Nagasaki High School Entrance Exam - Mathematics key terms

A counting number such as 1, 2, 3, and so on. In Japanese school mathematics, zero and negative numbers are usually excluded, so check how the problem uses the term.

A whole number that may be positive, negative, or zero, with no fractional or decimal part. Integers are closed under addition, subtraction, and multiplication.

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

Writing a natural number as a product of prime numbers, for example 84 = 2² × 3 × 7. It is useful for finding square roots and greatest common divisors.

The largest positive integer that divides two or more given integers exactly. For example, the greatest common divisor of 12 and 18 is 6.

The smallest positive integer that is a multiple of two or more given integers. For example, the least common multiple of 4 and 6 is 12.

The distance of a number from zero on the number line, always non-negative. The absolute value of both 5 and -5 is 5, written |5| = |-5| = 5.

A straight line on which every point corresponds to a number, with a fixed origin, unit length, and positive direction. It helps compare numbers and picture absolute value.

A number that, when multiplied by itself, gives the original number. Every positive number has a positive and a negative square root, written with the symbol √.

Rewriting a fraction so its denominator contains no square root, by multiplying numerator and denominator by a suitable radical. For example, 1/√2 becomes √2/2.

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

A number that can be written as a fraction a/b where a and b are integers and b is not zero. Its decimal form either terminates or repeats.

A decimal in which a digit or block of digits repeats forever, such as 0.333… or 0.121212…. A repeating decimal can always be converted into a fraction.

An algebraic expression that uses letters to stand for numbers, such as 3x + 2y. Letters follow the same operation rules as numbers.

An expression formed by multiplying numbers and letters only, with no addition or subtraction signs, such as 5ab or -x². Its degree is the sum of the letter exponents.

A sum of one or more monomials, such as x² - 4x + 7. Each separate monomial in it is called a term.

Terms that contain exactly the same letters raised to the same powers, such as 3x² and -5x². Like terms can be combined into a single term.

Removing brackets by multiplying out every pair of terms, turning a product into a sum. For example, (x + 2)(x - 3) expands to x² - x - 6.

Writing an expression as a product of simpler expressions. For example, x² - 9 factors as (x + 3)(x - 3). It is the reverse of expansion.

Standard identities used to expand or factor quickly, such as (a + b)² = a² + 2ab + b² and (a + b)(a - b) = a² - b². Memorizing them saves time.

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.

Two or more equations that must be satisfied at the same time by the same unknowns. Common solution methods are elimination (加減法) and substitution (代入法).

An equation whose highest power of the unknown is two, such as x² - 5x + 6 = 0. It can be 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, valid when a is not zero. It solves any quadratic equation.

The value b² - 4ac for a quadratic equation ax² + bx + c = 0. Its sign tells whether the equation has two, one, or no real solutions.

A statement comparing two expressions with signs such as <, >, ≤, or ≥. Solving it gives a range of values rather than a single value.

Two or more inequalities that must all hold at once. The solution is the overlap of their individual solution ranges, often shown on a number line.

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

A relationship in which y = a/x for a nonzero constant a. As x increases, y decreases, and the graph is a curve that never touches the axes.

A function written as y = ax + b, where a is the slope (変化の割合) and b is the y-intercept. Its graph is a straight line.

The rate of change of a linear function, equal to the change in y divided by the change in x between two points. It shows how steep the line is.

For a function, the change in y divided by the corresponding change in x between two points. For y = ax² it equals a(p + q) between x = p and x = q.

The basic quadratic function whose graph is a parabola with its vertex at the origin. The sign of a decides whether it opens upward or downward.

The U-shaped curve that is the graph of a quadratic function. It has a line of symmetry and a highest or lowest point called the vertex.

A relationship in which two figures have the same shape but possibly different sizes, with corresponding angles equal and corresponding sides in a fixed ratio.

The ratio of 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: a² + b² = c². Its converse tests for a right angle.

An angle inscribed in a circle is half the central angle standing on the same arc. Angles in the same segment are therefore equal.

A survey that studies only part of a population and uses the results to estimate the whole. Conclusions carry sampling error, unlike a full census.

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

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

A logical argument that starts from given conditions and known facts and reaches a conclusion step by step, giving a reason for every statement made.