Fukushima Entrance Mathematics key terms

A counting number such as 1, 2, 3, used to count objects; in Japanese school mathematics it usually excludes 0 and negative numbers.

A whole number that may be positive, negative, or zero, with no fractional part; the set includes values like -3, 0, and 7.

A natural number greater than 1 whose positive divisors are only 1 and itself, such as 2, 3, 5, and 7.

Writing a natural number as a product of prime numbers, for example 84 = 2^2 × 3 × 7; each factorization is unique apart from order.

The distance of a number from zero on the number line, written with vertical bars; it is never negative, so |-5| = 5.

A number whose square equals a given value; every positive number has a positive and a negative square root, written with the radical sign.

A real number that cannot be written as a fraction of two integers; its decimal expansion never ends and never repeats, as with √2 or π.

A number expressible as a fraction of two integers with a nonzero denominator; its decimal form either terminates or repeats a block of digits.

An algebraic expression containing variables, such as 3x + 2y, used to state relationships generally rather than for one specific number.

An expression formed only by multiplying numbers and variables, such as 5x or -2ab; a sum of such terms is called a polynomial.

Terms that contain exactly the same variables raised to the same powers, so their coefficients can be added or subtracted to simplify an expression.

Removing brackets by multiplying every term inside by the terms outside, rewriting a product of sums as a sum of terms.

Writing an expression as a product of simpler expressions; it is the reverse of expansion and is used to solve quadratic equations.

Standard identities for expanding products of binomials, such as (a+b)^2 = a^2 + 2ab + b^2 and (a+b)(a-b) = a^2 - b^2.

An equation in which the variable appears only to the first power, such as 3x - 5 = 7; it normally has exactly one solution.

Two or more equations that must hold at the same time; solving means finding values satisfying all of them together, often by elimination or substitution.

An equation whose highest power of the variable is two, such as ax^2 + bx + c = 0 with a not zero; it can have up to two real solutions.

A formula giving the solutions of ax^2 + bx + c = 0 as x = (-b ± √(b^2 - 4ac)) / 2a, usable when factorization is difficult.

The quantity b^2 - 4ac in a quadratic equation; its sign tells whether there are two, one, or no real solutions.

A statement comparing two expressions with symbols such as <, >, ≤, or ≥; its solution is a range of values rather than a single number.

A straight line on which numbers are placed by position, with positive values to the right; used to picture order, absolute value, and inequality solutions.

A relationship in which one quantity is a constant multiple of another, written y = ax; doubling x doubles y and the graph is a straight line through the origin.

A relationship written y = a/x in which the product of the two quantities stays constant; as one increases the other decreases, and the graph is a curve.

A function of the form y = ax + b whose graph is a straight line; a is the slope, showing how much y changes per unit increase of x.

The rate of change of a line, equal to the change in y divided by the change in x between two points; it measures steepness and direction.

The coordinate where a graph crosses an axis; the y-intercept of y = ax + b is b, the value of y when x is zero.

The curved graph of a quadratic function y = ax^2 + bx + c; it opens upward when a is positive and downward when a is negative.

The highest or lowest point of a parabola, where the curve changes direction; its coordinates are read from the completed-square form of the function.

The set of all input values for which a function is defined; for example, the domain of y = 1/x excludes zero.

A relation between two figures that can be made to coincide exactly by sliding, rotating, or flipping; corresponding sides and angles are then equal.

Criteria proving two triangles are congruent: three pairs of sides equal, two sides and the included angle equal, or two angles and the included side equal.

A relation in which one figure is an enlargement or reduction of another; corresponding angles are equal and corresponding sides share one fixed ratio.

The constant ratio between corresponding side lengths of two similar figures; areas are in the square of this ratio and volumes in its cube.

In a right triangle the square of the hypotenuse equals the sum of the squares of the other two sides, written a^2 + b^2 = c^2.

An angle inscribed in a circle is half the central angle standing on the same arc; angles standing on the same arc are equal.

A logical argument that starts from given conditions and known facts and reaches a conclusion step by step, each step justified by a reason.

The conditions assumed true at the start of a proof, taken as given; reasoning proceeds from these toward the statement to be shown.

The statement a proof must establish; it is the final claim reached after justified steps from the hypothesis.

A proof method that assumes the opposite of what is to be shown, then derives an impossibility, so the original statement must be true.

The sum of all data values divided by how many there are; it uses every value but is strongly affected by unusually large or small 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.

The value that appears most often in a data set; a data set may have no mode or several modes.

A number from 0 to 1 expressing how likely an event is, found for equally likely outcomes by dividing favourable outcomes by all possible outcomes.

The count of distinct possible outcomes of a situation, often found by systematic listing, tree diagrams, or the product rule for successive choices.

An arrangement of objects in which order matters; the number of ways to arrange r objects chosen from n distinct ones is written nPr.

A selection of objects in which order does not matter; the number of ways to choose r objects from n distinct ones is written nCr.

Investigating a whole population by examining only part of it, chosen so the sample fairly represents the population being studied.

A rule pairing each allowed input value with exactly one output value; the set of inputs is the domain and the set of outputs the range.

Rules allowing the same operation on both sides of an equation without changing its solutions: adding, subtracting, multiplying, or dividing by a nonzero number.

Replacing a variable with a number or expression; in solving systems it means expressing one variable in terms of another and inserting that into the other equation.

A function of the form y = ax^2 + bx + c with a not zero; its graph is a parabola whose shape and direction depend on a.

The set of all output values a function actually produces as the input runs through its domain.

An angle whose vertex is the centre of a circle and whose sides are radii; it equals twice an inscribed angle standing on the same arc.

The set of all possible outcomes of a random trial; an event is a subset of this set, and probability compares its size with the whole.