Hiroshima Entrance Math flashcards

44 cards. Read the term and the definition here. The deck at the top of this page is optional.

素因数分解 (prime factorization)

Writing a positive integer as a product of prime numbers only, for example 84 = 2² × 3 × 7. Used to find square roots, GCDs, and LCMs exactly.

絶対値 (absolute value)

The distance of a number from zero on the number line, always non-negative. Written |x|; equals x when x ≥ 0 and −x when x < 0.

平方根 (square root)

A number that, multiplied by itself, gives the original number. Every positive number has a positive and a negative square root; √a means the non-negative one.

有理化 (rationalizing the denominator)

Rewriting a fraction so no square root remains in the denominator, by multiplying numerator and denominator by a suitable root or conjugate, e.g. 1/√2 = √2/2.

展開 (expansion)

Removing brackets by multiplying every term inside by the factor outside, then collecting like terms. The reverse operation is factorization.

因数分解 (factorization)

Rewriting a polynomial as a product of simpler expressions, such as x² + 5x + 6 = (x + 2)(x + 3). Essential for solving quadratic equations.

乗法公式 (multiplication formulas)

Standard expansion identities to memorize: (a+b)² = a²+2ab+b², (a−b)² = a²−2ab+b², (a+b)(a−b) = a²−b², and (x+a)(x+b) = x²+(a+b)x+ab.

比例 (direct proportion)

A relationship where y = ax for a constant a ≠ 0; doubling x doubles y. The graph is a straight line through the origin.

反比例 (inverse proportion)

A relationship where y = a/x for a constant a ≠ 0; as x increases y decreases. The graph is a hyperbola with the axes as asymptotes.

一次関数 (linear function)

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

傾き (slope)

The constant a in y = ax + b, equal to the rise divided by the run between any two points on the line. Parallel lines share the same slope.

切片 (intercept)

The value of y where a graph crosses the y-axis, written b in y = ax + b. Found by setting x = 0.

変化の割合 (rate of change)

For a function, the change in y divided by the corresponding change in x. Constant for linear functions, but varying for quadratic functions unless an interval is specified.

連立方程式 (system of linear equations)

Two or more equations in the same variables solved simultaneously. The solution is the pair of values satisfying all equations, found by elimination or substitution.

代入 (substitution)

Replacing a variable with an equal expression or value. Used to solve simultaneous equations and to find intersections of graphs.

加減法 (elimination method)

Solving simultaneous equations by adding or subtracting the two equations, after scaling if needed, so one variable cancels out.

放物線 (parabola)

The U-shaped curve that is the graph of y = ax². It is symmetric about the y-axis and opens upward when a > 0, downward when a < 0.

軸 (axis of symmetry)

The vertical line about which a parabola is mirror-symmetric. For y = a(x − p)² + q the axis is the line x = p.

頂点 (vertex)

The highest or lowest point of a parabola, lying on its axis of symmetry. For y = a(x − p)² + q the vertex is (p, q).

平方完成 (completing the square)

Rewriting ax² + bx + c in the form a(x − p)² + q so the vertex and axis can be read off directly, and so the quadratic formula can be derived.

二次方程式 (quadratic equation)

An equation of the form ax² + bx + c = 0 with a ≠ 0. It has up to two real solutions, found by factoring, completing the square, or the quadratic formula.

解の公式 (quadratic formula)

The formula x = (−b ± √(b² − 4ac)) / 2a giving the solutions of ax² + bx + c = 0. It works for every quadratic equation.

判別式 (discriminant)

The quantity D = b² − 4ac for ax² + bx + c = 0. D > 0 gives two distinct real solutions, D = 0 one repeated solution, D < 0 no real solutions.

相似 (similarity)

Two figures have the same shape when corresponding angles are equal and corresponding side lengths are in a fixed ratio, the scale factor.

相似比 (similarity ratio)

The ratio of corresponding side lengths of two similar figures. Areas scale as the square of this ratio and volumes as its cube.

合同 (congruence)

Two figures coincide exactly when moved without resizing: all corresponding sides and angles are equal. Triangles are congruent by SSS, SAS, or ASA with corresponding parts.

対応する辺 (corresponding sides)

Sides that occupy the same relative position in two congruent or similar figures, matching equal angles. Their lengths are equal (congruent) or in a fixed ratio (similar).

平行線の錯角 (alternate interior angles)

Angles on opposite sides of a transversal cutting two parallel lines, between the lines. They are equal when the lines are parallel.

同位角 (corresponding angles)

Angles in matching positions at the two intersections made by a transversal cutting two lines. They are equal exactly when the two lines are parallel.

平行線の性質 (properties of parallel lines)

When two parallel lines are cut by a transversal, alternate interior angles are equal and corresponding angles are equal; conversely, equal angles prove the lines parallel.

三角形の内角の和 (angle sum of a triangle)

The three interior angles of any triangle add to 180°. An exterior angle equals the sum of the two non-adjacent interior angles.

多角形の内角の和 (angle sum of a polygon)

The interior angles of an n-sided polygon add to 180(n − 2) degrees. For a quadrilateral this is 360°.

外角 (exterior angle)

The angle formed between one side of a polygon and the extension of an adjacent side. The exterior angles of any convex polygon sum to 360°.

円周角の定理 (inscribed angle theorem)

An angle inscribed in a circle is half the central angle standing on the same arc. Inscribed angles on the same arc are equal, and an angle in a semicircle is 90°.

中心角 (central angle)

An angle whose vertex is the centre of a circle, measured by the arc it intercepts. It is twice any inscribed angle standing on the same arc.

接線 (tangent)

A line touching a circle at exactly one point. A tangent is perpendicular to the radius drawn to the point of contact.

三平方の定理 (Pythagorean theorem)

In a right triangle with legs a and b and hypotenuse c, a² + b² = c². The converse also holds and is used to prove a triangle right-angled.

平均値 (mean)

The sum of all data values divided by the number of values. Sensitive to extreme values, unlike the median.

中央値 (median)

The middle value when data are arranged in order; with an even number of values, the average of the two middle ones. Resistant to outliers.

最頻値 (mode)

The value or class that occurs most often in a data set. A data set may have no mode or several modes.

確率 (probability)

The likelihood of an event, between 0 and 1. For equally likely outcomes it equals the number of favourable outcomes divided by the total number of outcomes.

余事象 (complementary event)

The event that a given event does not happen. Its probability is 1 minus the probability of the event, often the easier route to an answer.

樹形図 (tree diagram)

A branching diagram listing all possible outcomes of a sequence of events, used to count cases and compute probabilities without missing any combination.

場合の数 (number of cases)

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