Praxis Math 5165 key terms

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 fixed block of digits.

A real number that cannot be expressed as a ratio of two integers. Its decimal expansion never terminates and never repeats, as happens with the square root of 2 or with pi.

A number of the form a + bi, where a and b are real numbers and i is the imaginary unit with i squared equal to negative one. The value a is the real part and b the imaginary part.

The conjugate of a + bi is a - bi. Multiplying a complex number by its conjugate produces the real number a squared plus b squared, which is used to divide complex numbers and find moduli.

The distance of a number from zero on the number line, written with vertical bars. It is always nonnegative: the absolute value of x equals x when x is at least zero and negative x otherwise.

Writing a whole number greater than 1 as a product of primes, for example 84 = 2 squared times 3 times 7. The result is unique apart from order and helps find greatest common divisors and least common multiples.

Two integers are congruent modulo n when their difference is a multiple of n, written a is congruent to b mod n. Clock arithmetic is the standard example, since 14 and 2 are congruent mod 12.

A quantity having both magnitude and direction, often drawn as an arrow and written in components such as the ordered pair 3, negative 2. Vectors add componentwise, and their length is computed with the distance formula.

A rule that assigns to each input value in its domain exactly one output value. Passing the vertical line test on a graph is the visual sign that a relation is a function.

The set of all input values for which a function or expression is defined. For a rational function, any value making the denominator zero is excluded from the domain.

A function that undoes another, sending each output back to its original input. It exists only when the original function is one-to-one, and its graph mirrors the original across the line y = x.

Combining two functions by feeding the output of one into the other, written f of g of x equals f of g of x. Order matters, and the input must lie in the domain of the inner function.

The steepness of a line, computed as the change in y divided by the change in x between two points. For a linear function it is the constant rate of change.

For a quadratic written as a x squared plus b x plus c, the value b squared minus 4ac. Its sign reveals the number of real solutions: positive gives two, zero gives one, negative gives none.

A line that a graph approaches ever more closely without necessarily touching. Rational functions often have vertical asymptotes where the denominator is zero and horizontal asymptotes determined by end behavior.

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 undo exponentials and turn products into sums.

A function of the form f of x equals a times b to the x, with b positive and not equal to 1. It models constant percent growth or decay, and its graph never crosses the horizontal axis.

A list of numbers in which each term after the first is obtained by multiplying the previous term by a fixed ratio r. The nth term equals the first term times r to the power n minus 1.

A rectangular array of numbers arranged in rows and columns. Matrices add entrywise, multiply when inner dimensions match, and can represent systems of equations or geometric transformations.

In a right triangle, the square of the hypotenuse equals the sum of the squares of the legs. It underlies the distance formula and the unit-circle definitions of sine and cosine.

Triangles with equal corresponding angles and proportional corresponding sides. The shared scale factor lets you find unknown lengths, and similarity is the basis for trigonometric ratios.

An angle whose vertex lies on a circle and whose sides are chords measures half of its intercepted arc. A special case gives a right angle for an angle inscribed in a semicircle.

An angle measure based on arc length, where one radian is the angle subtended by an arc equal in length to the radius. A full turn is 2 pi radians, or 360 degrees.

The circle of radius 1 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.

In any triangle, the ratio of a side to the sine of its opposite angle is the same for all three sides. It solves a triangle when two angles and a side are known.

A generalization of the Pythagorean theorem stating that c squared equals a squared plus b squared minus 2ab cosine C. It finds a missing side or angle when two sides and the included angle are known.

The equation sine squared theta plus cosine squared theta equals 1, which follows directly from the unit circle. It converts between sine and cosine and generates related identities involving tangent and secant.

A transformation that preserves all distances and angles, namely a translation, rotation, or reflection. Figures related by rigid motions are congruent, while dilations change size and produce similar figures.

The distance between two points is the square root of the sum of the squared differences of their x-coordinates and y-coordinates. It follows directly from the Pythagorean theorem.

A measure of how spread out data values are around the mean, equal to the square root of the variance. Larger values mean the data points are more dispersed.

A symmetric, bell-shaped distribution fully described by its mean and standard deviation. Roughly 68 percent of values lie within one standard deviation of the mean and about 95 percent within two.

The number of standard deviations a data value lies above or below the mean, computed as the value minus the mean, divided by the standard deviation. It allows comparison across different distributions.

A number between negative 1 and 1, usually written r, that measures the strength and direction of a linear relationship between two variables. A strong correlation alone does not establish causation.

The probability that event A occurs given that event B has already occurred, equal to the probability of both events divided by the probability of B. It restricts attention to the outcomes inside B.

Two events are independent when the occurrence of one does not change the probability of the other. In that case the probability of both equals the product of their separate probabilities.

The long-run average outcome of a random variable, found by multiplying each possible outcome by its probability and adding all those products together.

The value a function's outputs approach as the input approaches some point. Limits describe behavior near that point and form the foundation for continuity, derivatives, and integrals.

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.

The limit of a sum of areas of thin rectangles under a curve, interpreted as the signed area between a graph and the horizontal axis over a given interval.

The result linking the two branches of calculus: the definite integral of a continuous function can be evaluated by subtracting the values of any antiderivative at the interval's endpoints.