UNSA ETF Math Entrance Exam key terms

The property that one integer can be divided by another with zero remainder. Written a | b, meaning b = ak for some integer k. Used to factor numbers and simplify fractions.

An integer greater than 1 whose only positive divisors are 1 and itself. Primes such as 2, 3, 5, 7, 11 cannot be written as a product of smaller positive integers.

Writing a positive integer as a product of primes, for example 360 = 2 cubed times 3 squared times 5. The result is unique up to order and is used for divisors and lcm/gcd work.

The largest positive integer dividing each of two or more given integers. For 12 and 18 it is 6. It is found from prime factorizations by taking the lowest power of each shared prime.

The smallest positive integer that is a multiple of each of two or more given integers. For 4 and 6 it is 12. For two positive integers, lcm times gcd equals their product.

For integers a and m > 0, the remainder is the r with 0 ≤ r < m such that a = qm + r. We write a ≡ r (mod m) when two integers leave the same remainder.

A whole number, positive, negative, or zero, with no fractional part. The set is written Z. Integers are closed under addition, subtraction, and multiplication, but not under division.

A number expressible as a fraction p/q where p and q are integers and q is not zero. Decimals of rational numbers either terminate or repeat. Examples include 3/4, −7, and 0.25.

A real number that cannot be written as a fraction of two integers. Its decimal expansion is infinite and non-repeating. Familiar examples are the square root of 2, pi, and e.

The distance of a number from zero on the number line, written |x|. It equals x when x ≥ 0 and −x when x < 0. It is always non-negative and satisfies the triangle inequality.

Any number on the number line, whether rational or irrational. The real numbers are ordered and complete: every nonempty set bounded above has a least upper bound. They are written R.

A statement comparing two expressions with symbols such as <, ≤, >, or ≥. Solving an inequality gives a range of values; multiplying or dividing by a negative number reverses the direction.

An equation in which every variable appears to the first power, such as ax + b = 0 with a not zero. Its solution set is a single value in one variable, or a line in two variables.

An equation of the form ax² + bx + c = 0 with a not zero. Its solutions are given by the quadratic formula, and the discriminant b² − 4ac tells how many real roots exist.

The quantity b² − 4ac for a quadratic ax² + bx + c. Positive means two distinct real roots, zero means one repeated real root, and negative means no real roots but two complex roots.

Relations linking the coefficients of a polynomial to its roots. For ax² + bx + c, the roots sum to −b/a and multiply to c/a. For cubics, the pairwise product sum equals c/a.

An expression formed by adding terms of the form a·xⁿ where n is a non-negative integer and a is a constant coefficient. Its degree is the highest exponent with nonzero coefficient.

A polynomial p has factor (x − r) exactly when p(r) = 0. It converts root-finding into factoring and lets you divide out known roots to lower the degree.

When a polynomial p(x) is divided by (x − r), the remainder equals p(r). It gives a quick way to evaluate or test roots without performing long division.

Two or more equations that must hold at the same time. Solutions are values satisfying all equations simultaneously. Methods include substitution, elimination, and matrix techniques.

A rule assigning to each input in its domain exactly one output. Written f: A → B or y = f(x). Passing the vertical line test distinguishes functions from general curves.

The domain is the set of all permitted inputs of a function; the range is the set of all outputs it actually produces. Domain restrictions come from division by zero and even roots of negatives.

Combining two functions so the output of one becomes the input of the other, written (f ∘ g)(x) = f(g(x)). Order matters: composition is generally not commutative.

The function undoing another, written f⁻¹, satisfying f(f⁻¹(x)) = x on the appropriate domains. A function has an inverse only if it is one-to-one; its graph is the reflection in y = x.

Functions of the form aˣ and their inverses log_a(x). The number e gives the natural exponential and natural logarithm. They model growth, decay, and equations with unknowns in exponents.

The measure of steepness of a line, equal to the change in y divided by the change in x between two points. Parallel lines share a slope; perpendicular slopes multiply to −1.

An algebraic description of a straight line, commonly y = mx + b (slope-intercept) or ax + by + c = 0 (general). A point and slope, or two points, determine it uniquely.

The distance between points (x₁, y₁) and (x₂, y₂) in the plane, equal to the square root of (x₂ − x₁)² + (y₂ − y₁)². It follows from the Pythagorean theorem.

The coordinates of the point exactly halfway between two points, found by averaging the x-coordinates and averaging the y-coordinates separately. It is the center of the segment joining them.

In a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides: a² + b² = c². It underlies the distance formula and many trigonometric identities.

Ratios of side lengths in a right triangle: sine is opposite over hypotenuse, cosine is adjacent over hypotenuse, and tangent is opposite over adjacent. They extend to all angles via the unit circle.

The circle of radius 1 centered at the origin. A point at angle θ on it has coordinates (cos θ, sin θ), which defines sine and cosine for every real angle.

An angle measure based on arc length: one radian subtends an arc equal to the radius. A full turn is 2π radians, so 180 degrees equals π radians.

A sequence in which each term after the first is obtained by adding a fixed difference d. The nth term is a + (n − 1)d, and partial sums grow quadratically in n.

A sequence in which each term after the first is obtained by multiplying by a fixed ratio r. The nth term is a·rⁿ⁻¹. It converges to zero only when |r| < 1.

The sum of the terms of a sequence, written with sigma notation. A finite series has a last term; an infinite series converges when its partial sums approach a finite limit.

The value a function or sequence approaches as its input or index approaches a target. Written lim f(x) = L. Limits define continuity, derivatives, and convergence of infinite series.

A function is continuous at a point when its limit there equals its value there. Informally, the graph has no break, jump, or hole at that point.

An arrangement of objects in which order matters. The number of ordered arrangements of r objects chosen from n distinct objects is n!/(n − r)!, written P(n, r).

A selection of objects in which order does not matter. The number of ways to choose r objects from n distinct objects is n!/(r!(n − r)!), written C(n, r) or the binomial coefficient.

For a non-negative integer n, the product of all positive integers up to n, written n!. By convention 0! = 1. Factorials count orderings of n distinct objects.

A number between 0 and 1 measuring how likely an event is, equal to favorable outcomes divided by total equally likely outcomes. Complementary events sum to 1.

Two events are independent when the occurrence of one does not change the probability of the other. Then P(A and B) = P(A)·P(B).

The probability of event A given that event B has occurred, written P(A|B) and equal to P(A and B)/P(B) when P(B) is not zero.

The expansion of (a + b)ⁿ as a sum of terms involving binomial coefficients C(n, k)·aⁿ⁻ᵏ·bᵏ. It links algebra with counting and gives Pascal's triangle entries.

A number of the form a + bi where a and b are real and i² = −1. The real part is a and the imaginary part is b. Complex numbers add componentwise and multiply by distributing.

For z = a + bi, the conjugate is a − bi, written z̄. The product z·z̄ equals a² + b², a non-negative real number, which is useful for division and finding moduli.

For z = a + bi, the modulus |z| is the distance from the origin, equal to the square root of a² + b². The argument is the angle from the positive real axis.

A quantity with both magnitude and direction, represented by an ordered list of components or an arrow. Vectors add componentwise and can be multiplied by scalars.

The length of a vector, written |v|. For v = (x, y) it equals the square root of x² + y², generalizing the distance formula to any number of dimensions.

For vectors u and v, the sum of products of corresponding components. It equals |u||v|cos θ, so it measures alignment; it is zero exactly when the vectors are perpendicular.