Kyoto Entrance Mathematics key terms

A number on the continuous number line, including rationals such as 3/4 and irrationals such as √2 and π. In Kyoto entrance problems it is the default universe for variables unless a question restricts to integers.

A real number that cannot be written as a fraction of two integers. Its decimal expansion never terminates or repeats. Proofs of irrationality usually proceed by contradiction, assuming a reduced fraction p/q and deriving a divisibility conflict.

The distance of a number from zero on the number line, written |x|. It equals x when x ≥ 0 and −x when x < 0. Equations and inequalities with absolute values are solved by splitting into cases at the sign-change points.

Rewriting a polynomial as a product of lower-degree polynomials, for example x² − 5x + 6 = (x − 2)(x − 3). It is the standard first step for solving polynomial equations and simplifying rational expressions.

A polynomial P(x) has factor (x − a) exactly when P(a) = 0. It converts the search for factors into evaluating the polynomial at candidate values, typically divisors of the constant term.

When a polynomial P(x) is divided by x − a, the remainder equals P(a). Dividing by a linear expression ax + b gives remainder P(−b/a). It underlies both factor finding and synthetic division checks.

Multiplying out products of polynomials into a sum of monomials, using the distributive law and identities such as (a + b)³ = a³ + 3a²b + 3ab² + b³. Careful bookkeeping of like terms prevents sign and coefficient errors.

The identity (a + b)ⁿ = Σ C(n,k) a^(n−k) b^k for k from 0 to n, where C(n,k) is the binomial coefficient. It expands powers of sums and also yields identities for sums of binomial coefficients.

A statement comparing two expressions with <, ≤, >, or ≥. Solving means finding all values of the variable making it true; multiplying or dividing by a negative number reverses the direction of the sign.

Two or more inequalities that must hold simultaneously. The solution set is the intersection of the individual solution sets, often displayed on a number line or as a region in the plane.

The solutions of ax² + bx + c = 0 with a ≠ 0 are x = (−b ± √(b² − 4ac)) / (2a). It solves every quadratic over the reals or complexes, depending on the sign of the discriminant.

The quantity D = b² − 4ac for a quadratic ax² + bx + c. D > 0 gives two distinct real roots, D = 0 a repeated real root, and D < 0 two conjugate complex roots. It also decides the number of intersections of a parabola with a line.

For ax² + bx + c = 0 with roots α and β, the sum α + β = −b/a and the product αβ = c/a. These let you compute symmetric expressions in the roots without solving the equation.

Splitting a problem into disjoint cases that together cover all possibilities, then solving each separately. Typical triggers are absolute values, sign of a parameter, and the relative order of numbers in inequalities.

A proof technique that assumes the negation of the desired statement and derives an impossibility. It is the standard tool for proving irrationality, infinitude of primes, and uniqueness of certain representations.

The statement 'not Q implies not P', which is logically equivalent to 'P implies Q'. Proving the contrapositive is a valid way to prove the original implication, often easier when the conclusion is negative.

For 'P implies Q', P is sufficient for Q and Q is necessary for P. When both implications hold, P and Q are equivalent. Entrance problems often ask which condition is necessary, sufficient, or neither.

A collection of distinct objects, with union, intersection, difference, and complement as basic operations. Venn diagrams and De Morgan's laws help translate word problems into set equations and back.

The parabola y = ax² + bx + c, opening upward when a > 0 and downward when a < 0. Its axis of symmetry is x = −b/(2a) and its vertex, the extremum, is found by completing the square.

Rewriting ax² + bx + c as a(x − p)² + q, which reveals the vertex (p, q) and the range of the function. It is also the derivation route for the quadratic formula and for circle equations.

The largest and smallest values a function attains on a specified domain. For quadratics on a closed interval, compare values at the vertex (if inside) and at the endpoints; parameters may require case analysis.

The set of points (x, y) satisfying an inequality such as y > x² − 1, drawn as a shaded part of the plane with a solid or dashed boundary. Linear programming problems maximize a linear expression over a polygonal region.

For an acute angle in a right triangle, sine, cosine, and tangent are ratios of side lengths: opposite/hypotenuse, adjacent/hypotenuse, and opposite/adjacent. They extend to all angles through the unit circle.

In a triangle with sides a, b, c opposite angles A, B, C, we have a/sin A = b/sin B = c/sin C = 2R, where R is the circumradius. It links angles to sides when one pair is fully known.

In a triangle, a² = b² + c² − 2bc cos A, with analogous formulas for the other sides. It determines a side from two sides and the included angle, or an angle from all three sides.

Identities such as sin(α + β) = sin α cos β + cos α sin β and cos(α + β) = cos α cos β − sin α sin β. They generate double-angle and half-angle formulas and underpin trigonometric equation solving.

Writing a sin θ + b cos θ as r sin(θ + α) with r = √(a² + b²) and tan α = b/a. This reduces a sum to a single wave, making maxima, minima, and equation solutions immediate.

A function y = aˣ with a > 0 and a ≠ 1. It is increasing for a > 1 and decreasing for 0 < a < 1, always positive, and passes through (0, 1). Its inverse is the logarithm to base a.

The inverse of the exponential: y = log_a x means aʸ = x, defined only for x > 0. It converts products into sums and powers into multiples, which is why it solves exponential equations and inequalities.

The identity log_a b = log_c b / log_c a, valid for positive bases not equal to 1. It rewrites logarithms in a common base so that terms can be combined or compared in equations and inequalities.

The value a function or sequence approaches as its input approaches a point or infinity. Derivatives are defined as limits of difference quotients, and improper behavior such as division by zero is resolved by taking limits.

The limit of (f(a + h) − f(a))/h as h tends to 0, interpreted as the slope of the tangent to y = f(x) at x = a. It also gives the instantaneous rate of change of the quantity f.

The function f′(x) assigning to each x the derivative of f at x. Standard rules cover powers, products, quotients, and composites, and f′ tells where f increases, decreases, and attains extrema.

The straight line touching a curve at a point with the same slope as the curve there. For y = f(x) at x = a its equation is y − f(a) = f′(a)(x − a); tangents from an external point require solving for the contact point.

A table recording the sign of f′ on intervals between critical points, showing where f rises or falls and classifying each critical point as a local maximum, minimum, or neither. It is the standard tool for graphing and extremum problems.

The family of antiderivatives of a function, written ∫f(x)dx = F(x) + C where F′ = f. The arbitrary constant C matters when an initial condition or a definite integral later fixes it.

The signed area ∫ from a to b of f(x)dx, computed as F(b) − F(a) for any antiderivative F. It gives areas between curves, volumes of solids of revolution, and average values of functions.

The region between curves y = f(x) and y = g(x) from x = a to b has area ∫|f(x) − g(x)|dx. Finding intersection points first and splitting where the curves cross avoids sign errors.

Rotating a region about an axis produces a solid whose volume is ∫πy²dx (disk method) or ∫2πx·y dx (shell method). The choice of method depends on which axis the region is rotated around.

A sequence in which each term differs from the previous by a fixed common difference d. The n-th term is a + (n − 1)d and the sum of the first n terms is n(2a + (n − 1)d)/2.

A sequence in which each term is the previous term multiplied by a fixed common ratio r. The n-th term is a·r^(n−1); the infinite sum converges to a/(1 − r) exactly when |r| < 1.

An equation expressing a term of a sequence in terms of earlier terms, together with initial values. Standard types include linear recurrences solved by characteristic equations and those transformed by dividing or shifting.

A proof method for statements about all positive integers: verify the base case, assume the statement for n = k, and prove it for n = k + 1. Variants include strong induction and induction starting from a later integer.

A quantity with both magnitude and direction, represented by an arrow or by components. Addition follows the parallelogram law, and scalar multiplication stretches or reverses direction without changing the line of action.

For vectors a and b, the dot product is |a||b|cos θ, equal in components to a₁b₁ + a₂b₂ (plus a₃b₃ in space). It vanishes exactly when the vectors are perpendicular and gives the angle between them.

The vector from a fixed origin O to a point P, written with an arrow over OP. Expressing points as position vectors turns geometric conditions such as collinearity into linear relations among coefficients.

A number of the form a + bi with real a and b and i² = −1. Addition and multiplication follow ordinary algebra with i² replaced by −1; the real and imaginary parts are manipulated independently.

For z = a + bi, the conjugate is z̄ = a − bi, obtained by reflecting in the real axis. The product z·z̄ = a² + b² is real and nonnegative, which is how division by a complex number is performed.

The plane in which z = a + bi is plotted at coordinates (a, b). Multiplication by a complex number acts as rotation and scaling, so geometry problems about angles and distances become algebra on complex numbers.

An ordered arrangement of distinct objects. The number of ways to choose and order r objects from n is nPr = n!/(n − r)!. Circular arrangements and arrangements with repeated items need adjusted counts.

A selection of objects where order does not matter. The number of ways to choose r from n is nCr = n!/(r!(n − r)!), symmetric in r and n − r, and computed via Pascal's triangle for small n.

For equally likely outcomes, the probability of an event is the number of favorable outcomes divided by the total. Complementary events, unions, and conditional probabilities follow the standard addition and multiplication rules.

The probability-weighted average of a random variable: Σ x·P(X = x). It predicts long-run average outcomes in repeated trials and is used to judge whether a game or strategy is favorable.

The average of the squared deviations from the mean, computed as E(X²) − (E(X))². Its square root is the standard deviation; both measure how spread out data or a random variable is.

The largest positive integer dividing both of two integers. It is found by the Euclidean algorithm, repeatedly replacing the pair (a, b) with (b, a mod b) until the remainder is zero.

A procedure computing the greatest common divisor by repeated division with remainder: gcd(a, b) = gcd(b, r) where r is the remainder of a divided by b. It also yields integer solutions to linear Diophantine equations.

Two integers whose greatest common divisor is 1. Coprimality is the condition under which a fraction is in lowest terms and under which division by a number is legitimate in modular arithmetic.

The statement a ≡ b (mod m), meaning m divides a − b. Congruences may be added, subtracted, and multiplied freely; division requires the divisor to be coprime to the modulus, and solutions may form residue classes.