Contemporary Mathematics PDF Download – OpenStax (Quantitative Reasoning / Gen-Ed Math)

Contemporary Mathematics, published free by OpenStax, is a full liberal-arts / quantitative-reasoning textbook built for students who need one solid math course that is not calculus. Its 13 chapters run from sets, logic, and number systems through algebra, money math, probability, statistics, metric measurement, geometry, voting theory, and graph theory — the breadth a general-education “Quantitative Reasoning” or “Liberal Arts Math” course typically covers in a single semester.

Students taking a gen-ed math requirement alongside a BSCS, BSIT, BBA, or BS Math degree get the most out of this book — each chapter is written as a self-contained, stand-alone unit, so an instructor (or a student revising for a final) can pick exactly the chapters a specific syllabus covers, from budgeting and credit-card math in Money Management to Euler circuits in Graph Theory, without needing every earlier chapter first.

Book Overview

Chapter List

Chapter 1: Sets

Difficulty: Easy · Key topics: well-defined sets, roster & set-builder notation, subsets, cardinality, Venn diagrams, union & intersection, complement

This opening chapter treats a set as nothing more than a well-defined collection of objects — the book’s own example is a kitchen flatware drawer holding forks, spoons, and knives. It teaches two ways to describe a set, the roster (listing) method and set-builder notation, plus the standard symbols ℕ, ℤ, and ℚ for the natural numbers, integers, and rationals. It distinguishes finite, infinite, and empty sets, and equal sets from merely equivalent ones (same size, different members). It then covers subsets and proper subsets, Venn diagrams and complements, and finally union and intersection — first with two sets, then with three, where the diagram splits into eight regions.

Key Points:

• A set is a well-defined collection of objects — every member is either clearly in it or clearly not, with no ambiguity.
• ℕ = {1, 2, 3, …} (natural numbers), ℤ = {…, −2, −1, 0, 1, 2, …} (integers), ℚ = {p/q | p, q integers, q ≠ 0} (rationals).
• A subset (A ⊆ B) allows A = B; a proper subset (A ⊂ B) requires at least one member of B that is not in A.
• A finite set with n(A) members has exactly 2^n(A) subsets, including the empty set and the set itself.
• The complement A′ is every member of the universal set U that is not in A; a set and its complement are always disjoint.
• Union (A ∪ B) collects everything in either set; intersection (A ∩ B) keeps only what both sets share.

Memory Tip: Picture ∪ as a cup that scoops up everything from both sets (union), and ∩ as a cap that only holds what’s shared underneath (intersection) — the shape of each symbol matches what it does.

Common Mistake: Writing A ⊂ B when A could equal B — ⊂ specifically means a proper subset, so it’s wrong whenever the two sets might be identical; ⊆ is the safe symbol unless properness is confirmed.

Important Questions:

• How many subsets does a 4-element set have, and why? 2⁴ = 16, because each of the 4 elements is independently either in or out of a given subset.
• What is the complement of a set, and how do you find it? The complement A′ contains every element of the universal set U that is not in A; find it by removing A’s elements from U.

Chapter 2: Logic

Difficulty: Medium · Key topics: statements & quantifiers, compound statements & connectives, truth tables, conditional & biconditional, logical equivalence, De Morgan’s Laws, valid arguments

Logic is introduced as the study of building and testing reasoned arguments — the chapter opens with lawyers like Thurgood Marshall and Ruth Bader Ginsburg constructing legal arguments before the Supreme Court. It starts with statements (sentences that are definitely true or false) and their negations, then builds compound statements with and/but (∧), or (∨), if…then (→), and if-and-only-if (↔), plus the fixed reading order — parentheses, negation, disjunction/conjunction, conditional, biconditional. Truth tables formalize all of this, and the chapter closes with De Morgan’s Laws for negating compound statements and three valid argument forms: the law of detachment, denying the consequent, and the chain rule.

Key Points:

• A logical statement is a complete sentence with a definite truth value; its negation always has the opposite value.
• Conjunction (p ∧ q) is true only when both p and q are true; disjunction (p ∨ q) is false only when both are false.
• A conditional (p → q) is false in exactly one case — p true and q false — and true everywhere else.
• Connective reading order: parentheses first, then negation, then disjunction/conjunction (left to right), then conditional, then biconditional.
• De Morgan’s Laws: ¬(p ∨ q) ≡ ¬p ∧ ¬q, and ¬(p ∧ q) ≡ ¬p ∨ ¬q.
• The conditional (p → q) is logically equivalent to its contrapositive (¬q → ¬p); the converse (q → p) is equivalent to the inverse (¬p → ¬q) — but the conditional and its converse are not equivalent to each other.
• Law of Detachment: if p → q is true and p is true, then q must be true.

Memory Tip: “Break the bar, flip the connective, negate each piece” — for De Morgan’s Laws, distributing a negation across ∧ or ∨ always flips which connective you end up with.

Common Mistake: Assuming a conditional’s converse (q → p) says the same thing as the original (p → q) — it doesn’t; only the contrapositive (¬q → ¬p) is guaranteed logically equivalent to the original.

Important Questions:

• Write the contrapositive of “If it rains, the game is cancelled,” and say whether it must share the original’s truth value. Contrapositive: “If the game is not cancelled, it did not rain.” Yes — the contrapositive is always logically equivalent to the original conditional.
• Given p → q is true and p is true, what can you conclude, and by what rule? q must be true, by the Law of Detachment.

Chapter 3: Real Number Systems and Number Theory

Difficulty: Medium · Key topics: primes & composites, GCD/LCM, integers, order of operations, rational & irrational numbers, real numbers, clock (modulo) arithmetic, exponents, scientific notation, arithmetic & geometric sequences

This chapter builds the full real-number system from the ground up, opening with why large primes matter for encryption — breaking a modern encryption key by factoring would take a fast computer roughly 300 trillion years. It covers prime and composite numbers, the greatest common divisor and least common multiple via prime factorization, the integers and order of operations, then rational numbers (fractions of integers, always reducible to lowest terms) and irrational numbers (like π, that can’t be written as one integer over another). It combines both into the real numbers and their arithmetic properties, then adds clock/modulo arithmetic, exponent rules, scientific notation, and arithmetic and geometric sequences with their sum formulas.

Key Points:

• Every composite number has a unique prime factorization, which is how the GCD and LCM of a set of numbers are found.
• Between any two rational numbers there is always another rational number — this is the density property of the rationals.
• √(a×b) = √a × √b lets you simplify a square root by pulling out perfect-square factors, and rationalize a denominator using a conjugate.
• Clock arithmetic is modulo 12 (modulo 7 for days): divide the result by 12 (or 7) and keep only the remainder.
• Any nonzero number to the 0 power equals 1, and a negative exponent means the reciprocal of the base to the positive power: a⁻ⁿ = 1/aⁿ.
• Scientific notation writes a number as a value from 1 up to (but not including) 10, times 10 to a power; the exponent tracks how far the decimal point moved.
• An arithmetic sequence has a constant difference between terms; a geometric sequence has a constant ratio — each has its own formula for summing the first n terms.

Memory Tip: For clock arithmetic, just keep dividing by 12 (or 7 for days) and hang on to the remainder only — that remainder is the whole answer, nothing else from the division matters.

Common Mistake: Simplifying √(a+b) as √a + √b — square roots do not distribute over addition; they only split cleanly over multiplication and division: √(a×b) = √a × √b.

Important Questions:

• Find the GCD and LCM of 24 and 36 using prime factorization. 24 = 2³×3, 36 = 2²×3²; GCD = 2²×3 = 12, LCM = 2³×3² = 72.
• Simplify √50. 50 = 25×2, and 25 is a perfect square, so √50 = √25 × √2 = 5√2.

Chapter 4: Number Representation and Calculation

Difficulty: Easy · Key topics: Hindu-Arabic positional system, early numeration systems (Babylonian base-60, Mayan base-20, Roman additive), base conversion, arithmetic in other bases

This chapter steps back from the number system everyone takes for granted to ask how humans have recorded quantity across history — from tally marks and Roman numerals (an additive system with no place value) to the Babylonian base-60 and Mayan base-20 systems, both of which combine positional value with some additive elements. It shows how expanded form — writing a number as digits times powers of the base — both explains the familiar Hindu-Arabic base-10 system and generalizes to convert numbers into and out of other bases, including base 2, the language of computers. It closes by rebuilding addition, subtraction, multiplication, and division from scratch inside a non-base-10 system, using that base’s own addition and multiplication tables.

Key Points:

• An additive system (like Roman numerals) repeats symbols to build larger values; a positional system (like Hindu-Arabic) gives the same digit different values depending on where it sits.
• The Babylonian system used base 60 and the Mayan system used base 20, both blending positional value with additive elements.
• Expanded form (each digit × its place value, a power of the base) is what makes converting any base-b number into base 10 straightforward.
• To convert a base-10 number into another base, divide repeatedly by the new base — the remainders, read from last to first, become the new digits.
• Bases above 10 need extra symbols (A, B, C, …) for digit values 10 and higher, since only 0–9 exist as numerals.
• Addition, subtraction, multiplication, and division in another base use the same procedures as base 10, but with that base’s own addition/multiplication table, not the base-10 one.

Memory Tip: When converting base 10 into another base by repeated division, the remainders come out in reverse order — the first remainder computed is the last digit of the answer, so read them bottom to top.

Common Mistake: Reading the remainders from a base-conversion division in the order they were computed instead of reversing them — that produces the digits backwards.

Important Questions:

• Convert 45 (base 10) to binary (base 2). 45÷2=22 r1, 22÷2=11 r0, 11÷2=5 r1, 5÷2=2 r1, 2÷2=1 r0, 1÷2=0 r1. Reading remainders bottom to top: 101101 in base 2.
• Convert the Roman numeral XLIV to a Hindu-Arabic number. XL = 40 (X before L subtracts), IV = 4 (I before V subtracts); XLIV = 44.

Chapter 5: Algebra

Difficulty: Medium · Key topics: algebraic expressions, linear equations & inequalities, ratios & proportions, graphing lines, quadratic equations, functions, graphing functions, systems of equations & inequalities, linear programming

This chapter makes the jump from arithmetic’s fixed answers to algebra’s variables, where 5+3 always equals 8 but 5x+3 depends on what x represents. It covers algebraic expressions and evaluating them, solving linear equations and inequalities in one variable (including applications), ratios and proportions, and graphing lines and inequalities on the coordinate plane — where two lines can meet at one point, never meet, or coincide everywhere. It introduces quadratic equations (up to two solutions, solvable by factoring, the square-root method, graphing, or the quadratic formula) and functions, including the vertical line test, before covering how to graph a linear function using its slope and intercepts. It ends with systems of two linear equations and inequalities, and linear programming — maximizing or minimizing an objective function subject to constraints.

Key Points:

• A linear equation in one variable can have exactly one solution, no solution, or infinitely many solutions — not just one.
• A ratio compares part-to-part, part-to-whole, or whole-to-part; a fraction only ever compares part-to-whole — every fraction is a ratio, but not every ratio is a fraction.
• A relation is a function only when every x-value maps to exactly one y-value; the vertical line test checks this graphically.
• Slope is rise over run: m = (y₂−y₁)/(x₂−x₁) for any two points on a line.
• A quadratic equation (highest power 2) can have two, one, or zero real solutions, found by factoring, the square-root method, graphing, or the quadratic formula.
• Two linear equations in two variables intersect at exactly one point, never intersect (parallel lines), or intersect everywhere (the same line).
• Linear programming maximizes or minimizes an objective function subject to a set of constraints that define a feasible region.

Memory Tip: The vertical line test in one sentence: if a vertical line can ever cross the graph twice, it isn’t a function — a function lets each x-value answer to only one y-value.

Common Mistake: Treating every ratio as a fraction and forcing a part-to-whole reading on it — a ratio like “3 boys to 5 girls” is part-to-part, not the fraction 3/5 of anything.

Important Questions:

• Solve 3x − 7 = 11. Add 7 to both sides: 3x = 18; divide by 3: x = 6.
• Find the slope of the line through (2, 3) and (6, 11). m = (11−3)/(6−2) = 8/4 = 2.

Chapter 6: Money Management

Difficulty: Easy · Key topics: percent, discounts/markups/sales tax, simple & compound interest, budgeting, savings, investments, loans, student loans, credit cards, buying/leasing a car, renting vs. homeownership, income tax

Framed around the finding that the average American carried $96,371 in consumer debt in 2021, with fewer than one in four Americans debt-free, this chapter is a practical tour of personal finance built on one core skill: percent. It covers discounts, markups, and sales tax; simple interest (I=Prt) and compound interest, where growth is small early on and large later; building a budget using the 50-30-20 rule (needs, wants, savings); the three main savings vehicles (savings accounts, money market accounts, and CDs) and how they rank by interest rate; and investments — bonds, stocks, mutual funds, and retirement accounts. It closes with loans and amortization, federal student loans (subsidized versus unsubsidized), credit cards and minimum payments, buying versus leasing a car, renting versus owning a home, and how progressive federal income tax brackets actually work.

Key Points:

• Simple interest: I = P × r × t (principal × rate × time); compound interest lets each period’s interest earn interest too, so growth accelerates over time.
• The 50-30-20 budgeting rule allocates roughly 50% of income to needs, 30% to wants, and 20% to savings and debt reduction.
• Savings accounts typically pay the lowest interest, money market accounts more, and CDs the most — in exchange for less access to the money.
• Federal subsidized student loans don’t accrue interest until after graduation; unsubsidized loans start accruing interest as soon as the money is disbursed.
• Paying only a credit card’s minimum payment keeps the account current but stretches the payoff over years and multiplies the total interest paid.
• A mortgage is a loan like any other, but the monthly payment usually also includes an escrow portion for property taxes and insurance.
• Federal income tax is progressive and marginal: only the income inside each bracket is taxed at that bracket’s rate, not all of it.

Memory Tip: Being “in the 32% bracket” does not mean 32% of all income is taxed — only the slice of income that falls inside that bracket is; every lower slice is still taxed at its own, lower rate.

Common Mistake: Assuming a tax bracket is an average rate applied to the whole income — it’s a marginal rate applied only to the income within that bracket, so the effective (average) rate is always lower than the top bracket’s rate.

Important Questions:

• Find the simple interest on a $2,000 loan at 5% annual interest for 3 years. I = P×r×t = 2000 × 0.05 × 3 = $300.
• What is the key difference between a subsidized and an unsubsidized federal student loan? A subsidized loan doesn’t accrue interest until after graduation; an unsubsidized loan starts accruing interest immediately after the funds are disbursed.

Chapter 7: Probability

Difficulty: Medium · Key topics: multiplication rule for counting, permutations & combinations, tree diagrams & sample spaces, basic probability, odds, addition rule, conditional probability, binomial distribution, expected value

Opening with the fact that U.S. commercial casinos brought in over $43 billion in revenue in 2019, this chapter builds the counting and probability tools needed to know the odds of an outcome — starting with the multiplication rule for counting, then permutations (order matters) and combinations (order doesn’t), both computable with factorials. It covers tree diagrams and tables for organizing a sample space, then the three types of probability — theoretical, empirical, and subjective — and how permutations and combinations feed into probability calculations. It ends with odds versus probability, the addition rule for combining events, conditional probability, the binomial distribution for repeated independent trials, and expected value as the long-run average outcome.

Key Points:

• The Multiplication Rule for Counting multiplies the number of choices at each stage to count a large set of outcomes.
• Permutations count arrangements where order matters; combinations count selections where it doesn’t — both are computed using factorials.
• Theoretical probability is a ratio of equally-likely outcomes; empirical probability comes from repeating an experiment; subjective probability is an informed judgment call when the experiment can’t be repeated.
• The probability of an event’s complement is 1 minus the probability of the event.
• Odds are the ratio of an event’s probability to its complement’s probability — not the same number as the probability itself.
• The Addition Rule finds P(A or B); for events that aren’t mutually exclusive, the Inclusion/Exclusion Principle subtracts the double-counted overlap.
• Expected value is the sum of every possible outcome multiplied by its probability — the long-run average result.

Memory Tip: Permutation and Position both start with P — order (position) matters for a permutation; for a combination, it doesn’t.

Common Mistake: Treating “odds” and “probability” as the same number — probability is P(event) out of the whole sample space, while odds compare P(event) to P(not event), so 3-to-1 odds is a 3/4 probability, not 3/1.

Important Questions:

• How many ways can you choose a committee of 3 people from a group of 8, if order doesn’t matter? Combination: C(8,3) = 8!/(3!×5!) = 56.
• A single die is rolled once. What is the expected value of the roll? E = (1+2+3+4+5+6)/6 = 21/6 = 3.5.

Chapter 8: Statistics

Difficulty: Medium · Key topics: gathering/organizing/visualizing data, mean/median/mode, range & standard deviation, percentiles, normal distribution & z-scores, scatter plots/correlation/regression

Framed by how a WNBA team decides on a player’s salary using measurable performance data, this chapter is the mathematics of gathering, organizing, summarizing, and drawing conclusions from data. It distinguishes categorical from quantitative data and covers visualizing each (pie/bar charts versus histograms), warns how manipulated axes or bin widths can misrepresent a distribution, and covers the three measures of central tendency (mean, median, mode) along with range, standard deviation, and percentiles. It introduces the normal distribution and the 68-95-99.7 rule, z-scores for comparing values across different datasets, and closes with scatter plots, the correlation coefficient r, and regression lines for predicting one variable from another.

Key Points:

• The median resists outliers, but the mean is pulled toward them — this affects which measure of center best represents a skewed dataset.
• The 68-95-99.7 Rule: for normally distributed data, about 68% of values fall within 1 standard deviation of the mean, 95% within 2, and 99.7% within 3.
• A z-score measures how many standard deviations a value sits above (positive) or below (negative) the mean, letting you compare values from different datasets.
• The correlation coefficient r ranges from −1 to 1: positive means the variables rise together, negative means one rises as the other falls, and values near 0 mean a weak relationship.
• A regression line is the best-fit line through a scatter plot, used to predict the response variable from the explanatory variable.
• Areas in a histogram represent the proportion of data in each bin; changing bin widths or axis scales can misrepresent the same dataset.
• The range is just the largest value minus the smallest; the standard deviation is closer to the average distance every value sits from the mean.

Memory Tip: 68-95-99.7 pairs with 1-2-3: one standard deviation from the mean covers 68%, two covers 95%, three covers 99.7% — the pattern grows fast because the normal curve’s tails are so thin.

Common Mistake: Reporting the mean for a dataset with extreme outliers (like household income) instead of the median — a handful of very high values drags the mean well above what a “typical” value actually looks like.

Important Questions:

• A dataset has mean 70 and standard deviation 5. What is the z-score for a value of 82? z = (82−70)/5 = 2.4.
• What does a correlation coefficient of r = −0.85 tell you about two variables? A strong negative relationship — as one variable increases, the other tends to decrease, and the points cluster tightly around a downward-sloping line.

Chapter 9: Metric Measurement

Difficulty: Easy · Key topics: metric system & prefixes, area, volume, weight/mass, temperature

Using a Finnish road sign as its opening image, this short chapter makes the case for the metric system (the International System of Units, SI) as the measurement standard used in every country except the United States and two others. It covers the decimal structure of metric prefixes, where each unit is exactly 10 times the next, and a memorable phrase for ordering them from largest to smallest. It then works through area (square meters), volume (cubic meters, with the liter defined as exactly one cubic decimeter), weight and mass — pointing out mass is constant while weight changes with gravity — and temperature on the Celsius scale, anchored at 0°C (water freezes) and 100°C (water boils).

Key Points:

• The metric system is fully decimal: every successive unit is exactly 10 times larger than the one before it, so converting between units is just moving a decimal point.
• Metric prefixes run largest to smallest as kilo, hecto, deka, (base unit), deci, centi, milli.
• The metric base unit for area is the square meter (m²); for volume it’s the cubic meter (m³).
• One liter equals exactly one cubic decimeter — the volume of a 10 cm × 10 cm × 10 cm cube.
• Mass (the amount of matter in an object) never changes; weight (the force gravity exerts on that mass) does — the same object weighs less on the Moon.
• The Celsius scale is anchored at water’s freezing point (0°C) and boiling point (100°C), 100 degrees apart.

Memory Tip: “King Henry Died Drinking Chocolate Milk” holds the metric prefix order from largest to smallest — kilo, hecto, deka, (base), deci, centi, milli — without memorizing each one separately.

Common Mistake: Using “weight” and “mass” interchangeably — mass (in grams/kilograms) stays fixed everywhere, while weight is a force that changes with the local strength of gravity.

Important Questions:

• Convert 2.5 meters to centimeters. Centi means 1/100, so multiply by 100: 2.5 m = 250 cm.
• Why does an astronaut’s mass stay the same on the Moon while their weight decreases? Mass measures the amount of matter in their body, which doesn’t change; weight is the force of gravity on that mass, and the Moon’s gravity is weaker than Earth’s.

Chapter 10: Geometry

Difficulty: Medium · Key topics: points/lines/planes, angles & transversals, triangles & congruence, polygons/perimeter/circumference, tessellations, area, volume & surface area, right-triangle trigonometry

Opening with Raphael’s painting The School of Athens and a nod to Euclid, “the father of geometry,” this chapter builds geometry from its most basic objects — points, lines, and planes — up through angles (acute, obtuse, right, straight, complementary, supplementary, and the angle pairs formed when a transversal crosses parallel lines) and triangles, including the four congruence theorems (SAS, ASA, SSS, AAS) and similarity. It covers polygons, perimeter, and circumference; tessellations as repeating, gap-free patterns; the area formulas for triangles, parallelograms, trapezoids, rhombi, regular polygons, and circles; and volume and surface area for prisms and cylinders. It closes with right-triangle trigonometry — the Pythagorean theorem, the two special right triangles, and the sine, cosine, and tangent ratios.

Key Points:

• The interior angles of any triangle always sum to 180°; two triangles are congruent when their corresponding angles and side lengths all match (proven via SAS, ASA, SSS, or AAS).
• A transversal crossing two parallel lines creates four types of equal-angle pairs: alternate interior, alternate exterior, corresponding, and vertical angles.
• The sum of a regular polygon’s interior angles is (n−2)×180°, and its exterior angles always sum to 360° regardless of how many sides it has.
• Circumference: C = 2πr (or C = πd); circle area: A = πr².
• Area formulas differ by shape: triangle A = ½bh, rectangle A = lw, parallelogram A = bh, trapezoid A = ½h(b₁+b₂).
• The Pythagorean Theorem, a² + b² = c², applies only to right triangles, where c is the hypotenuse.
• The primary trig ratios: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent.

Memory Tip: “SOH-CAH-TOA” — Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent — holds all three trig ratios in one word.

Common Mistake: Applying the Pythagorean Theorem (a²+b²=c²) to a triangle that isn’t a right triangle — the formula only holds when one angle is exactly 90°.

Important Questions:

• A right triangle has legs of 6 and 8. Find the hypotenuse. a²+b²=c²: 6²+8²=36+64=100, so c=√100=10.
• Find the area of a trapezoid with height 4, and parallel sides 5 and 9. A = ½h(b₁+b₂) = ½×4×(5+9) = 2×14 = 28.

Chapter 11: Voting and Apportionment

Difficulty: Medium · Key topics: voting methods (plurality, runoff, ranked-choice, Borda count, pairwise comparison, approval), fairness criteria, Arrow’s Impossibility Theorem, standard divisor & quota, apportionment methods (Hamilton, Jefferson, Adams, Webster), apportionment paradoxes, Balinski-Young theorem

Framed as the task of founding the constitution of an imaginary new democracy, this chapter treats voting and resource-apportionment as genuine mathematical problems, not just civics. It compares plurality, runoff, ranked-choice (instant runoff), Borda count, pairwise comparison, and approval voting, then tests each against fairness criteria such as the majority and head-to-head criteria — Arrow’s Impossibility Theorem proves that no ranked voting method can satisfy every fairness criterion at once. It then turns to apportionment: dividing a fixed number of seats fairly using the standard divisor and standard quota, comparing Hamilton’s, Jefferson’s, Adams’s, and Webster’s methods, and the paradoxes (Alabama, population, new-state) that can make even a mathematically defined method feel unfair — the Balinski-Young Impossibility Theorem proves no apportionment method can satisfy every fairness criterion either.

Key Points:

• Plurality voting simply awards the win to whoever gets the most votes, even without a majority; ranked-choice (instant runoff) and Borda count use voters’ full rankings instead.
• Arrow’s Impossibility Theorem: any ranked-preference voting method will violate at least one recognized fairness criterion.
• The standard divisor is total population divided by house size; the standard quota is how many seats each group’s population entitles it to.
• Hamilton’s method distributes leftover seats to the largest fractional remainders and satisfies the quota rule; Jefferson’s and Adams’s methods use a modified divisor and can violate it, favoring larger or smaller groups respectively.
• Webster’s method rounds the modified quota normally and favors neither larger nor smaller groups, but can still violate the quota rule.
• The Alabama paradox: increasing the total number of seats available can actually cause one group to lose a seat under some methods.
• The Balinski-Young Impossibility Theorem: no apportionment method can simultaneously satisfy the quota rule and avoid every paradox.

Memory Tip: Jefferson favors the big (states), Adams favors the small, Webster plays it straight down the middle, and Hamilton just hands out leftovers by size of the remainder — four methods, four different biases.

Common Mistake: Assuming a “fair” voting or apportionment method must exist if it’s designed carefully enough — Arrow’s and Balinski-Young’s Impossibility Theorems both prove mathematically that no method can satisfy every fairness criterion simultaneously.

Important Questions:

• What is the key difference between Hamilton’s and Jefferson’s apportionment methods? Hamilton’s method distributes leftover seats by the size of each group’s fractional remainder and satisfies the quota rule; Jefferson’s method uses a modified divisor instead, favors larger groups, and can violate the quota rule.
• What is the Alabama paradox? A situation where increasing the total number of seats to be apportioned causes a group to lose a seat it would otherwise have kept, under certain apportionment methods.

Chapter 12: Graph Theory

Difficulty: Medium-Hard · Key topics: graphs & multigraphs, vertices/edges/degree, subgraphs & cycles, graph isomorphism, walks/trails/paths, Euler circuits & trails, Hamilton cycles & paths, Traveling Salesperson Problem, trees

Introduced through the idea of a graph as a stripped-down map — keeping only the connections between objects and discarding size, shape, and distance — this chapter builds graph theory from vertices and edges up through real navigation and optimization problems. It covers a vertex’s degree, complete graphs, subgraphs, cycles, and when two graphs are isomorphic (structurally identical). It distinguishes walks, trails, and paths through a graph, then Euler circuits and trails (which travel every edge exactly once, existing only under specific degree conditions) from Hamilton cycles and paths (which visit every vertex exactly once instead). It closes with the Traveling Salesperson Problem — comparing the always-optimal but impractical brute-force method against the fast but not-always-optimal nearest-neighbor greedy algorithm — and trees, the connected, cycle-free graphs that underlie network and spanning-tree problems.

Key Points:

• The sum of every vertex’s degree in a graph always equals twice the number of edges (the Handshake Lemma).
• An Euler circuit — a route that uses every edge exactly once and returns to its start — exists only when a graph is connected and every vertex has even degree.
• An Euler trail (every edge once, but start and end can differ) exists only when a graph has exactly two vertices of odd degree.
• A Hamilton cycle visits every vertex exactly once and returns to the start; unlike an Euler circuit, it says nothing about edges, and the two don’t always coincide.
• The brute-force method always finds the lowest-weight Hamilton cycle for the Traveling Salesperson Problem but becomes impractical fast as vertices increase; the nearest-neighbor method is a fast greedy algorithm that usually doesn’t find the true optimum.
• Two graphs are isomorphic only if they match on vertex count, edge count, the multiset of vertex degrees, and subgraph structure — mismatching any of these rules out an isomorphism immediately.
• A tree is a connected graph with no cycles; a tree on n vertices always has exactly n−1 edges, which is what makes a spanning tree the minimum-edge way to connect every vertex in a network.

Memory Tip: Euler goes edge by edge, Hamilton goes vertex by vertex — an Euler circuit/trail must use every edge exactly once, while a Hamilton cycle/path must visit every vertex exactly once; the two requirements can overlap, but neither guarantees the other.

Common Mistake: Confusing an Euler circuit with a Hamilton cycle because both “visit everything and return to the start” — an Euler circuit’s rule is about edges, a Hamilton cycle’s rule is about vertices, and a graph can easily have one without the other.

Important Questions:

• A connected graph has all vertices of even degree. Does it have an Euler circuit? Yes — the Euler Circuit Theorem guarantees one exists whenever a graph is connected and every vertex has even degree.
• What’s the practical trade-off between the brute-force and nearest-neighbor methods for the Traveling Salesperson Problem? Brute force checks every possible route and guarantees the shortest one, but the number of routes grows factorially, making it impractical for more than a handful of vertices; nearest-neighbor is fast at any size but usually returns a route close to optimal, not the true shortest one.

Chapter 13: Math and…

Difficulty: Easy · Key topics: golden ratio & Fibonacci sequence, water/energy conservation math, drug-dosage & vaccine-validation math, sound frequency & music theory, tournament math

This closing chapter is a tour of five places math shows up outside a typical syllabus. In art, it covers the golden ratio (φ) and the Fibonacci sequence, both found repeatedly in nature and used deliberately in design and composition. In the environment, it works through the measurable impact of water and energy conservation, including calculating electrical demand from wattage and the benefits of residential solar power. In medicine, it covers computing liquid and pill dosages and how mathematics is used to validate a new drug’s effectiveness and model the spread of a virus. In music, it connects frequency (measured in hertz) to pitch and covers half-steps, whole-steps, and octaves. In sports, it closes with the mathematics of tournament formats — how many games a round-robin or single-elimination bracket actually requires.

Key Points:

• The golden ratio, φ ≈ 1.618, and the Fibonacci sequence (1, 1, 2, 3, 5, 8, 13, …, where each term is the sum of the two before it) both appear repeatedly in natural patterns and in art and design.
• Electrical demand is calculated directly from wattage, which is how a household’s solar-power needs or savings get estimated.
• Mathematical modeling is used to track and predict how a virus spreads through a population, and statistics are used to validate whether a new drug or vaccine is actually effective.
• Frequency (measured in hertz) determines pitch — higher frequency means higher pitch — while decibels measure a sound’s intensity, not its pitch.
• A round-robin tournament with n teams requires n(n−1)/2 total games, since every team plays every other team exactly once.
• A single-elimination tournament with n teams requires exactly n−1 games, since every game eliminates exactly one team until only the champion remains.

Memory Tip: Round-robin games = n(n−1)/2 is really just “n choose 2” — the number of unique pairs you can make from n teams, since every pair plays exactly once.

Common Mistake: Mixing up round-robin and single-elimination game counts — round-robin needs n(n−1)/2 games (everyone plays everyone), while single-elimination needs only n−1 (half the field is cut every round).

Important Questions:

• How many games are played in a round-robin tournament with 8 teams? n(n−1)/2 = 8×7/2 = 28 games.
• How many games are played in a single-elimination tournament with 16 teams? n−1 = 16−1 = 15 games, since exactly one team is eliminated per game until one champion remains.

Download Contemporary Mathematics PDF (Free)

Contemporary Mathematics is free directly from its publisher — the official OpenStax book page offers both a free PDF download and a free online read of the complete, current text. Click below to go to the official page and choose the format you want.

How to Study This Book

Unlike a sequential calculus text, Contemporary Mathematics is explicitly built as 13 stand-alone chapters — the book’s own preface describes “stand-alone sections” by design — so an instructor or self-study student can pick exactly the chapters a syllabus needs without reading start to finish. A few chapters do lean on earlier ones, though, and it’s worth knowing which before skipping ahead.

Chapters 1-2 (Sets, Logic) are the foundation for Chapter 12 (Graph Theory) and Chapter 11 (Voting and Apportionment): set notation (∪, ∩, complement) reappears directly in Graph Theory’s vertex/edge language, and the argument forms from Logic underpin how Voting and Apportionment frames its fairness proofs. Read 1 and 2 first if a syllabus includes 11 or 12.

Chapter 3 (Real Number Systems and Number Theory) is the one true prerequisite chapter: its order-of-operations, exponent, and scientific-notation sections are used without re-explanation in Chapter 4 (Number Representation), Chapter 5 (Algebra), and Chapter 6 (Money Management)’s interest formulas. A student rusty on exponents should work through Appendix A (Co-Req Appendix: Integer Powers of 10) first — a short, self-contained remedial refresher the book includes for exactly this gap — before Chapter 3 or Chapter 4, not after.

Chapters 5 (Algebra) and 6 (Money Management) pair naturally: Chapter 6’s simple- and compound-interest formulas are direct applications of the linear- and exponential-style equations Chapter 5 teaches how to solve, so working through 5 before 6 makes the finance chapter’s formulas click instead of feeling like memorization.

Chapters 7 (Probability) and 8 (Statistics) are sequential by design — Chapter 7’s combinations and probability rules resurface directly in Chapter 8’s binomial-distribution and percentile sections, so this is one place in the book where reading in order genuinely pays off, even though the book presents them as separate stand-alone chapters.

Chapters 9 (Metric Measurement) and 10 (Geometry) share almost no dependency on the rest of the book and can be read in either order, or skipped entirely by a syllabus that doesn’t cover measurement or geometry — Chapter 10’s right-triangle trigonometry is the one exception, leaning lightly on Chapter 5’s function and graphing language.

Chapters 11 (Voting and Apportionment) and 13 (Math and…) are genuinely self-contained and make good “extra topic” or capstone-project chapters for a course with room for one enrichment unit; Chapter 13 in particular is built as five independent mini-topics (Art, Environment, Medicine, Music, Sports), so a class can assign just one sub-section as a project rather than the whole chapter.

What this book never covers: calculus, matrix algebra, or trigonometric identities beyond the right-triangle basics in Chapter 10. A student who needs those should move on to Precalculus or College Algebra rather than expect Contemporary Mathematics to build toward them — it is a terminal gen-ed course text, not a pipeline into a calculus sequence.

Used In These Programs

Contemporary Mathematics is a General Education / Quantitative Reasoning book — it is not tied to one specific degree. It commonly satisfies a university’s math gen-ed or “Quant Reasoning” requirement across BS Computer Science, BS Information Technology, BS Software Engineering, BBA, BS Mathematics, and other programs with a general math requirement outside the calculus track. Browse all General Mathematics books or all Mathematics books.

Who Should Read This

This book is written for a student who needs one solid general-education math course that isn’t calculus — someone fulfilling a Quantitative Reasoning or Liberal Arts Math requirement alongside a BSCS, BSIT, BBA, or other non-math-heavy degree, or anyone who wants a broad, practical math refresher covering everything from logic and probability to personal finance and geometry in one book. It assumes comfort with basic arithmetic (Appendix A and Chapter 3 rebuild anything shakier than that) but does not assume any prior algebra beyond what Chapter 5 itself teaches. It is not the right book for a student who needs to build toward calculus — that track needs the proof-and-logic foundation in Discrete Mathematics: An Open Introduction, or a calculus text once algebra is solid.

Applicable Universities

This book is useful for gen-ed / Quantitative Reasoning requirements at Pakistani universities and colleges offering BS programs in Computer Science, Information Technology, Software Engineering, Business Administration, and Mathematics, including Punjab University, Virtual University, COMSATS, FAST, UET, NUST, and other HEC-recognized institutions.

FAQs

Related Books

Contemporary Mathematics stands apart from freebooks.pk’s other published Mathematics titles, which are mostly on the calculus track. These are the closest already-published books — useful as a follow-on once this course is done, not as peers at the same level:

• Discrete Mathematics: An Open Introduction – Oscar Levin (the closest topical match — sets, logic, and graph theory all reappear here in more depth, with formal proofs)
• Calculus Volume 1 – Strang & Herman (OpenStax) (the natural next step for a student whose degree does require calculus, once this course is finished)
• Active Calculus – Matthew Boelkins (2nd Edition) (a second free calculus option, useful for the same continuing-on-to-calculus student)
• Calculus Volume 2 – Strang & Herman (OpenStax) (for later reference, once Volume 1 is complete)

Contemporary Mathematics is a free, complete way to satisfy a Quantitative Reasoning or Liberal Arts Math requirement — covering sets and logic through probability, statistics, money management, geometry, voting, and graph theory in one course. Browse more Mathematics books to continue building from here.

Contemporary Mathematics, by Donna Kirk (Senior Contributing Author) with contributing authors Barbara Boschmans, Brian Beaudrie, Matthew Cathey, Valeree Falduto, Maureen Gerlofs, Quin Hearn, and Ian Walters. Published by OpenStax / Rice University. Licensed under CC BY-NC-SA 4.0 (Creative Commons Attribution-NonCommercial-ShareAlike International). Free PDF download and online read at https://openstax.org/details/books/contemporary-mathematics