OSAT Middle Level/Intermediate Mathematics (225) flashcards

32 cards. Read the term and the definition here. The deck at the top of this page is optional.

Rational number

A number expressible as a/b where a and b are integers and b is not zero. Its decimal form terminates or repeats, and every integer is rational.

Irrational number

A real number that cannot be written as a ratio of integers. Its decimal neither terminates nor repeats, as with √2, π, and e.

Terminating decimal test

A fraction in lowest terms has a terminating decimal exactly when its denominator has no prime factors other than 2 and 5.

Negative exponent rule

For a nonzero base a and integer n, a to the power negative n equals 1 divided by a to the power n.

Rationalizing a denominator

Rewriting a fraction so no radical remains in the denominator, usually by multiplying numerator and denominator by the radical or its conjugate.

Successive percent change

Percent changes applied in sequence multiply as factors rather than add. A 20% increase then a 20% decrease leaves 96% of the original amount.

Absolute value as distance

The absolute value of a difference gives the distance between two numbers on the number line, so the equation absolute value of x equals k has two solutions.

Scientific notation

A number written as a times 10 to the n, where a is at least 1 and less than 10; exponents add when multiplying and subtract when dividing.

Slope formula

The slope through two points equals the change in y divided by the change in x. Horizontal lines have slope zero; vertical lines have undefined slope.

Point-slope form

A linear equation written as y minus y-one equals m times the quantity x minus x-one, using a known point and the slope.

Quadratic formula

For ax squared plus bx plus c equals zero with a nonzero, x equals negative b plus or minus the square root of b squared minus 4ac, all over 2a.

Discriminant

The expression b squared minus 4ac in a quadratic. Positive means two real roots, zero means one repeated root, negative means no real roots.

Vertex of a parabola

For y equals ax squared plus bx plus c, the vertex's x-coordinate is negative b over 2a; it is the maximum when a is negative.

Difference of squares

The factoring pattern in which a squared minus b squared equals the product of a minus b and a plus b.

Inverse function

A function that reverses another: composing them in either order returns the input. Its graph is the original reflected over the line y equals x.

Exponential decay

A model of the form y equals a times b to the x with base b between 0 and 1, so values shrink by a constant percent each period.

Compound interest formula

A equals P times the quantity 1 plus r over n raised to nt, where P is principal, r the annual rate, n the compounding periods per year, t years.

Perpendicular slopes

Two nonvertical lines are perpendicular exactly when their slopes are negative reciprocals, so the product of the slopes equals negative one.

Triangle angle sum

The three interior angles of any triangle add to 180 degrees, which forces each exterior angle to equal the sum of the two remote interior angles.

Pythagorean theorem

In a right triangle the sum of the squares of the legs equals the square of the hypotenuse; the converse tests whether an angle is right.

Trapezoid area

The area equals half the height times the sum of the two parallel bases, written as one-half times b-one plus b-two, times h.

Circle area and circumference

A circle's area is pi times the radius squared, and its circumference is 2 pi times the radius, equivalently pi times the diameter.

Cone volume

A cone's volume is one-third the base area times the height, so it is one-third of the cylinder with the same base and height.

Similar figure ratios

Corresponding angles are equal and sides are proportional. Areas compare as the square of the scale factor and volumes as its cube.

Midpoint formula

The midpoint of a segment averages the endpoints' coordinates: the x-coordinate is their average and so is the y-coordinate.

Transversal angle pairs

When parallel lines are cut by a transversal, alternate interior, alternate exterior, and corresponding angles are congruent, while same-side interior angles are supplementary.

Median of a data set

The middle value when data are ordered from least to greatest; with an even count, average the two central values. It resists outliers.

Interquartile range

The difference between the third quartile and the first quartile, measuring the spread of the middle half of the data and ignoring extreme values.

Standard deviation

A measure of how far data values typically lie from the mean; it is large for widely spread data and zero when all values are equal.

Addition rule of probability

The probability of A or B equals the probability of A plus the probability of B minus the probability of both, removing the double count.

Conditional probability

The probability of event A given that B occurred equals the probability of both events divided by the probability of B, provided B is possible.

Expected value

The sum of each outcome multiplied by its probability. It represents the long-run average per trial and need not be an actual outcome.