ILTS Mathematics (208) key terms

A number expressible 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 forever.

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

The set of all rational and irrational numbers, ordered on a number line and closed under the four basic operations except division by zero. It contains every point of the line.

The distance of a number from zero on the number line, always nonnegative. Formally, the absolute value of x equals x when x is nonnegative and its opposite when x is negative.

The property that between any two distinct real numbers there is a rational number, and also an irrational one. No number has an immediate successor on the number line.

The unique expression of a whole number greater than one as a product of primes, apart from order. It underlies greatest common factors, least common multiples, and divisibility tests.

The largest whole number dividing each number in a given set with no remainder. It can be found from prime factorizations by taking each shared prime to its lowest power.

The smallest positive whole number that is a multiple of each number in a given set. For fractions, the least common multiple of the denominators gives the least common denominator.

Rules for adding, subtracting, multiplying, and dividing positive and negative whole numbers. Subtraction is adding the opposite, and a product or quotient is negative when signs differ.

The number that sums with a given number to zero, also called its opposite. Every real number has exactly one additive inverse, and it lies the same distance from zero on the other side.

A way of writing numbers as a product of a number between 1 and 10 and a power of ten. It makes very large or very small quantities compact and easy to compare by order of magnitude.

Laws governing powers: multiplying like bases adds exponents, dividing subtracts them, a power of a power multiplies them, and negative or zero exponents follow from these patterns.

The connection that the nth root of a number equals that number raised to the power 1/n, with numerator and denominator of a fractional exponent giving power and root respectively.

Solving problems by recognizing that two quantities change at a constant ratio. It supports unit rates, scale drawings, percent change, and setting up and solving proportions.

A function whose graph is a straight line, writable as f(x) = mx + b. The constant m gives the constant rate of change and b gives the value at x = 0.

The constant rate of change of a line, computed as the change in output divided by the change in input between two points. Horizontal lines have slope zero; vertical lines have undefined slope.

Two or more linear equations considered together, solved by finding values satisfying all at once. Graphically the solution is the intersection point, and parallel lines mean no solution.

A function of the form f(x) = ax squared + bx + c with a not zero. Its graph is a parabola whose vertex, axis of symmetry, and intercepts follow from the coefficients.

The form f(x) = a(x - h) squared + k for a quadratic, showing the vertex at (h, k) directly. It reveals the maximum or minimum value and the direction of opening.

Rewriting a quadratic expression as a perfect square plus a constant by adding and subtracting the square of half the linear coefficient. It derives the quadratic formula and converts to vertex form.

The quantity b squared minus 4ac in the quadratic formula. Its sign tells how many real solutions exist: positive for two, zero for one repeated, negative for none.

A number of the form a + bi where i squared equals negative one. Real and imaginary parts are combined, and conjugates help divide and find magnitudes.

A function of the form f(x) = a times b to the x power with b positive and not one. It models constant percent growth or decay, changing by a fixed factor per unit.

The exponent to which a base must be raised to produce a given number. Logarithms invert exponential functions and convert multiplication into addition, enabling equation solving.

A function that undoes another, so composing them returns the original input. Its graph is the reflection of the original graph across the line y = x.

How much one quantity changes per unit change in another, averaged over an interval or taken instantaneously. Linear functions have constant rates; other functions vary.

Shifts, stretches, compressions, and reflections that change a parent graph. Adding outside the function shifts vertically, adding inside shifts horizontally, and negatives reflect across axes.

A relationship in which two figures have equal corresponding angles and proportional corresponding side lengths. Scale factor relates lengths, its square relates areas, its cube relates volumes.

A relationship in which two figures have the same shape and size, related by rigid motions. Triangle congruence can be established by side-side-side, side-angle-side, and angle-side-angle criteria.

In a right triangle, the square of the hypotenuse equals the sum of the squares of the legs. Its converse identifies right triangles, and it underlies the distance formula.

The distance between two points in the coordinate plane, found by applying the Pythagorean theorem to the horizontal and vertical differences between their coordinates.

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 one centered at the origin. A point on it at angle theta has coordinates cosine theta and sine theta, giving a consistent definition of trigonometric functions.

An angle measure defined as the arc length cut on a unit circle. A full turn is two pi radians, connecting arc length, sector area, and angular speed to linear quantities.

The amount of space a three-dimensional figure occupies, measured in cubic units. Formulas for prisms, cylinders, pyramids, cones, and spheres follow from base area and height.

The total area of all faces or curved surfaces of a solid, measured in square units. Nets help derive formulas and distinguish lateral area from total area.

Summary values describing a typical data point: mean, median, and mode. The mean responds to every value, the median resists outliers, and the mode names the most frequent value.

Values describing how dispersed data are: range, interquartile range, variance, and standard deviation. They complement center measures and are reported in original or squared units.

The typical distance of data values from their mean, equal to the square root of the variance. Small values indicate tightly clustered data; large values indicate wide spread.

A data value far from the rest of the distribution, often flagged as lying more than 1.5 interquartile ranges beyond a quartile. Outliers strongly affect the mean but little the median.

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 95 percent within two.

The number of standard deviations a value lies from the mean, computed as the difference divided by the standard deviation. It allows comparison across different distributions.

A graph plotting paired data as points, and a number from negative one to one summarizing the strength and direction of a linear association between the two variables.

A straight line chosen to approximate a linear trend in paired data, often by least squares. Its slope estimates change per unit and it supports prediction within the observed range.

A number from zero to one expressing how likely an event is, either from counting equally likely outcomes or from long-run relative frequency. Probabilities of all outcomes sum to one.

The set of all possible outcomes of a random experiment. Listing it systematically makes counting favorable outcomes and computing theoretical probabilities straightforward.

Events whose occurrence does not change each other's probability, so the probability of both equals the product of their individual probabilities. Sampling with replacement often yields independence.

The probability that an event occurs given that another event has occurred, computed as the probability of both divided by the probability of the given event. It underlies Bayes-style reasoning.

Counting methods for selecting items. Permutations count arrangements where order matters, while combinations count groups where order does not, differing by a factor of the factorial of the number chosen.

The long-run average outcome of a random variable, found by multiplying each outcome by its probability and summing. It supports decisions under uncertainty and fair-game analysis.

The iterative process of translating a real situation into mathematics, solving, interpreting results in context, validating, and revising. Assumptions and units are checked at each stage.

A chain of justified statements establishing a conclusion, using definitions, axioms, and prior results. Direct proof, contradiction, and induction are common forms expected in reasoning tasks.

A single case that satisfies a statement's hypotheses but violates its conclusion, thereby disproving a universal claim. Finding one is often easier than constructing a full proof.

A general statement proposed as true based on observed patterns but not yet proved. Testing many examples and seeking a counterexample are the first steps toward proof or refutation.

The statement that any side of a triangle is shorter than the sum of the other two, and that the absolute value of a sum is at most the sum of absolute values.