OSAT Advanced Mathematics (211) key terms

A single specific case that satisfies the conditions of a statement but makes its conclusion false, which is enough to disprove a universal claim such as 'all primes are odd.'

A proof technique showing a statement holds for a smallest base case, then that whenever it holds for one case it must hold for the next, thereby covering all positive integers.

A proof strategy that assumes the statement is false, derives a logical impossibility from that assumption, and concludes the original statement must therefore be true.

A number written a + bi, where a and b are real and i squared equals negative one; a is the real part and b the imaginary part, allowing solutions to equations such as x squared plus one equals zero.

Two integers are congruent modulo n when their difference is divisible by n; for example 17 and 5 are congruent modulo 6 because 17 minus 5 equals 12.

Every integer greater than 1 factors into primes in exactly one way apart from the order of the factors, which underpins work with divisibility, greatest common factors, and fractions.

A real number that cannot be written as a ratio of two integers; its decimal expansion neither terminates nor repeats, as with the square root of two and pi.

The combined function f of g, written f(g(x)), applies g first and feeds its output into f; its domain excludes inputs where g is undefined or g(x) lies outside f's domain.

A function that undoes another: f inverse of y equals x exactly when f of x equals y. It exists only when f is one-to-one, and its graph mirrors f across the line y equals x.

The domain is the set of all allowable inputs of a function; the range is the set of all outputs the function actually produces as inputs vary across that domain.

A polynomial p of x has factor x minus c exactly when p of c equals zero, linking polynomial division, remainders, and the zeros of a function.

The exponent to which a base must be raised to produce a given number: log base b of x equals y means b to the y equals x. Logarithms reverse exponentiation and turn products into sums.

A list of numbers in which each term after the first is a constant multiple, called the ratio, of the previous term; the nth term equals a times r to the n minus one.

The quantity b squared minus 4ac taken from the quadratic formula; its sign tells how many real roots the equation a x squared plus b x plus c equals zero has: two, one, or none.

A line that a graph approaches arbitrarily closely as the input or output grows without bound, without necessarily ever touching it, as with y equals 1 over x near the axes.

Figures with equal corresponding angles and proportional corresponding side lengths; their areas compare as the square of the scale factor and their volumes as its cube.

An angle whose vertex lies on a circle and whose sides are chords; it measures half the intercepted arc, so angles subtending the same arc are equal.

A transformation that stretches or shrinks a figure about a fixed center by a scale factor, preserving shape and angle measures while changing all distances proportionally.

In any triangle the sum of the lengths of any two sides exceeds the length of the third; stated with vectors, the magnitude of a plus b is at most the sum of their magnitudes.

An angle measure defined as intercepted arc length divided by radius; a full turn is 2 pi radians, so pi radians equals 180 degrees and one radian is about 57.3 degrees.

The circle of radius one centered at the origin; a point at angle theta on it has coordinates cosine theta and sine theta, which defines the trigonometric functions for all angles.

Sine squared theta plus cosine squared theta equals one for every angle theta, a direct consequence of the unit circle; dividing by either square gives related identities involving tangent, secant, and cotangent.

The equation c squared equals a squared plus b squared minus 2ab cosine C relates the three sides of a triangle to one angle, generalizing the Pythagorean theorem to non-right triangles.

The value L that f of x approaches as x approaches a specified point, whether or not f is defined there; it is the foundation for continuity, derivatives, and integrals.

A function is continuous at a point when its limit there exists, the function is defined there, and those two values agree; informally, the graph has no break, jump, or hole.

The instantaneous rate of change of a function, defined as the limit of a difference quotient; geometrically it gives the slope of the tangent line at a point.

A differentiation rule for composite functions: the derivative of f of g of x equals f prime at g of x times g prime of x, letting you differentiate nested expressions such as sine of 3x squared.

The limit of Riemann sums over an interval, interpreted as signed area between a graph and the x-axis and used to accumulate rates such as distance from velocity.

It links the two branches of calculus: the derivative of an accumulated area function returns the original integrand, and definite integrals can be evaluated using any antiderivative.

The probability that event A occurs given that event B has occurred, written P of A given B and computed as the probability of both divided by the probability of B.

Two events are independent when knowing that one occurred does not change the probability of the other, equivalently when the probability of both equals the product of their probabilities.

The long-run average outcome of a random variable, found for discrete variables by multiplying each possible value by its probability and summing the products.

A measure of how far values typically sit from their mean, calculated as the square root of the variance; larger values indicate more spread-out data.

For sufficiently large samples, the distribution of sample means is approximately normal regardless of the population's shape, with mean mu and standard error sigma divided by the square root of n.

An idealized mathematical description of a real situation, built from variables, assumptions, and equations, used to explain behavior or predict outcomes and later validated against data.

A pattern in which a quantity changes by a constant percentage per unit time, modeled by A of t equals A-zero times e to the kt with k positive, or decay when k is negative.

A method that fits a straight line to paired data by minimizing squared vertical distances, used to describe trend, estimate slope, and predict values of one variable from another.

Checking or converting units by treating them as algebraic quantities that cancel; it catches setup errors and converts between measurement systems without memorizing every conversion factor.