Okayama Public High School Entrance Examination - Mathematics key terms

A positive whole counting number such as 1, 2, 3, and so on, used for counting objects. In Japanese school mathematics zero is usually not included when the word 自然数 is used.

Any whole number that can be written without a fractional or decimal part, including positive numbers, zero, and negative numbers such as -3, 0, and 7.

A natural number greater than 1 whose only positive divisors are 1 and itself, such as 2, 3, 5, 7, and 11. The number 1 is not treated as prime.

Writing a natural number as a product of prime numbers only, for example 60 = 2² × 3 × 5. This form makes divisors, multiples, and square roots easy to find.

A number that divides another number exactly with no remainder. For 12 the positive divisors are 1, 2, 3, 4, 6, and 12.

A number obtained by multiplying a given integer by another integer, such as 3, 6, 9, and 12 for 3. Zero is also a multiple of every integer.

The largest positive integer that divides two or more given integers exactly. For 18 and 24 it is 6, found by comparing their prime factorizations.

The smallest positive integer that is a multiple of each of two or more given integers. For 4 and 6 it is 12, useful when adding fractions with different denominators.

The distance of a number from zero on the number line, always zero or positive. It is written with vertical bars, so |−5| = 5 and |5| = 5.

A number that, when multiplied by itself, gives the original number. For example 7 and −7 are square roots of 49, written ±√49 = ±7.

Rewriting a fraction so that no square root remains in the denominator, by multiplying numerator and denominator by a suitable root, as 1/√2 = √2/2.

Removing brackets by multiplying out every term, turning a product into a sum. For example (x + 3)(x − 5) expands to x² − 2x − 15.

Writing an expression as a product of simpler expressions, the reverse of expansion. For example x² − 5x + 6 factors as (x − 2)(x − 3).

Standard identities used to expand or factor expressions quickly, such as (a + b)² = a² + 2ab + b² and (a + b)(a − b) = a² − b².

An equation in which the highest power of the unknown is two, written in the form ax² + bx + c = 0 with a ≠ 0. It can have zero, one, or two real solutions.

A formula giving the solutions of ax² + bx + c = 0 as x = (−b ± √(b² − 4ac)) / 2a. It works even when factorization is not obvious.

The value b² − 4ac for ax² + bx + c = 0. Its sign tells whether the equation has two distinct real solutions, one repeated solution, or none.

A relationship where y = ax with a nonzero constant a, so that doubling x doubles y. Its graph is a straight line through the origin.

A relationship where y = a/x with a nonzero constant a, so the product xy stays fixed. Its graph is a pair of curves that never touch the axes.

A function written y = ax + b with a ≠ 0. Its graph is a straight line with slope a and y-intercept b.

The quotient (change in y) ÷ (change in x) between two points. For a linear function it equals the slope a and stays constant everywhere.

A number describing how steeply a line rises or falls, equal to the vertical change divided by the horizontal change between two points on it.

The coordinate where a graph crosses an axis. The y-intercept b is where the line y = ax + b meets the y-axis, at the point (0, b).

The curved graph of a quadratic function y = ax². It is symmetric about the y-axis and opens upward when a > 0, downward when a < 0.

The function in which y equals a constant a times the square of x. Its graph is a parabola through the origin, and the ratio of change grows as x grows.

A point shared by y = ax² and y = mx + n. Its x-coordinates are found by solving ax² = mx + n, a quadratic equation.

The set of all values the input variable x may take in a given function, sometimes restricted by the problem's situation, such as a length that cannot be negative.

The set of values a variable can take, written with inequalities such as 1 ≤ x ≤ 4. In function problems it often means the range of y for a stated x interval.

Two figures that can be made to coincide exactly by sliding, rotating, or flipping. Congruent triangles have all corresponding sides and angles equal.

Three standard sets of facts that guarantee two triangles are congruent: three pairs of sides, two sides and the included angle, or two angles and the included side.

Two figures with the same shape but possibly different size, where corresponding angles are equal and corresponding lengths share a constant ratio.

Three ways to show two triangles are similar: two pairs of equal angles, two pairs of proportional sides with equal included angles, or three pairs of proportional sides.

The constant ratio between corresponding lengths of two similar figures. Areas compare as the square of this ratio and volumes as its cube.

A logical argument that starts from given facts and known theorems and reaches a conclusion step by step, stating the reason for each step.

The conditions assumed true at the start of a proof, taken from the problem statement or a figure's given markings. Reasoning begins from these.

The statement a proof must establish, such as a particular angle equality or parallelism. It is the final claim reached after the reasoning.

An angle whose vertex lies on a circle is half the central angle standing on the same arc. Angles standing on the same arc are equal.

In a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides, written a² + b² = c².

A study that collects data from part of a larger group and uses the results to estimate features of the whole group, instead of examining every member.

The entire group of objects or people a survey aims to describe. Conclusions from a sample are meant to apply to this full group.

The part of a population actually observed in a sample survey. Its size and how it is chosen affect how reliable the estimate will be.

The middle value when data are arranged in order. With an even number of values it is the mean of the two middle values.

The value that appears most often in a data set. It is especially useful for categorical data or when a typical value is wanted.

The frequency of a class divided by the total number of data values, showing what fraction of the data falls in that class.

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

The event that a given event does not happen. Its probability equals 1 minus the probability of the event, often easier to compute.