EJU Mathematics Course 1 key terms

Any number that can be placed on the number line, including integers, fractions, and irrational numbers such as √2 and π. Course 1 works entirely within this set.

A number expressible as a fraction of two integers with a nonzero denominator. Its decimal expansion either terminates or repeats periodically.

A real number that cannot be written as a fraction of integers, so its decimal expansion is infinite and non-repeating, for example √2 or π.

The distance of a number from zero on the number line, always nonnegative. Written |x|, it equals x when x ≥ 0 and −x when x < 0.

A value whose square equals a given nonnegative number. The symbol √a denotes the nonnegative root, and √a is defined only for a ≥ 0.

Rewriting a fraction so the denominator contains no radical, typically by multiplying numerator and denominator by a conjugate expression such as √a − √b.

An expression with a root inside a root, such as √(a + √b). It can often be simplified to √x + √y when a = x + y and b = 4xy.

An inequality in one variable of degree one. Solving it isolates the variable, remembering that multiplying or dividing by a negative number reverses the inequality sign.

Two or more inequalities that must hold simultaneously. The solution is the intersection of the individual solution sets, usually written as a compound interval on the number line.

A straight line on which each point corresponds to a real number, used to picture intervals, distances, and the solution sets of inequalities.

A collection of distinct objects called elements. Standard notation includes ∈ for membership, ⊂ for subset, and ∪ and ∩ for union and intersection.

The intersection A ∩ B contains elements in both A and B; the union A ∪ B contains elements in at least one of them.

For a universal set U, the complement of A is the set of elements in U that are not in A, written with an overline or as Aᶜ.

A proposition is a statement that is either true or false. A condition contains a variable and becomes a proposition once values are assigned.

For P ⇒ Q, P is sufficient for Q and Q is necessary for P. When both implications hold, the conditions are equivalent and written P ⇔ Q.

The contrapositive of P ⇒ Q is (not Q) ⇒ (not P). A statement and its contrapositive are always logically equivalent, which enables indirect proofs.

A proof technique that assumes the negation of the claim and derives a contradiction, thereby establishing the original statement, such as the irrationality of √2.

A function of the form y = ax² + bx + c with a ≠ 0. Its graph is a parabola opening upward when a > 0 and downward when a < 0.

Rewriting ax² + bx + c as a(x − p)² + q, which reveals the vertex (p, q) and the axis of symmetry of the parabola.

The highest or lowest point of a parabola, located at x = −b/(2a) for y = ax² + bx + c. It gives the maximum or minimum value of the function.

The greatest and least values a function takes on a specified domain. For quadratics, check the vertex when it lies inside the domain and the endpoints otherwise.

The set of input values for which a function is defined. Restrictions arise from denominators, square roots, and any stated interval.

Shifting a graph without changing its shape. Replacing x by x − p and y by y − q moves the graph p units horizontally and q units vertically.

Flipping a graph across an axis or line. Reflecting across the x-axis replaces y by −y; across the y-axis replaces x by −x; across y = x swaps x and y.

For ax² + bx + c = 0 with a ≠ 0, the solutions are x = (−b ± √(b² − 4ac)) / (2a). It solves every quadratic equation.

The quantity D = b² − 4ac for a quadratic equation. D > 0 gives two distinct real roots, D = 0 one repeated root, and D < 0 no real roots.

For ax² + bx + c = 0 with roots α and β, the sum α + β = −b/a and the product αβ = c/a, useful for building equations from roots.

The number of intersection points of y = ax² + bx + c with the x-axis equals the number of real solutions of ax² + bx + c = 0.

An inequality such as ax² + bx + c > 0. Its solution set is read from the parabola, using the roots and the sign of the leading coefficient.

Ratios of side lengths in a right triangle: sine, cosine, and tangent. They connect an acute angle to the proportions of the triangle.

In a triangle with sides a, b, c opposite angles A, B, C and circumradius R, a/sin A = b/sin B = c/sin C = 2R.

In a triangle, a² = b² + c² − 2bc·cos A, relating all three sides to one angle. It determines a side from two sides and the included angle.

Given two sides b, c and included angle A, the area is (1/2)bc·sin A. Heron's formula gives the area from three side lengths.

Relations such as sin²θ + cos²θ = 1 and tan θ = sin θ / cos θ, which allow one ratio to be computed from another.

Extensions of the ratios to angles from 0° to 180°. For obtuse θ, sin θ stays positive while cos θ and tan θ become negative.

Summary values describing a data set's center: mean (average), median (middle value when ordered), and mode (most frequent value).

The first and third quartiles split ordered data into quarters; the interquartile range is their difference and measures spread resistant to outliers.

A diagram showing minimum, first quartile, median, third quartile, and maximum. It displays the spread and skew of a data set at a glance.

Variance is the mean of squared deviations from the mean; standard deviation is its square root and expresses spread in the original units.

A number between −1 and 1 measuring the strength and direction of a linear relationship between two variables; values near ±1 indicate a strong linear trend.

A graph plotting paired data as points on a coordinate plane, used to inspect whether two variables show a positive, negative, or no association.

An ordered arrangement of objects. The number of ways to arrange r objects chosen from n distinct objects is nPr = n! / (n − r)!.

A selection of objects where order does not matter. The number of ways to choose r from n is nCr = n! / (r!(n − r)!).

The product of all positive integers up to n, written n! and defined with 0! = 1. It counts orderings of n distinct objects.

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

The event that a given event does not occur. Since an event and its complement cover all outcomes, their probabilities sum to 1.

Events that cannot occur at the same time. For mutually exclusive events, the probability of either occurring is the sum of their probabilities.

P(A ∪ B) = P(A) + P(B) − P(A ∩ B). The subtraction corrects for outcomes counted in both events.

Trials whose outcomes do not affect each other. For independent events A and B, the probability of both is P(A) × P(B).

Divisibility, multiples, divisors, and the division algorithm. These ideas underlie prime factorization and the Euclidean algorithm for greatest common divisor.

The greatest common divisor is the largest integer dividing both numbers; the least common multiple is the smallest positive integer both divide.

A repeated-division method for finding the greatest common divisor: replace the pair (a, b) with (b, r) where r is the remainder of a divided by b.

Two integers whose only positive common divisor is 1. Coprimality is central to proofs about fractions in lowest terms and to Diophantine equations.