WEST-E Middle Level Mathematics (NES 203) cheat sheet

Rational numbers are exactly those expressible as p/q with integers p, q and q ≠ 0; their decimal expansions either terminate or eventually repeat.

Irrational numbers have non-terminating, non-repeating decimal expansions (e.g., √2, π); the square root of a non-perfect-square integer is irrational.

Absolute value |x| gives distance from 0 on the number line, so |a − b| is the distance between a and b.

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

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

For positive a, b: √(ab) = √a·√b and √(a/b) = √a/√b; these rules fail when both radicands are negative.

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

Percentage change = (new − old)/old × 100%; a p% increase followed by a p% decrease yields a net decrease of p²/100 percent.

Proportional reasoning: quantities are proportional iff their ratio is constant; scale factors apply multiplicatively, not additively, to lengths.

Slope between (x₁, y₁) and (x₂, y₂) is m = (y₂ − y₁)/(x₂ − x₁); parallel lines share m, perpendicular lines have m₁·m₂ = −1.

Point-slope form y − y₁ = m(x − x₁); slope-intercept y = mx + b; standard form Ax + By = C with slope −A/B.

A linear function f(x) = mx + b has constant first differences; an exponential f(x) = a·b^x has constant successive ratios.

Quadratic f(x) = ax² + bx + c has vertex x = −b/(2a); the discriminant b² − 4ac gives 0, 1, or 2 real roots when negative, zero, or positive.

Quadratic formula: x = (−b ± √(b² − 4ac))/(2a); factored form a(x − r₁)(x − r₂) reveals the zeros r₁, r₂.

Laws of exponents: a^m·a^n = a^(m+n), (a^m)^n = a^(mn), a^(−n) = 1/a^n, and a^0 = 1 for a ≠ 0.

A function has exactly one output per input; it is one-to-one iff it passes both the vertical and horizontal line tests, guaranteeing an inverse function.

Arithmetic sequences have aₙ = a₁ + (n − 1)d and sum n(a₁ + aₙ)/2; geometric sequences have aₙ = a₁·r^(n−1) and sum a₁(1 − rⁿ)/(1 − r) for r ≠ 1.

Solving systems by substitution or elimination yields one solution (intersecting lines), none (parallel), or infinitely many (coincident lines).

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

Pythagorean theorem a² + b² = c² for right triangles; the converse holds, and common triples include 3-4-5, 5-12-13, and 8-15-17.

Area formulas: triangle ½bh; parallelogram bh; trapezoid ½(b₁ + b₂)h; circle πr². Circumference is 2πr.

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

Similar figures have equal corresponding angles; corresponding lengths scale by k, areas by k², and volumes by k³.

Sum of interior angles of an n-gon is (n − 2)·180°; each interior angle of a regular n-gon is (n − 2)·180°/n.

A central angle equals its intercepted arc; an inscribed angle equals half its intercepted arc.

Distance formula d = √((x₂ − x₁)² + (y₂ − y₁)²); midpoint is ((x₁ + x₂)/2, (y₁ + y₂)/2).

Conversions: 1 in = 2.54 cm, 1 km ≈ 0.621 mi, 1 L = 1000 cm³; a dilation by factor k multiplies area by k².

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

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

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); for dependent events multiply by the conditional probability P(B|A).

Permutations count ordered arrangements: nPr = n!/(n − r)!; combinations count unordered selections: nCr = n!/(r!(n − r)!).

The Fundamental Counting Principle: if choices are made in stages, multiply the number of options at each stage.

Complement rule: P(not A) = 1 − P(A); probabilities always satisfy 0 ≤ P ≤ 1 and total 1 across a sample space.

A fair coin flipped n times has 2ⁿ equally likely outcome sequences; the binomial probability of k successes is nCk·p^k·(1 − p)^(n−k).

In a simple graph, the sum of all vertex degrees equals twice the number of edges, so that sum is always even.

Rational numbers are exactly those expressible as p/q with integers p, q and q ≠ 0; their decimal expansions either terminate or eventually repeat.

Irrational numbers have non-terminating, non-repeating decimal expansions (e.g., √2, π); the square root of a non-perfect-square integer is irrational.

Absolute value |x| gives distance from 0 on the number line, so |a − b| is the distance between a and b.

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

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

For positive a, b: √(ab) = √a·√b and √(a/b) = √a/√b; these rules fail when both radicands are negative.

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

Percentage change = (new − old)/old × 100%; a p% increase followed by a p% decrease yields a net decrease of p²/100 percent.

Proportional reasoning: quantities are proportional iff their ratio is constant; scale factors apply multiplicatively, not additively, to lengths.

Slope between (x₁, y₁) and (x₂, y₂) is m = (y₂ − y₁)/(x₂ − x₁); parallel lines share m, perpendicular lines have m₁·m₂ = −1.

Point-slope form y − y₁ = m(x − x₁); slope-intercept y = mx + b; standard form Ax + By = C with slope −A/B.

A linear function f(x) = mx + b has constant first differences; an exponential f(x) = a·b^x has constant successive ratios.

Quadratic f(x) = ax² + bx + c has vertex x = −b/(2a); the discriminant b² − 4ac gives 0, 1, or 2 real roots when negative, zero, or positive.

Quadratic formula: x = (−b ± √(b² − 4ac))/(2a); factored form a(x − r₁)(x − r₂) reveals the zeros r₁, r₂.

Laws of exponents: a^m·a^n = a^(m+n), (a^m)^n = a^(mn), a^(−n) = 1/a^n, and a^0 = 1 for a ≠ 0.

A function has exactly one output per input; it is one-to-one iff it passes both the vertical and horizontal line tests, guaranteeing an inverse function.

Arithmetic sequences have aₙ = a₁ + (n − 1)d and sum n(a₁ + aₙ)/2; geometric sequences have aₙ = a₁·r^(n−1) and sum a₁(1 − rⁿ)/(1 − r) for r ≠ 1.

Solving systems by substitution or elimination yields one solution (intersecting lines), none (parallel), or infinitely many (coincident lines).

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

Pythagorean theorem a² + b² = c² for right triangles; the converse holds, and common triples include 3-4-5, 5-12-13, and 8-15-17.

Area formulas: triangle ½bh; parallelogram bh; trapezoid ½(b₁ + b₂)h; circle πr². Circumference is 2πr.

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

Similar figures have equal corresponding angles; corresponding lengths scale by k, areas by k², and volumes by k³.

Sum of interior angles of an n-gon is (n − 2)·180°; each interior angle of a regular n-gon is (n − 2)·180°/n.

A central angle equals its intercepted arc; an inscribed angle equals half its intercepted arc.

Distance formula d = √((x₂ − x₁)² + (y₂ − y₁)²); midpoint is ((x₁ + x₂)/2, (y₁ + y₂)/2).

Conversions: 1 in = 2.54 cm, 1 km ≈ 0.621 mi, 1 L = 1000 cm³; a dilation by factor k multiplies area by k².

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

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

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); for dependent events multiply by the conditional probability P(B|A).

Permutations count ordered arrangements: nPr = n!/(n − r)!; combinations count unordered selections: nCr = n!/(r!(n − r)!).

The Fundamental Counting Principle: if choices are made in stages, multiply the number of options at each stage.

Complement rule: P(not A) = 1 − P(A); probabilities always satisfy 0 ≤ P ≤ 1 and total 1 across a sample space.

A fair coin flipped n times has 2ⁿ equally likely outcome sequences; the binomial probability of k successes is nCk·p^k·(1 − p)^(n−k).

In a simple graph, the sum of all vertex degrees equals twice the number of edges, so that sum is always even.