Nara Public High School Entrance Examination - Mathematics key terms

A positive whole counting number such as 1, 2, 3, used to count objects. In Japanese school mathematics zero is usually not included and the set is written with N.

A whole number with no fractional part, including positive numbers, negative numbers, and zero. Results of adding, subtracting, and multiplying integers are always integers.

A natural number greater than 1 whose only positive divisors are 1 and itself, for example 2, 3, 5, 7, 11. The number 1 is not prime by definition.

Writing a natural number as a product of prime numbers, for example 84 = 2² × 3 × 7. This form makes divisors, multiples, and square roots easy to compute.

An integer that divides another integer with no remainder. For 12 the positive divisors are 1, 2, 3, 4, 6, 12, and the number of divisors can be found from the prime factorization.

A number obtained by multiplying an integer by another integer, such as 3, 6, 9 for multiples of 3. Common multiples and the least common multiple are found from prime factorization.

The largest positive integer that divides two or more integers exactly. It is computed by multiplying the shared prime factors, each taken to the smallest exponent appearing.

The smallest positive integer that is a multiple of two or more integers. It equals the product of all prime factors, each taken to the largest exponent appearing.

The distance of a number from zero on the number line, always zero or positive. The absolute value of −5 is 5, and of 5 is also 5.

A straight line on which points correspond to numbers, with a fixed origin, unit length, and positive direction. It is used to compare numbers and to picture absolute value and inequalities.

A number whose square equals a given non-negative number. The positive square root of a is written √a, and √a × √a = a holds for a ≥ 0.

Rewriting a fraction so that no square root remains in the denominator, for example turning 1/√2 into √2/2. Multiply numerator and denominator by the same root or conjugate.

A mathematical expression containing letters that stand for numbers, such as 3x + 2y. Rules for simplifying, expanding, and factoring apply to these expressions.

Removing parentheses by multiplying out every term, for example (x + 2)(x + 3) = x² + 5x + 6. The distributive law is the underlying rule.

Writing an expression as a product of simpler expressions, for example x² − 9 = (x + 3)(x − 3). It is the reverse operation of expansion and is used to solve quadratic equations.

A statement that two expressions are equal, containing one or more unknown letters. Solving means finding the values of the unknown that make the statement true.

Two or more equations that must be satisfied at the same time by the same unknowns. They are solved by elimination or substitution to find a common pair of values.

An equation in which the highest power of the unknown is two, written in the form ax² + bx + c = 0 with a ≠ 0. It is solved by factoring, completing the square, or the quadratic formula.

The formula giving the solutions of ax² + bx + c = 0 as x = (−b ± √(b² − 4ac)) / 2a, where a ≠ 0. It solves any quadratic equation, even when factoring is difficult.

The quantity D = b² − 4ac in a quadratic equation. If D > 0 there are two distinct real solutions, if D = 0 one repeated solution, and if D < 0 no real solutions.

A statement comparing two expressions using symbols such as <, >, ≤, or ≥. Solving an inequality gives a range of values, and multiplying or dividing by a negative number reverses the sign.

Replacing a letter in an expression or equation with a specific number or another expression. It is used to check solutions and to solve simultaneous equations.

A relationship in which y = ax with a nonzero constant a, so doubling x doubles y. The graph is a straight line through the origin.

A relationship in which y = a/x with a nonzero constant a, so the product xy stays constant. The graph is a curve with two branches that never touches the axes.

An ordered pair of numbers (x, y) locating a point on a plane relative to two perpendicular axes. The first value gives horizontal position and the second gives vertical position.

For a straight line y = ax + b, the constant a showing how much y changes when x increases by 1. It measures steepness and direction of the line.

The value where a graph crosses an axis. In y = ax + b the y-intercept is b, the point where the line meets the vertical axis.

The quotient of the change in y divided by the change in x between two points. For a linear function it equals the slope; for y = ax² it varies with the interval chosen.

A relationship in which y is a constant a times the square of x. Its graph is a parabola with vertex at the origin, symmetric about the y-axis, opening upward when a > 0.

The curved graph of a quadratic function such as y = ax². It has an axis of symmetry and a vertex, and appears in problems about a line intersecting the curve.

A relation between two figures that can be made to coincide exactly by translation, rotation, or reflection. Congruent triangles have equal corresponding sides and angles.

Three standard criteria: three pairs of corresponding sides equal; two sides and the included angle equal; or one side and two angles equal. Any one condition guarantees congruent triangles.

A relation in which two figures have the same shape, with corresponding angles equal and corresponding sides in a fixed ratio. Areas are in the square of that ratio and volumes in its cube.

The constant ratio between corresponding side lengths of two similar figures. If the ratio is m : n, the ratio of areas is m² : n² and of volumes is m³ : n³.

In a right triangle the square of the hypotenuse equals the sum of the squares of the other two sides, written a² + b² = c². It is used to find unknown lengths and distances.

An angle inscribed in a circle equals half the central angle standing on the same arc. Angles in the same segment are equal, and an angle in a semicircle is a right angle.

A logical argument starting from given conditions and known facts, reaching a conclusion step by step with stated reasons. In geometry it typically proves congruence, similarity, or a property of figures.

The conditions assumed true at the start of a proof, taken from the problem statement. Reasoning proceeds from the hypothesis to the conclusion using definitions and proven theorems.

The statement that a proof aims to establish. It must follow necessarily from the hypothesis through valid reasoning, not from examples or measurement.

The sum of all data values divided by the number of values. It represents the balance point of the data but is strongly affected by extreme values.

The middle value when data are arranged in order; with an even number of values it is the mean of the two middle ones. It is less affected by extreme values than the mean.

The value that appears most often in a data set. It can be used for categorical data as well as numerical data and may not be unique.

A number from 0 to 1 expressing how likely an event is. For equally likely outcomes it equals the number of favorable outcomes divided by the total number of outcomes.

The count of all possible distinct outcomes in a situation. It is found by systematic listing, tree diagrams, or the multiplication principle, and forms the denominator of a probability.

The event that a given event does not happen. Since an event and its complement together cover all outcomes, their probabilities add to 1.

Investigating a subset drawn from a larger population in order to estimate properties of the whole. Conclusions are stated with awareness that sampling introduces variation and error.