Toyama Prefectural Public High School Entrance Examination - Mathematics key terms

A whole number with no fractional or decimal part, such as 0, 7, or −3. It may be positive, negative, or zero, and is used when counting or describing exact quantities.

A counting number such as 1, 2, 3, and so on. In Japanese school mathematics it usually excludes zero and is used when the context requires positive whole counts.

The distance of a number from zero on the number line, written with vertical bars. It is never negative, so the absolute value of both 5 and −5 is 5.

A straight line on which every point corresponds to a number, with zero marked and positive numbers to the right. It is used to compare numbers and to picture absolute value and inequalities.

Writing a whole number greater than 1 as a product of prime numbers only. For example, 60 = 2 × 2 × 3 × 5. It helps find greatest common divisors and square roots.

A whole 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 considered prime.

A number that, when multiplied by itself, gives a specified number. The square root of 9 is 3 because 3 × 3 = 9; positive and negative roots both exist.

A number that cannot be written as a fraction of two integers. Its decimal form never ends and never repeats, as with the square root of 2 or the value of pi.

A number that can be expressed as a fraction whose numerator and denominator are integers, with a nonzero denominator. Its decimal form either ends or repeats a block of digits.

An expression that uses letters to stand for numbers, such as 3x + 2y. It allows general rules to be written compactly and manipulated by the usual rules of arithmetic.

Terms that have exactly the same letter part, such as 5x and −2x. Only like terms may be combined by adding or subtracting their coefficients to simplify an expression.

The numerical factor multiplying a letter in a term. In the term 7x the coefficient is 7, and in −3ab the coefficient is −3; it tells how many of the letter part there are.

Rewriting a product of brackets as a sum of terms by multiplying everything out. For example, (x + 2)(x + 3) expands to x² + 5x + 6.

Rewriting a sum or difference as a product of factors. For example, x² + 5x + 6 factorizes to (x + 2)(x + 3); it is the reverse process of expansion.

A standard identity used to expand or factorize quickly, such as (a + b)² = a² + 2ab + b² or (a + b)(a − b) = a² − b². These save time in calculations.

An equation in which the unknown appears only to the first power, such as 3x − 5 = 10. Solving it means finding the value of the unknown that makes both sides equal.

The rules that adding, subtracting, multiplying, or dividing both sides of an equation by the same nonzero number keeps it true. These rules justify each step in solving an equation.

Moving a term from one side of an equation to the other while changing its sign. For example, x + 4 = 9 becomes x = 9 − 4, giving x = 5.

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

Replacing a letter with a number or with an expression it equals. It is used to check solutions and to solve simultaneous equations by expressing one unknown in terms of the other.

An equation in which the highest power of the unknown is two, such as x² − 5x + 6 = 0. It is solved by factorization, completing the square, or the quadratic formula.

A formula giving the solutions of ax² + bx + c = 0 as x = (−b ± √(b² − 4ac)) ÷ 2a, where a is not zero. It solves any quadratic equation.

A statement comparing two quantities that are not necessarily equal, using signs such as <, >, ≤, or ≥. Its solution is a range of values rather than a single number.

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

A relationship in which the product of two quantities is constant, written y = a ÷ x. As one quantity increases the other decreases, and the graph is a curve that never touches the axes.

The fixed number a linking two proportionally related quantities, as in y = ax or y = a ÷ x. It can be found when one matching pair of values is known.

A function of the form y = ax + b whose graph is a straight line. The value a gives the steepness of the line and b gives where it crosses the vertical axis.

A measure of how steep a line is, equal to the change in y divided by the change in x between two points. It is the coefficient a in y = ax + b.

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

The curved graph of a quadratic function such as y = ax². It is symmetric about a vertical axis and opens upward when a is positive and downward when a is negative.

A function in which the variable is squared, of the form y = ax² with a not zero. Its graph is a parabola, and the rate of change is not constant.

The highest or lowest point of a parabola, where the curve turns around. It lies on the axis of symmetry and marks the maximum or minimum value of the function.

An ordered pair of numbers that locates a point on a plane, written as (x, y). The first number gives the horizontal position and the second the vertical position.

The two perpendicular number lines, horizontal and vertical, that set up a coordinate plane. They cross at the origin, where both coordinates are zero.

A relationship in which two figures have exactly the same shape and size, so one can be moved onto the other by sliding, turning, or flipping. Corresponding sides and angles are equal.

A relationship in which two figures have the same shape but not necessarily the same size. Corresponding angles are equal and corresponding sides share a fixed ratio.

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

Sides that occupy matching positions in two congruent or similar figures. In congruent figures they are equal in length; in similar figures they are in a fixed ratio.

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

A triangle containing one angle of exactly 90 degrees. The side opposite the right angle is the hypotenuse, and it is the longest side.

An angle formed at a point on a circle by two chords drawn from that point. It is half the size of the central angle standing on the same arc.

An angle whose vertex is the centre of a circle and whose sides are radii. It is twice any inscribed angle that stands on the same arc.

The amount of space a solid occupies, measured in cubic units. For prisms and cylinders it equals base area times height; for pyramids and cones it is one third of that.

The total area of all the faces of a solid figure, measured in square units. It is found by adding the areas of each face, including curved surfaces where present.

The sum of all data values divided by how many values there are. It describes the centre of the data but is strongly affected by unusually large or small values.

The middle value when the data are arranged in order of size. With an even number of values it is the mean of the two middle ones, and it is less affected by extremes than the mean.

The value that appears most often in a set of data. A data set may have no mode or several modes, and it is useful for non-numerical categories.

A number between 0 and 1 expressing how likely an event is. For equally likely outcomes it is the number of favourable outcomes divided by the total number of outcomes.

The count of all the different ways something can happen. It is found by systematic listing, tree diagrams, or multiplication when choices are made in stages.

The set of all possible outcomes of a trial or experiment. Each outcome in it is equally likely when probability is calculated by counting favourable cases.

A logical chain of statements, each justified by a definition, given fact, or previously proved result, that establishes a conclusion. In geometry it often shows triangles are congruent.

The conditions assumed to be true at the start of a proof or problem. Reasoning proceeds from these given facts toward the conclusion that must be shown.

The statement that a proof or argument is trying to establish. It must follow logically from the hypothesis through justified steps.

The statement formed by swapping the hypothesis and conclusion of a conditional. A true statement does not guarantee that its converse is true, so each must be checked separately.

A single case that satisfies the conditions of a statement but makes the conclusion false, showing the statement is not always true. One counterexample is enough to disprove a general claim.