DSST Math for Liberal Arts flashcards

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

Rational number test

A real number is rational exactly when its decimal expansion terminates or repeats. Non-terminating, non-repeating decimals such as √2 = 1.41421356… and π are irrational.

Order of operations

Evaluate grouped symbols first, then exponents, then multiplication and division from left to right, and finally addition and subtraction from left to right.

Absolute value as distance

|x| gives the distance from x to 0 on the number line, so |a − b| is the distance between a and b and is never negative.

Comparing fractions by cross-multiplication

For positive denominators, a/b < c/d exactly when ad < bc. Multiply each numerator by the opposite denominator and compare the two products.

Cardinality of a union

For two finite sets, n(A ∪ B) = n(A) + n(B) − n(A ∩ B). Subtract the overlap because those elements were counted twice.

De Morgan's laws

The complement of a union is the intersection of the complements, (A ∪ B)′ = A′ ∩ B′, and the complement of an intersection is the union of the complements.

Conditional statement truth

p → q is false only in the case where p is true and q is false. In every other combination of truth values the conditional is true.

Contrapositive equivalence

The contrapositive of p → q is ∼q → ∼p, and it always has the same truth value as the original. Converse and inverse do not.

Number of subsets

A set containing n distinct elements has exactly 2ⁿ subsets, counting the empty set and the set itself. For example, a 4-element set has 16 subsets.

Metric prefix steps

Moving one prefix place left multiplies by 10, and one place right divides by 10. Kilo, hecto, deka, base, deci, centi, milli run in that order.

Metric volume and mass link

One cubic centimeter equals one milliliter, 1000 cubic centimeters equal one liter, and one kilogram of water occupies about one liter.

Celsius to Fahrenheit

Convert with °F = (9/5)°C + 32. Going the other way, °C = (5/9)(°F − 32). Kelvin equals Celsius plus 273.15.

Circle and triangle area

A circle of radius r has area πr² and circumference 2πr. A triangle with base b and height h has area ½bh.

Similar figure scaling

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

Slope formula

The slope through (x₁, y₁) and (x₂, y₂) is m = (y₂ − y₁)/(x₂ − x₁). A horizontal line has slope 0; a vertical line has undefined slope.

Perpendicular slope rule

Two nonvertical lines are perpendicular exactly when their slopes are negative reciprocals, so the product m₁·m₂ equals −1.

Vertical line test

A graph represents a function only if no vertical line crosses it more than once, because each input may have just one output.

Vertex of a parabola

For f(x) = ax² + bx + c, the vertex occurs at x = −b/(2a). The parabola opens upward when a > 0 and downward when a < 0.

Average rate of change

On the interval from a to b, the average rate of change of f is (f(b) − f(a))/(b − a), which is the slope of the secant line.

Inequality sign reversal

Adding or subtracting the same number leaves an inequality unchanged, but multiplying or dividing both sides by a negative number reverses the direction of the sign.

Number of solutions to a system

Two linear equations have one solution if their slopes differ, none if slopes match but intercepts differ, and infinitely many if the equations describe the same line.

Graphing a linear inequality

Plot the boundary line solid when equality is included and dashed when it is not, then shade the half-plane containing a test point that satisfies the inequality.

Absolute value inequalities

For a > 0, |x| < a is equivalent to −a < x < a, while |x| > a is equivalent to x < −a or x > a.

Negative and zero exponents

Any nonzero base to the zero power equals 1, and a negative exponent means a reciprocal: a⁻ⁿ = 1/aⁿ. Thus 2⁻³ = 1/8.

Logarithm definition

log_b x = y means bʸ = x, defined only for b > 0, b ≠ 1, and x > 0. The natural logarithm ln x uses base e.

Change of base formula

log_b x = (log x)/(log b), using logarithms of any convenient base. For example, log₂ 7 = (log 7)/(log 2) ≈ 2.807.

Compound interest formula

A = P(1 + r/n)^(nt) gives the amount after t years when principal P earns annual rate r compounded n times yearly. Continuous compounding uses A = Pe^(rt).

Rule of 72

An investment doubling time in years is approximately 72 divided by the annual percentage rate. At 6% a year, money doubles in about 12 years.

Permutation versus combination

nPr = n!/(n − r)! counts ordered arrangements of r items chosen from n. nCr = n!/(r!(n − r)!) counts unordered selections, so nCr is smaller.

Fundamental counting principle

When a task happens in stages, multiply the number of choices at each stage to get the total number of possible outcomes.

Circular arrangements

The number of distinct ways to seat n different objects around a circle is (n − 1)!, because rotations of the same arrangement are not counted separately.

Conditional probability formula

The probability of B given A is P(B|A) = P(A and B)/P(A), provided P(A) > 0. Rearranging gives the multiplication rule.