = ⁿ√(a^m).

Log rules: log(xy) = log x + log y; log(x/y) = log x − log y; log(x^k) = k log x; log_b b = 1, log_b 1 = 0.

Change of base: log_b x = ln x / ln b; exponential y = a·b^x has growth rate b − 1 per period.

Arithmetic sequence: a_n = a_1 + (n − 1)d, sum = n(a_1 + a_n)/2. Geometric: a_n = a_1 r^(n−1), sum = a_1(1 − r^n)/(1 − r).

Function composition is not commutative: (f∘g)(x) = f(g(x)); inverse functions satisfy f(f⁻¹(x)) = x and swap domain and range.

Rational function f(x) = p(x)/q(x) has vertical asymptotes where q(x) = 0 (unless the factor cancels) and horizontal asymptote from degree comparison.

Triangle angle sum is 180°; exterior angle equals the sum of the two remote interior angles.

Pythagorean theorem a² + b² = c²; triples include 3-4-5, 5-12-13, 8-15-17, and their multiples.

Area formulas: triangle ½bh; trapezoid ½(b₁ + b₂)h; circle πr²; sector (θ/360)πr².

Similar figures: corresponding sides proportional; areas scale by k² and volumes by k³ for linear scale factor k.

Circle: inscribed angle is half its intercepted arc; a tangent is perpendicular to the radius at the point of tangency.

Distance formula d = √[(x₂ − x₁)² + (y₂ − y₁)²]; midpoint is the average of coordinates.

Slopes: parallel lines have equal slopes; perpendicular lines have slopes whose product is −1.

Volume: prism/cylinder Bh; pyramid/cone ⅓Bh; sphere (4/3)πr³; surface area of sphere 4πr².

Regular polygon with n sides: interior angle = (n − 2)180/n; sum of exterior angles is always 360°.

Probability of an event = favorable outcomes / total equally likely outcomes; P(A or B) = P(A) + P(B) − P(A and B).

Independent events: P(A and B) = P(A)·P(B); conditional probability P(A|B) = P(A and B)/P(B).

Permutations count order: nPr = n!/(n − r)!; combinations ignore order: nCr = n!/[r!(n − r)!].

Mean = sum/n; median is the middle value of ordered data; mode is most frequent; the mean is pulled toward skew.

Standard deviation measures typical distance from the mean; about 68% of normal data lies within 1 SD, 95% within 2 SD.

z-score = (x − μ)/σ; it states how many standard deviations a value lies from the mean.

Expected value of a discrete variable = Σ x·P(x); a fair game has expected net gain of zero.

Correlation r ranges from −1 to 1 and measures linear association only; correlation does not imply causation.

Sampling distribution of the mean has standard error σ/√n; larger samples narrow the spread.

Limit definition of derivative: f′(x) = lim(h→0) [f(x + h) − f(x)]/h; the derivative is the slope of the tangent line.

Power rule d/dx x^n = n x^(n−1); product rule (uv)′ = u′v + uv′; quotient rule (u/v)′ = (u′v − uv′)/v².

Chain rule d/dx f(g(x)) = f′(g(x))·g′(x); derivative of sin x is cos x, of cos x is −sin x, of e^x is e^x, of ln x is 1/x.

A function has a local extremum where f′ = 0 or is undefined; sign change of f′ confirms max or min.

Second derivative f″ > 0 means concave up (local minimum); f″ < 0 means concave down (local maximum).

Fundamental Theorem: ∫_a^b f(x) dx = F(b) − F(a) where F′ = f.

Area between curves y = f(x) and y = g(x) on [a, b] is ∫_a^b |f − g| dx.

Riemann sums with n rectangles approximate the definite integral; the limit as n → ∞ is exact.

Mean Value Theorem: if f is continuous on [a, b] and differentiable on (a, b), some c has f′(c) = [f(b) − f(a)]/(b − a).

Euclid's Elements (c. 300 BCE) organized geometry as an axiomatic-deductive system with five postulates.

Pythagoras and his school (6th century BCE) linked number and geometry; the discovery of irrational √2 is attributed to that circle.

Archimedes (c. 287–212 BCE) used exhaustion to approximate π and derived areas and volumes of curved figures.

Al-Khwarizmi (c. 820 CE) wrote on solving quadratic equations; his name yields the word algorithm, and al-jabr yields algebra.

Descartes (1637) joined algebra and geometry with coordinate axes; Fermat independently developed analytic geometry.

Newton and Leibniz independently developed calculus in the late 1600s; Newton used fluxions, Leibniz the dy/dx notation.

Euler (1707–1783) introduced e, the function notation f(x), and the identity e^(iπ) + 1 = 0.

Cantor (late 1800s) founded set theory and proved the reals are uncountably infinite.

Non-Euclidean geometry was developed in the 1800s by Lobachevsky, Bolyai, and Riemann, showing the parallel postulate is independent.

Direct proof assumes the hypothesis and derives the conclusion through valid inferences.

Proof by contradiction assumes the negation of the statement and derives a contradiction, establishing the original.

Proof by contrapositive proves ¬Q → ¬P, which is logically equivalent to P → Q.

Mathematical induction: base case P(1), then assume P(k) and prove P(k + 1); valid only for statements indexed by integers.

A counterexample disproves a universal statement; one example never proves a universal claim.

Converse (Q → P) and inverse (¬P → ¬Q) are equivalent to each other but not to the original conditional.

A statement is a biconditional (iff) only when both the conditional and its converse hold.

Van Hiele levels describe geometry learning: visualization, analysis, informal deduction, formal deduction, rigor.

Manipulatives and multiple representations build conceptual understanding before procedural fluency; formative assessment guides instruction.

Rational numbers are exactly the terminating or eventually repeating decimals; irrationals have non-terminating, non-repeating expansions.

Closure: integers are closed under +, −, × but not ÷; rationals are closed under all four operations except division by zero.

Order of operations: grouping symbols, exponents/roots, multiplication and division left to right, then addition and subtraction left to right.

Field axioms for rationals and reals: commutativity, associativity, distributivity, additive and multiplicative identities and inverses.

Absolute value satisfies |ab| = |a||b| and the triangle inequality |a + b| ≤ |a| + |b|; equality holds when ab ≥ 0.

Scientific notation: a × 10^n with 1 ≤ |a| < 10; multiplying adds exponents, dividing subtracts them.

Complex numbers: i² = −1; |a + bi| = √(a² + b²); the conjugate a − bi gives quotient (a + bi)(c − di)/(c² + d²).

Percent change = (new − old)/old × 100; successive percent changes multiply rather than add.

Divisibility: a number is divisible by 3 or 9 iff its digit sum is; by 4 iff its last two digits form a multiple of 4.

Quadratic formula: x = [−b ± √(b² − 4ac)]/(2a); the discriminant b² − 4ac tells 2, 1, or 0 real roots.

Vertex of y = ax² + bx + c occurs at x = −b/(2a); in vertex form y = a(x − h)² + k the vertex is (h, k).

Sum and product of roots of ax² + bx + c = 0 are −b/a and c/a.

Exponent rules: a^m · a^n = a^(m+n); (a^m)^n = a^(mn); a^(−n) = 1/a^n; a^(m/n) = ⁿ√(a^m).

Log rules: log(xy) = log x + log y; log(x/y) = log x − log y; log(x^k) = k log x; log_b b = 1, log_b 1 = 0.

Change of base: log_b x = ln x / ln b; exponential y = a·b^x has growth rate b − 1 per period.

Arithmetic sequence: a_n = a_1 + (n − 1)d, sum = n(a_1 + a_n)/2. Geometric: a_n = a_1 r^(n−1), sum = a_1(1 − r^n)/(1 − r).

Function composition is not commutative: (f∘g)(x) = f(g(x)); inverse functions satisfy f(f⁻¹(x)) = x and swap domain and range.

Rational function f(x) = p(x)/q(x) has vertical asymptotes where q(x) = 0 (unless the factor cancels) and horizontal asymptote from degree comparison.

Triangle angle sum is 180°; exterior angle equals the sum of the two remote interior angles.

Pythagorean theorem a² + b² = c²; triples include 3-4-5, 5-12-13, 8-15-17, and their multiples.

Area formulas: triangle ½bh; trapezoid ½(b₁ + b₂)h; circle πr²; sector (θ/360)πr².

Similar figures: corresponding sides proportional; areas scale by k² and volumes by k³ for linear scale factor k.

Circle: inscribed angle is half its intercepted arc; a tangent is perpendicular to the radius at the point of tangency.

Distance formula d = √[(x₂ − x₁)² + (y₂ − y₁)²]; midpoint is the average of coordinates.

Slopes: parallel lines have equal slopes; perpendicular lines have slopes whose product is −1.

Volume: prism/cylinder Bh; pyramid/cone ⅓Bh; sphere (4/3)πr³; surface area of sphere 4πr².

Regular polygon with n sides: interior angle = (n − 2)180/n; sum of exterior angles is always 360°.

Probability of an event = favorable outcomes / total equally likely outcomes; P(A or B) = P(A) + P(B) − P(A and B).

Independent events: P(A and B) = P(A)·P(B); conditional probability P(A|B) = P(A and B)/P(B).

Permutations count order: nPr = n!/(n − r)!; combinations ignore order: nCr = n!/[r!(n − r)!].

Mean = sum/n; median is the middle value of ordered data; mode is most frequent; the mean is pulled toward skew.

Standard deviation measures typical distance from the mean; about 68% of normal data lies within 1 SD, 95% within 2 SD.

z-score = (x − μ)/σ; it states how many standard deviations a value lies from the mean.

Expected value of a discrete variable = Σ x·P(x); a fair game has expected net gain of zero.

Correlation r ranges from −1 to 1 and measures linear association only; correlation does not imply causation.

Sampling distribution of the mean has standard error σ/√n; larger samples narrow the spread.

Limit definition of derivative: f′(x) = lim(h→0) [f(x + h) − f(x)]/h; the derivative is the slope of the tangent line.

Power rule d/dx x^n = n x^(n−1); product rule (uv)′ = u′v + uv′; quotient rule (u/v)′ = (u′v − uv′)/v².

Chain rule d/dx f(g(x)) = f′(g(x))·g′(x); derivative of sin x is cos x, of cos x is −sin x, of e^x is e^x, of ln x is 1/x.

A function has a local extremum where f′ = 0 or is undefined; sign change of f′ confirms max or min.

Second derivative f″ > 0 means concave up (local minimum); f″ < 0 means concave down (local maximum).

Fundamental Theorem: ∫_a^b f(x) dx = F(b) − F(a) where F′ = f.

Area between curves y = f(x) and y = g(x) on [a, b] is ∫_a^b |f − g| dx.

Riemann sums with n rectangles approximate the definite integral; the limit as n → ∞ is exact.

Mean Value Theorem: if f is continuous on [a, b] and differentiable on (a, b), some c has f′(c) = [f(b) − f(a)]/(b − a).

Euclid's Elements (c. 300 BCE) organized geometry as an axiomatic-deductive system with five postulates.

Pythagoras and his school (6th century BCE) linked number and geometry; the discovery of irrational √2 is attributed to that circle.

Archimedes (c. 287–212 BCE) used exhaustion to approximate π and derived areas and volumes of curved figures.

Al-Khwarizmi (c. 820 CE) wrote on solving quadratic equations; his name yields the word algorithm, and al-jabr yields algebra.

Descartes (1637) joined algebra and geometry with coordinate axes; Fermat independently developed analytic geometry.

Newton and Leibniz independently developed calculus in the late 1600s; Newton used fluxions, Leibniz the dy/dx notation.

Euler (1707–1783) introduced e, the function notation f(x), and the identity e^(iπ) + 1 = 0.

Cantor (late 1800s) founded set theory and proved the reals are uncountably infinite.

Non-Euclidean geometry was developed in the 1800s by Lobachevsky, Bolyai, and Riemann, showing the parallel postulate is independent.

Direct proof assumes the hypothesis and derives the conclusion through valid inferences.

Proof by contradiction assumes the negation of the statement and derives a contradiction, establishing the original.

Proof by contrapositive proves ¬Q → ¬P, which is logically equivalent to P → Q.

Mathematical induction: base case P(1), then assume P(k) and prove P(k + 1); valid only for statements indexed by integers.

A counterexample disproves a universal statement; one example never proves a universal claim.

Converse (Q → P) and inverse (¬P → ¬Q) are equivalent to each other but not to the original conditional.

A statement is a biconditional (iff) only when both the conditional and its converse hold.

Van Hiele levels describe geometry learning: visualization, analysis, informal deduction, formal deduction, rigor.

Manipulatives and multiple representations build conceptual understanding before procedural fluency; formative assessment guides instruction.

Rational numbers are exactly the terminating or eventually repeating decimals; irrationals have non-terminating, non-repeating expansions.

Closure: integers are closed under +, −, × but not ÷; rationals are closed under all four operations except division by zero.

Order of operations: grouping symbols, exponents/roots, multiplication and division left to right, then addition and subtraction left to right.

Field axioms for rationals and reals: commutativity, associativity, distributivity, additive and multiplicative identities and inverses.

Absolute value satisfies |ab| = |a||b| and the triangle inequality |a + b| ≤ |a| + |b|; equality holds when ab ≥ 0.

Scientific notation: a × 10^n with 1 ≤ |a| < 10; multiplying adds exponents, dividing subtracts them.

Complex numbers: i² = −1; |a + bi| = √(a² + b²); the conjugate a − bi gives quotient (a + bi)(c − di)/(c² + d²).

Percent change = (new − old)/old × 100; successive percent changes multiply rather than add.

Divisibility: a number is divisible by 3 or 9 iff its digit sum is; by 4 iff its last two digits form a multiple of 4.

Quadratic formula: x = [−b ± √(b² − 4ac)]/(2a); the discriminant b² − 4ac tells 2, 1, or 0 real roots.

Vertex of y = ax² + bx + c occurs at x = −b/(2a); in vertex form y = a(x − h)² + k the vertex is (h, k).

Sum and product of roots of ax² + bx + c = 0 are −b/a and c/a.

Exponent rules: a^m · a^n = a^(m+n); (a^m)^n = a^(mn); a^(−n) = 1/a^n; a^(m/n) = ⁿ√(a^m).

Log rules: log(xy) = log x + log y; log(x/y) = log x − log y; log(x^k) = k log x; log_b b = 1, log_b 1 = 0.

Change of base: log_b x = ln x / ln b; exponential y = a·b^x has growth rate b − 1 per period.

Arithmetic sequence: a_n = a_1 + (n − 1)d, sum = n(a_1 + a_n)/2. Geometric: a_n = a_1 r^(n−1), sum = a_1(1 − r^n)/(1 − r).

Function composition is not commutative: (f∘g)(x) = f(g(x)); inverse functions satisfy f(f⁻¹(x)) = x and swap domain and range.

Rational function f(x) = p(x)/q(x) has vertical asymptotes where q(x) = 0 (unless the factor cancels) and horizontal asymptote from degree comparison.

Triangle angle sum is 180°; exterior angle equals the sum of the two remote interior angles.

Pythagorean theorem a² + b² = c²; triples include 3-4-5, 5-12-13, 8-15-17, and their multiples.

Area formulas: triangle ½bh; trapezoid ½(b₁ + b₂)h; circle πr²; sector (θ/360)πr².

Similar figures: corresponding sides proportional; areas scale by k² and volumes by k³ for linear scale factor k.

Circle: inscribed angle is half its intercepted arc; a tangent is perpendicular to the radius at the point of tangency.

Distance formula d = √[(x₂ − x₁)² + (y₂ − y₁)²]; midpoint is the average of coordinates.

Slopes: parallel lines have equal slopes; perpendicular lines have slopes whose product is −1.

Volume: prism/cylinder Bh; pyramid/cone ⅓Bh; sphere (4/3)πr³; surface area of sphere 4πr².

Regular polygon with n sides: interior angle = (n − 2)180/n; sum of exterior angles is always 360°.

Probability of an event = favorable outcomes / total equally likely outcomes; P(A or B) = P(A) + P(B) − P(A and B).

Independent events: P(A and B) = P(A)·P(B); conditional probability P(A|B) = P(A and B)/P(B).

Permutations count order: nPr = n!/(n − r)!; combinations ignore order: nCr = n!/[r!(n − r)!].

Mean = sum/n; median is the middle value of ordered data; mode is most frequent; the mean is pulled toward skew.

Standard deviation measures typical distance from the mean; about 68% of normal data lies within 1 SD, 95% within 2 SD.

z-score = (x − μ)/σ; it states how many standard deviations a value lies from the mean.

Expected value of a discrete variable = Σ x·P(x); a fair game has expected net gain of zero.

Correlation r ranges from −1 to 1 and measures linear association only; correlation does not imply causation.

Sampling distribution of the mean has standard error σ/√n; larger samples narrow the spread.

Limit definition of derivative: f′(x) = lim(h→0) [f(x + h) − f(x)]/h; the derivative is the slope of the tangent line.

Power rule d/dx x^n = n x^(n−1); product rule (uv)′ = u′v + uv′; quotient rule (u/v)′ = (u′v − uv′)/v².

Chain rule d/dx f(g(x)) = f′(g(x))·g′(x); derivative of sin x is cos x, of cos x is −sin x, of e^x is e^x, of ln x is 1/x.

A function has a local extremum where f′ = 0 or is undefined; sign change of f′ confirms max or min.

Second derivative f″ > 0 means concave up (local minimum); f″ < 0 means concave down (local maximum).

Fundamental Theorem: ∫_a^b f(x) dx = F(b) − F(a) where F′ = f.

Area between curves y = f(x) and y = g(x) on [a, b] is ∫_a^b |f − g| dx.

Riemann sums with n rectangles approximate the definite integral; the limit as n → ∞ is exact.

Mean Value Theorem: if f is continuous on [a, b] and differentiable on (a, b), some c has f′(c) = [f(b) − f(a)]/(b − a).

Euclid's Elements (c. 300 BCE) organized geometry as an axiomatic-deductive system with five postulates.

Pythagoras and his school (6th century BCE) linked number and geometry; the discovery of irrational √2 is attributed to that circle.

Archimedes (c. 287–212 BCE) used exhaustion to approximate π and derived areas and volumes of curved figures.

Al-Khwarizmi (c. 820 CE) wrote on solving quadratic equations; his name yields the word algorithm, and al-jabr yields algebra.

Descartes (1637) joined algebra and geometry with coordinate axes; Fermat independently developed analytic geometry.

Newton and Leibniz independently developed calculus in the late 1600s; Newton used fluxions, Leibniz the dy/dx notation.

Euler (1707–1783) introduced e, the function notation f(x), and the identity e^(iπ) + 1 = 0.

Cantor (late 1800s) founded set theory and proved the reals are uncountably infinite.

Non-Euclidean geometry was developed in the 1800s by Lobachevsky, Bolyai, and Riemann, showing the parallel postulate is independent.

Direct proof assumes the hypothesis and derives the conclusion through valid inferences.

Proof by contradiction assumes the negation of the statement and derives a contradiction, establishing the original.

Proof by contrapositive proves ¬Q → ¬P, which is logically equivalent to P → Q.

Mathematical induction: base case P(1), then assume P(k) and prove P(k + 1); valid only for statements indexed by integers.

A counterexample disproves a universal statement; one example never proves a universal claim.

Converse (Q → P) and inverse (¬P → ¬Q) are equivalent to each other but not to the original conditional.

A statement is a biconditional (iff) only when both the conditional and its converse hold.

Van Hiele levels describe geometry learning: visualization, analysis, informal deduction, formal deduction, rigor.

Manipulatives and multiple representations build conceptual understanding before procedural fluency; formative assessment guides instruction.

Rational numbers are exactly the terminating or eventually repeating decimals; irrationals have non-terminating, non-repeating expansions.

Closure: integers are closed under +, −, × but not ÷; rationals are closed under all four operations except division by zero.

Order of operations: grouping symbols, exponents/roots, multiplication and division left to right, then addition and subtraction left to right.

Field axioms for rationals and reals: commutativity, associativity, distributivity, additive and multiplicative identities and inverses.

Absolute value satisfies |ab| = |a||b| and the triangle inequality |a + b| ≤ |a| + |b|; equality holds when ab ≥ 0.

Scientific notation: a × 10^n with 1 ≤ |a| < 10; multiplying adds exponents, dividing subtracts them.

Complex numbers: i² = −1; |a + bi| = √(a² + b²); the conjugate a − bi gives quotient (a + bi)(c − di)/(c² + d²).

Percent change = (new − old)/old × 100; successive percent changes multiply rather than add.

Divisibility: a number is divisible by 3 or 9 iff its digit sum is; by 4 iff its last two digits form a multiple of 4.

Quadratic formula: x = [−b ± √(b² − 4ac)]/(2a); the discriminant b² − 4ac tells 2, 1, or 0 real roots.

Vertex of y = ax² + bx + c occurs at x = −b/(2a); in vertex form y = a(x − h)² + k the vertex is (h, k).

Sum and product of roots of ax² + bx + c = 0 are −b/a and c/a.

Exponent rules: a^m · a^n = a^(m+n); (a^m)^n = a^(mn); a^(−n) = 1/a^n; a^(m/n) = ⁿ√(a^m).

Log rules: log(xy) = log x + log y; log(x/y) = log x − log y; log(x^k) = k log x; log_b b = 1, log_b 1 = 0.

Change of base: log_b x = ln x / ln b; exponential y = a·b^x has growth rate b − 1 per period.

Arithmetic sequence: a_n = a_1 + (n − 1)d, sum = n(a_1 + a_n)/2. Geometric: a_n = a_1 r^(n−1), sum = a_1(1 − r^n)/(1 − r).

Function composition is not commutative: (f∘g)(x) = f(g(x)); inverse functions satisfy f(f⁻¹(x)) = x and swap domain and range.

Rational function f(x) = p(x)/q(x) has vertical asymptotes where q(x) = 0 (unless the factor cancels) and horizontal asymptote from degree comparison.

Triangle angle sum is 180°; exterior angle equals the sum of the two remote interior angles.

Pythagorean theorem a² + b² = c²; triples include 3-4-5, 5-12-13, 8-15-17, and their multiples.

Area formulas: triangle ½bh; trapezoid ½(b₁ + b₂)h; circle πr²; sector (θ/360)πr².

Similar figures: corresponding sides proportional; areas scale by k² and volumes by k³ for linear scale factor k.

Circle: inscribed angle is half its intercepted arc; a tangent is perpendicular to the radius at the point of tangency.

Distance formula d = √[(x₂ − x₁)² + (y₂ − y₁)²]; midpoint is the average of coordinates.

Slopes: parallel lines have equal slopes; perpendicular lines have slopes whose product is −1.

Volume: prism/cylinder Bh; pyramid/cone ⅓Bh; sphere (4/3)πr³; surface area of sphere 4πr².

Regular polygon with n sides: interior angle = (n − 2)180/n; sum of exterior angles is always 360°.

Probability of an event = favorable outcomes / total equally likely outcomes; P(A or B) = P(A) + P(B) − P(A and B).

Independent events: P(A and B) = P(A)·P(B); conditional probability P(A|B) = P(A and B)/P(B).

Permutations count order: nPr = n!/(n − r)!; combinations ignore order: nCr = n!/[r!(n − r)!].

Mean = sum/n; median is the middle value of ordered data; mode is most frequent; the mean is pulled toward skew.

Standard deviation measures typical distance from the mean; about 68% of normal data lies within 1 SD, 95% within 2 SD.

z-score = (x − μ)/σ; it states how many standard deviations a value lies from the mean.

Expected value of a discrete variable = Σ x·P(x); a fair game has expected net gain of zero.

Correlation r ranges from −1 to 1 and measures linear association only; correlation does not imply causation.

Sampling distribution of the mean has standard error σ/√n; larger samples narrow the spread.

Limit definition of derivative: f′(x) = lim(h→0) [f(x + h) − f(x)]/h; the derivative is the slope of the tangent line.

Power rule d/dx x^n = n x^(n−1); product rule (uv)′ = u′v + uv′; quotient rule (u/v)′ = (u′v − uv′)/v².

Chain rule d/dx f(g(x)) = f′(g(x))·g′(x); derivative of sin x is cos x, of cos x is −sin x, of e^x is e^x, of ln x is 1/x.

A function has a local extremum where f′ = 0 or is undefined; sign change of f′ confirms max or min.

Second derivative f″ > 0 means concave up (local minimum); f″ < 0 means concave down (local maximum).

Fundamental Theorem: ∫_a^b f(x) dx = F(b) − F(a) where F′ = f.

Area between curves y = f(x) and y = g(x) on [a, b] is ∫_a^b |f − g| dx.

Riemann sums with n rectangles approximate the definite integral; the limit as n → ∞ is exact.

Mean Value Theorem: if f is continuous on [a, b] and differentiable on (a, b), some c has f′(c) = [f(b) − f(a)]/(b − a).

Euclid's Elements (c. 300 BCE) organized geometry as an axiomatic-deductive system with five postulates.

Pythagoras and his school (6th century BCE) linked number and geometry; the discovery of irrational √2 is attributed to that circle.

Archimedes (c. 287–212 BCE) used exhaustion to approximate π and derived areas and volumes of curved figures.

Al-Khwarizmi (c. 820 CE) wrote on solving quadratic equations; his name yields the word algorithm, and al-jabr yields algebra.

Descartes (1637) joined algebra and geometry with coordinate axes; Fermat independently developed analytic geometry.

Newton and Leibniz independently developed calculus in the late 1600s; Newton used fluxions, Leibniz the dy/dx notation.

Euler (1707–1783) introduced e, the function notation f(x), and the identity e^(iπ) + 1 = 0.

Cantor (late 1800s) founded set theory and proved the reals are uncountably infinite.

Non-Euclidean geometry was developed in the 1800s by Lobachevsky, Bolyai, and Riemann, showing the parallel postulate is independent.

Direct proof assumes the hypothesis and derives the conclusion through valid inferences.

Proof by contradiction assumes the negation of the statement and derives a contradiction, establishing the original.

Proof by contrapositive proves ¬Q → ¬P, which is logically equivalent to P → Q.

Mathematical induction: base case P(1), then assume P(k) and prove P(k + 1); valid only for statements indexed by integers.

A counterexample disproves a universal statement; one example never proves a universal claim.

Converse (Q → P) and inverse (¬P → ¬Q) are equivalent to each other but not to the original conditional.

A statement is a biconditional (iff) only when both the conditional and its converse hold.

Van Hiele levels describe geometry learning: visualization, analysis, informal deduction, formal deduction, rigor.

Manipulatives and multiple representations build conceptual understanding before procedural fluency; formative assessment guides instruction.

Rational numbers are exactly the terminating or eventually repeating decimals; irrationals have non-terminating, non-repeating expansions.

Closure: integers are closed under +, −, × but not ÷; rationals are closed under all four operations except division by zero.

Order of operations: grouping symbols, exponents/roots, multiplication and division left to right, then addition and subtraction left to right.

Field axioms for rationals and reals: commutativity, associativity, distributivity, additive and multiplicative identities and inverses.

Absolute value satisfies |ab| = |a||b| and the triangle inequality |a + b| ≤ |a| + |b|; equality holds when ab ≥ 0.

Scientific notation: a × 10^n with 1 ≤ |a| < 10; multiplying adds exponents, dividing subtracts them.

Complex numbers: i² = −1; |a + bi| = √(a² + b²); the conjugate a − bi gives quotient (a + bi)(c − di)/(c² + d²).

Percent change = (new − old)/old × 100; successive percent changes multiply rather than add.

Divisibility: a number is divisible by 3 or 9 iff its digit sum is; by 4 iff its last two digits form a multiple of 4.

Quadratic formula: x = [−b ± √(b² − 4ac)]/(2a); the discriminant b² − 4ac tells 2, 1, or 0 real roots.

Vertex of y = ax² + bx + c occurs at x = −b/(2a); in vertex form y = a(x − h)² + k the vertex is (h, k).

Sum and product of roots of ax² + bx + c = 0 are −b/a and c/a.

Exponent rules: a^m · a^n = a^(m+n); (a^m)^n = a^(mn); a^(−n) = 1/a^n; a^(m/n) = ⁿ√(a^m).

Log rules: log(xy) = log x + log y; log(x/y) = log x − log y; log(x^k) = k log x; log_b b = 1, log_b 1 = 0.

Change of base: log_b x = ln x / ln b; exponential y = a·b^x has growth rate b − 1 per period.

Arithmetic sequence: a_n = a_1 + (n − 1)d, sum = n(a_1 + a_n)/2. Geometric: a_n = a_1 r^(n−1), sum = a_1(1 − r^n)/(1 − r).

Function composition is not commutative: (f∘g)(x) = f(g(x)); inverse functions satisfy f(f⁻¹(x)) = x and swap domain and range.

Rational function f(x) = p(x)/q(x) has vertical asymptotes where q(x) = 0 (unless the factor cancels) and horizontal asymptote from degree comparison.

Triangle angle sum is 180°; exterior angle equals the sum of the two remote interior angles.

Pythagorean theorem a² + b² = c²; triples include 3-4-5, 5-12-13, 8-15-17, and their multiples.

Area formulas: triangle ½bh; trapezoid ½(b₁ + b₂)h; circle πr²; sector (θ/360)πr².

Similar figures: corresponding sides proportional; areas scale by k² and volumes by k³ for linear scale factor k.

Circle: inscribed angle is half its intercepted arc; a tangent is perpendicular to the radius at the point of tangency.

Distance formula d = √[(x₂ − x₁)² + (y₂ − y₁)²]; midpoint is the average of coordinates.

Slopes: parallel lines have equal slopes; perpendicular lines have slopes whose product is −1.

Volume: prism/cylinder Bh; pyramid/cone ⅓Bh; sphere (4/3)πr³; surface area of sphere 4πr².

Regular polygon with n sides: interior angle = (n − 2)180/n; sum of exterior angles is always 360°.

Probability of an event = favorable outcomes / total equally likely outcomes; P(A or B) = P(A) + P(B) − P(A and B).

Independent events: P(A and B) = P(A)·P(B); conditional probability P(A|B) = P(A and B)/P(B).

Permutations count order: nPr = n!/(n − r)!; combinations ignore order: nCr = n!/[r!(n − r)!].

Mean = sum/n; median is the middle value of ordered data; mode is most frequent; the mean is pulled toward skew.

Standard deviation measures typical distance from the mean; about 68% of normal data lies within 1 SD, 95% within 2 SD.

z-score = (x − μ)/σ; it states how many standard deviations a value lies from the mean.

Expected value of a discrete variable = Σ x·P(x); a fair game has expected net gain of zero.

Correlation r ranges from −1 to 1 and measures linear association only; correlation does not imply causation.

Sampling distribution of the mean has standard error σ/√n; larger samples narrow the spread.

Limit definition of derivative: f′(x) = lim(h→0) [f(x + h) − f(x)]/h; the derivative is the slope of the tangent line.

Power rule d/dx x^n = n x^(n−1); product rule (uv)′ = u′v + uv′; quotient rule (u/v)′ = (u′v − uv′)/v².

Chain rule d/dx f(g(x)) = f′(g(x))·g′(x); derivative of sin x is cos x, of cos x is −sin x, of e^x is e^x, of ln x is 1/x.

A function has a local extremum where f′ = 0 or is undefined; sign change of f′ confirms max or min.

Second derivative f″ > 0 means concave up (local minimum); f″ < 0 means concave down (local maximum).

Fundamental Theorem: ∫_a^b f(x) dx = F(b) − F(a) where F′ = f.

Area between curves y = f(x) and y = g(x) on [a, b] is ∫_a^b |f − g| dx.

Riemann sums with n rectangles approximate the definite integral; the limit as n → ∞ is exact.

Mean Value Theorem: if f is continuous on [a, b] and differentiable on (a, b), some c has f′(c) = [f(b) − f(a)]/(b − a).

Euclid's Elements (c. 300 BCE) organized geometry as an axiomatic-deductive system with five postulates.

Pythagoras and his school (6th century BCE) linked number and geometry; the discovery of irrational √2 is attributed to that circle.

Archimedes (c. 287–212 BCE) used exhaustion to approximate π and derived areas and volumes of curved figures.

Al-Khwarizmi (c. 820 CE) wrote on solving quadratic equations; his name yields the word algorithm, and al-jabr yields algebra.

Descartes (1637) joined algebra and geometry with coordinate axes; Fermat independently developed analytic geometry.

Newton and Leibniz independently developed calculus in the late 1600s; Newton used fluxions, Leibniz the dy/dx notation.

Euler (1707–1783) introduced e, the function notation f(x), and the identity e^(iπ) + 1 = 0.

Cantor (late 1800s) founded set theory and proved the reals are uncountably infinite.

Non-Euclidean geometry was developed in the 1800s by Lobachevsky, Bolyai, and Riemann, showing the parallel postulate is independent.

Direct proof assumes the hypothesis and derives the conclusion through valid inferences.

Proof by contradiction assumes the negation of the statement and derives a contradiction, establishing the original.

Proof by contrapositive proves ¬Q → ¬P, which is logically equivalent to P → Q.

Mathematical induction: base case P(1), then assume P(k) and prove P(k + 1); valid only for statements indexed by integers.

A counterexample disproves a universal statement; one example never proves a universal claim.

Converse (Q → P) and inverse (¬P → ¬Q) are equivalent to each other but not to the original conditional.

A statement is a biconditional (iff) only when both the conditional and its converse hold.

Van Hiele levels describe geometry learning: visualization, analysis, informal deduction, formal deduction, rigor.

Manipulatives and multiple representations build conceptual understanding before procedural fluency; formative assessment guides instruction.

Rational numbers are exactly the terminating or eventually repeating decimals; irrationals have non-terminating, non-repeating expansions.

Closure: integers are closed under +, −, × but not ÷; rationals are closed under all four operations except division by zero.

Order of operations: grouping symbols, exponents/roots, multiplication and division left to right, then addition and subtraction left to right.

Field axioms for rationals and reals: commutativity, associativity, distributivity, additive and multiplicative identities and inverses.

Absolute value satisfies |ab| = |a||b| and the triangle inequality |a + b| ≤ |a| + |b|; equality holds when ab ≥ 0.

Scientific notation: a × 10^n with 1 ≤ |a| < 10; multiplying adds exponents, dividing subtracts them.

Complex numbers: i² = −1; |a + bi| = √(a² + b²); the conjugate a − bi gives quotient (a + bi)(c − di)/(c² + d²).

Percent change = (new − old)/old × 100; successive percent changes multiply rather than add.

Divisibility: a number is divisible by 3 or 9 iff its digit sum is; by 4 iff its last two digits form a multiple of 4.

Quadratic formula: x = [−b ± √(b² − 4ac)]/(2a); the discriminant b² − 4ac tells 2, 1, or 0 real roots.

Vertex of y = ax² + bx + c occurs at x = −b/(2a); in vertex form y = a(x − h)² + k the vertex is (h, k).

Sum and product of roots of ax² + bx + c = 0 are −b/a and c/a.

Exponent rules: a^m · a^n = a^(m+n); (a^m)^n = a^(mn); a^(−n) = 1/a^n; a^(m/n) = ⁿ√(a^m).

Log rules: log(xy) = log x + log y; log(x/y) = log x − log y; log(x^k) = k log x; log_b b = 1, log_b 1 = 0.

Change of base: log_b x = ln x / ln b; exponential y = a·b^x has growth rate b − 1 per period.

Arithmetic sequence: a_n = a_1 + (n − 1)d, sum = n(a_1 + a_n)/2. Geometric: a_n = a_1 r^(n−1), sum = a_1(1 − r^n)/(1 − r).

Function composition is not commutative: (f∘g)(x) = f(g(x)); inverse functions satisfy f(f⁻¹(x)) = x and swap domain and range.

Rational function f(x) = p(x)/q(x) has vertical asymptotes where q(x) = 0 (unless the factor cancels) and horizontal asymptote from degree comparison.

Triangle angle sum is 180°; exterior angle equals the sum of the two remote interior angles.

Pythagorean theorem a² + b² = c²; triples include 3-4-5, 5-12-13, 8-15-17, and their multiples.

Area formulas: triangle ½bh; trapezoid ½(b₁ + b₂)h; circle πr²; sector (θ/360)πr².

Similar figures: corresponding sides proportional; areas scale by k² and volumes by k³ for linear scale factor k.

Circle: inscribed angle is half its intercepted arc; a tangent is perpendicular to the radius at the point of tangency.

Distance formula d = √[(x₂ − x₁)² + (y₂ − y₁)²]; midpoint is the average of coordinates.

Slopes: parallel lines have equal slopes; perpendicular lines have slopes whose product is −1.

Volume: prism/cylinder Bh; pyramid/cone ⅓Bh; sphere (4/3)πr³; surface area of sphere 4πr².

Regular polygon with n sides: interior angle = (n − 2)180/n; sum of exterior angles is always 360°.

Probability of an event = favorable outcomes / total equally likely outcomes; P(A or B) = P(A) + P(B) − P(A and B).

Independent events: P(A and B) = P(A)·P(B); conditional probability P(A|B) = P(A and B)/P(B).

Permutations count order: nPr = n!/(n − r)!; combinations ignore order: nCr = n!/[r!(n − r)!].

Mean = sum/n; median is the middle value of ordered data; mode is most frequent; the mean is pulled toward skew.

Standard deviation measures typical distance from the mean; about 68% of normal data lies within 1 SD, 95% within 2 SD.

z-score = (x − μ)/σ; it states how many standard deviations a value lies from the mean.

Expected value of a discrete variable = Σ x·P(x); a fair game has expected net gain of zero.

Correlation r ranges from −1 to 1 and measures linear association only; correlation does not imply causation.

Sampling distribution of the mean has standard error σ/√n; larger samples narrow the spread.

Limit definition of derivative: f′(x) = lim(h→0) [f(x + h) − f(x)]/h; the derivative is the slope of the tangent line.

Power rule d/dx x^n = n x^(n−1); product rule (uv)′ = u′v + uv′; quotient rule (u/v)′ = (u′v − uv′)/v².

Chain rule d/dx f(g(x)) = f′(g(x))·g′(x); derivative of sin x is cos x, of cos x is −sin x, of e^x is e^x, of ln x is 1/x.

A function has a local extremum where f′ = 0 or is undefined; sign change of f′ confirms max or min.

Second derivative f″ > 0 means concave up (local minimum); f″ < 0 means concave down (local maximum).

Fundamental Theorem: ∫_a^b f(x) dx = F(b) − F(a) where F′ = f.

Area between curves y = f(x) and y = g(x) on [a, b] is ∫_a^b |f − g| dx.

Riemann sums with n rectangles approximate the definite integral; the limit as n → ∞ is exact.

Mean Value Theorem: if f is continuous on [a, b] and differentiable on (a, b), some c has f′(c) = [f(b) − f(a)]/(b − a).

Euclid's Elements (c. 300 BCE) organized geometry as an axiomatic-deductive system with five postulates.

Pythagoras and his school (6th century BCE) linked number and geometry; the discovery of irrational √2 is attributed to that circle.

Archimedes (c. 287–212 BCE) used exhaustion to approximate π and derived areas and volumes of curved figures.

Al-Khwarizmi (c. 820 CE) wrote on solving quadratic equations; his name yields the word algorithm, and al-jabr yields algebra.

Descartes (1637) joined algebra and geometry with coordinate axes; Fermat independently developed analytic geometry.

Newton and Leibniz independently developed calculus in the late 1600s; Newton used fluxions, Leibniz the dy/dx notation.

Euler (1707–1783) introduced e, the function notation f(x), and the identity e^(iπ) + 1 = 0.

Cantor (late 1800s) founded set theory and proved the reals are uncountably infinite.

Non-Euclidean geometry was developed in the 1800s by Lobachevsky, Bolyai, and Riemann, showing the parallel postulate is independent.

Direct proof assumes the hypothesis and derives the conclusion through valid inferences.

Proof by contradiction assumes the negation of the statement and derives a contradiction, establishing the original.

Proof by contrapositive proves ¬Q → ¬P, which is logically equivalent to P → Q.

Mathematical induction: base case P(1), then assume P(k) and prove P(k + 1); valid only for statements indexed by integers.

A counterexample disproves a universal statement; one example never proves a universal claim.

Converse (Q → P) and inverse (¬P → ¬Q) are equivalent to each other but not to the original conditional.

A statement is a biconditional (iff) only when both the conditional and its converse hold.

Van Hiele levels describe geometry learning: visualization, analysis, informal deduction, formal deduction, rigor.

Manipulatives and multiple representations build conceptual understanding before procedural fluency; formative assessment guides instruction.