WEST-E Middle Level Mathematics (NES 203) 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 p/q where p and q are integers and q is not zero. Its decimal form either terminates or eventually repeats.

Irrational number

A real number that cannot be written as a ratio of two integers. Its decimal expansion is non-terminating and non-repeating, such as √2 or π.

Absolute value as distance

The value |a − b| equals the distance between points a and b on the number line, and |x| is the distance from x to zero.

Divisibility rule for 3

An integer is divisible by 3 exactly when the sum of its digits is divisible by 3, so 4,812 qualifies because 4+8+1+2 = 15.

Scientific notation

Writing a number as a × 10^n where 1 ≤ |a| < 10 and n is an integer. Multiplying such numbers adds the exponents.

Percent change formula

Percent change equals (new value − old value) divided by old value, times 100. A p% rise then p% fall gives net change −p²/100 percent.

Unit rate

A rate expressed per one unit of the second quantity, found by dividing the first quantity by the second, such as 60 miles per hour.

Slope formula

For two points (x₁, y₁) and (x₂, y₂), slope m = (y₂ − y₁)/(x₂ − x₁), measuring vertical change per unit of horizontal change.

Perpendicular line slopes

Two nonvertical lines are perpendicular exactly when the product of their slopes is −1, meaning each slope is the negative reciprocal of the other.

Quadratic formula

For ax² + bx + c = 0 with a ≠ 0, the solutions are x = (−b ± √(b² − 4ac)) / (2a).

Discriminant

The quantity b² − 4ac for a quadratic. It is negative for no real roots, zero for one repeated root, and positive for two distinct real roots.

Vertex of a parabola

For f(x) = ax² + bx + c, the vertex occurs at x = −b/(2a); the parabola opens upward when a is positive and downward when a is negative.

Function definition

A relation assigning exactly one output to each input in its domain. Passing the vertical line test confirms a graph represents a function.

One-to-one function

A function giving distinct outputs for distinct inputs. Its graph passes both vertical and horizontal line tests, and it has an inverse function.

Linear versus exponential growth

A linear function changes by a constant difference per step; an exponential function changes by a constant ratio, so it eventually outgrows any linear function.

Arithmetic sequence nth term

For an arithmetic sequence with first term a₁ and common difference d, the nth term is aₙ = a₁ + (n − 1)d.

Geometric sequence nth term

For a geometric sequence with first term a₁ and constant ratio r, the nth term is aₙ = a₁·r^(n−1), where r may be fractional or negative.

Zero exponent and negative exponent

For any nonzero base a, a⁰ = 1 and a^(−n) = 1/aⁿ. These follow from the quotient rule for exponents.

Triangle angle sum

The three interior angles of any Euclidean triangle add to 180 degrees, and an exterior angle equals the sum of the two remote interior angles.

Pythagorean theorem

In a right triangle with legs a and b and hypotenuse c, a² + b² = c². Its converse identifies right triangles from side lengths.

Area of a trapezoid

The area equals one-half the sum of the parallel bases times the height: A = ½(b₁ + b₂)h, where h is perpendicular distance between bases.

Circle circumference and area

For a circle of radius r, circumference is 2πr and area is πr². Diameter is twice the radius, so circumference also equals πd.

Volume of a cone

A cone's volume is one-third the base area times height: V = ⅓πr²h. A pyramid uses the same one-third rule with its base area.

Similar figure scaling

If two figures are similar with length scale factor k, corresponding areas scale by k² and corresponding volumes scale by k³.

Interior angle sum of polygon

A polygon with n sides has interior angles summing to (n − 2)·180 degrees; a regular n-gon has each interior angle equal to that sum divided by n.

Inscribed angle theorem

An angle inscribed in a circle measures half the intercepted arc, while a central angle equals its intercepted arc; an angle subtending a diameter is right.

Distance and midpoint formulas

Distance between two points is √((x₂ − x₁)² + (y₂ − y₁)²); the midpoint is the average of the x-coordinates and of the y-coordinates.

Mean versus median

The mean is the sum divided by the count; the median is the middle ordered value. Outliers shift the mean far more than the median.

Standard deviation

A measure of spread giving typical distance of data from the mean. In normal distributions about 68% of values fall within one standard deviation.

Addition rule for probability

For any two events, P(A or B) = P(A) + P(B) − P(A and B); subtract the overlap so outcomes counted twice are corrected.

Independent events

Events A and B are independent when P(A and B) = P(A)·P(B), meaning the occurrence of one does not change the other's probability.

Permutation versus combination

Permutations count ordered arrangements, nPr = n!/(n − r)!; combinations count unordered selections, nCr = n!/(r!(n − r)!). Order distinguishes them.