Yale requires at least two course credits in classes that involve “quantitative reasoning,” designated “QR” in the Yale College Bluebook. The mental rigor resulting from quantitative study has been celebrated since ancient times, and applications of quantitative methods have proven critical to many different disciplines. Mathematics and statistics are basic tools for the natural and social sciences, and they have become useful in many of the humanities as well. Information technology and the rigorous dissection of logical arguments in any discipline depend on algorithms and formal logical constructs. An educated person must be able to use quantitative information to make, understand, and evaluate arguments.

Many quantitative reasoning courses are taught through the departments of Mathematics, Statistics, and Computer Science. Such courses may also be found in Architecture; Astronomy; Chemistry; Economics; Engineering; Environmental Studies; Geology and Geophysics; Global Affairs; Linguistics; Molecular, Cellular, and Developmental Biology; Operations Research; Philosophy; Physics; Political Science; Psychology; and Sociology. Below is a list of popular QR courses at Yale as well as some student evaluations.

### Popular QR Courses at Yale

**AMTH 160b: The Structure of Networks.** Network structures and network dynamics described through examples and applications ranging from marketing to epidemics and the world climate. Study of social and biological networks as well as networks in the humanities. Mathematical graphs provide a simple common language to describe the variety of networks and their properties.

**APHY 110b: The Technological World.** An exploration of modern technologies that play a role in everyday life, including the underlying science, current applications, and future prospects. Examples include solar cells, light-emitting diodes (LEDs), computer displays, the global positioning system, fiber-optic communication systems, and the application of technological advances to medicine. For students not committed to a major in science or engineering; no college-level science or mathematics required.

**ARCH 161: Introduction to Structures**. Basic principles governing the behavior of building structures. Developments in structural form combined with the study of force systems, laws of statics, and mechanics of materials and members and their application to a variety of structural systems.

*“It was a great venture into a field most of us are unfamiliar with–structural logic behind how buildings stand. It was exhilarating. I couldn’t recommend it more highly. Don’t be scared by the quantitative and scientific aspect of it, AT ALL! Every single person in my 22-person class, including those who are very shaky with math, felt good about the class. Jump in.”*

**ASTR**** 110: Planets and Stars**. An introduction to stars and planetary systems. Topics include the solar system and extrasolar planets, planet and stellar formation, and the evolution of stars from birth to death.

**ASTR**** 120: Galaxies and the Universe**. An introduction to stars and stellar evolution; the structure and evolution of the Milky Way galaxy and other galaxies; quasars, active galactic nuclei, and supermassive black holes; cosmology and the expanding universe.

*“It’s a great class! You learn about the universe and its evolution and about things that sound like science fiction but are actually true. It’s not completely easy, but it’s extremely enjoyable and I loved everything we learned.”*

**ASTR 160b: Frontiers and Controversies in Astrophysics. **A detailed study of three fundamental areas in astrophysics that are currently subjects of intense research and debate: (1) planetary systems around stars other than the sun; (2) pulsars, black holes, and the relativistic effects associated with them; (3) the age and ultimate fate of the universe. No prerequisite other than a working knowledge of elementary algebra.

**ASTR 170a: Introduction to Cosmology. **An introduction to modern cosmological theories and observations. Topics include aspects of special and general relativity; curved space-time; the Big Bang; inflation; primordial element synthesis; the cosmic microwave background; the formation of galaxies; and large-scale structure.

**ASTR 210b: Stars and Their Evolution.**An intensive introduction to stars. Nuclear processes and element production, stellar evolution, stellar deaths and supernova explosions, and stellar remnants including white dwarfs, neutron stars, and black holes. A close look at our nearest star, the sun. Prerequisite: a strong background in high school mathematics and physics.

**ASTR 220a: Galaxies and Cosmology.**An intensive introduction to extragalactic astronomy. The structure and contents of galaxies, evolution of galaxies, observational cosmology, and the history of the universe.

**ASTR 255a: Research Methods in Astrophysics.**The acquisition and analysis of astrophysical data, including the design and use of ground- and space-based telescopes, computational manipulation of digitized images and spectra, and confrontation of data with theoretical models. Examples taken from current research at Yale and elsewhere. Use of the Python programming language. Includes an optional field trip during October recess to the Arecibo 300-meter radio telescope. No previous programming experience required.

**CHEM 112a: Chemistry with Problem Solving.**For beginning students in chemistry or for those whose exposure to the subject has been moderate. Special emphasis on scientific problem-solving skills through an additional discussion section devoted to quantitative reasoning. Enrollment by placement only and limited to freshmen.

**CHEM 114a or b: Comprehensive General Chemistry.**A comprehensive survey of modern descriptive, inorganic, and physical chemistry for students with a good secondary school exposure to general chemistry. Enrollment by placement only.

**CHEM 118a: Quantitative Foundations of General Chemistry.**An advanced course emphasizing conceptual aspects and physical principles in general chemistry. Fulfills the general chemistry prerequisite for organic chemistry. Attendance at a weekly discussion section required. Must be taken concurrently with CHEM 119L. Enrollment by placement only.

**CPSC 101b: Great Ideas in Computer Science.**An introduction for nonmajors to some of the most important ideas in computer science: what the computer is; how it works; what it can do and what it cannot do, now and in the future. Topics include algorithms, elementary programming, hardware, language interpretation, software engineering, complexity, models of computation, and artificial intelligence. No previous programming experience required.

**CPSC 112a or b: Introduction to Programming.**Development on the computer of programming skills, problem-solving methods, and selected applications.

**CPSC**

**191: Great Ideas in Computer Science**. An introduction for non-majors to some of the most important ideas in computer science: what the computer is; how it works; what it can do and what it cannot do, now and in the future. Topics include algorithms, elementary programming, hardware, language interpretation, software engineering, complexity, models of computation, and artificial intelligence.

**CPSC**** 112 and 201: Introduction to Programming & Computer Science**. Introduction to the concepts, techniques, and applications of computer science, Topics include computer systems; theoretical foundations of computing; and artificial intelligence. Examples stress the important of different problem-solving methods.

*“AMAZING AMAZING AMAZING. One of the funnest classes I’ve taken at Yale. I loved doing the problem sets. Even though they were sometimes frustrating at first, finishing one and getting everything to compile and run properly was one of the most rewarding feelings. Before taking this class I knew nothing about computers or programming. Now I can get my computer to do stuff for me!!! TOO COOL!! Great great great class.”*

**CPSC 202a: Mathematical Tools for Computer Science. **Introduction to formal methods for reasoning and to mathematical techniques basic to computer science. Topics include propositional logic, discrete mathematics, and linear algebra. Emphasis on applications to computer science: recurrences, sorting, graph traversal, Gaussian elimination.

**ECON 108a or b: Quantitative Foundations of Microeconomics. **Introductory microeconomics with a special emphasis on quantitative methods and examples. Intended for students with limited or no prior exposure to calculus. Preference to freshmen. Permission of Economics DUS required.

**ECON 110 and 115: Introductory Microeconomics**. An introduction the basic tools of microeconomics to provide a rigorous framework for understanding how individuals, firms, markets, and governments allocate scarce resources. The design and evaluation of public policy.

**ECON 251: Financial Theory**. Capital asset pricing model, arbitrage pricing theory, option pricing, social security, operation of security exchanges, investment banks, securitization, mortgage derivatives, interest rate derivatives, hedge funds, financial crises, agency theory, and financial incentives.

**EENG 201b: Introduction to Computer Engineering. **Introduction to the theoretical principles underlying the design and programming of simple processors that can perform algorithmic computational tasks. Topics include data representation in digital form, combinational logic design and Boolean algebra, sequential logic design and finite state machines, and basic computer architecture principles. Hands-on laboratory involving the active design, construction, and programming of a simple processor.

**ENAS 110b: The Technological World.**An exploration of modern technologies that play a role in everyday life, including the underlying science, current applications, and future prospects. Examples include solar cells, light-emitting diodes (LEDs), computer displays, the global positioning system, fiber-optic communication systems, and the application of technological advances to medicine. For students not committed to a major in science or engineering; no college-level science or mathematics required.

**ENAS 120b: Introduction to Environmental Engineering.**Introduction to engineering principles related to the environment, with emphasis on causes of problems and technologies for abatement. Topics include air and water pollution, global climate change, hazardous chemical and emerging environmental technologies. Prerequisites: high school calculus and chemistry or CHEM 114, 115 (may be taken concurrently) or permission of instructor.

**EVST 201a: Atmosphere, Ocean, and Environmental Change.**Physical processes that control Earth’s atmosphere, ocean, and climate. Quantitative methods for constructing energy and water budgets. Topics include clouds, rain, severe storms, regional climate, the ozone layer, air pollution, ocean currents and productivity, the seasons, El Niño, the history of Earth’s climate, global warming, energy, and water resources. Must be taken concurrently with EVST 202La.

**GLBL 121a: Applied Quantitative Analysis.**Mathematical fundamentals that underlie analytical approaches in public policy and the social sciences. Development of mathematical skills in areas such as linear functions, single and multiple variable differentiation, exponential functions, and optimization. Statistical approaches include descriptive statistics, principles of sampling, hypothesis tests, simple linear regression, multiple regression, and models for analyzing categorical outcomes.

**G&G 140: Atmosphere, Ocean, and Environmental Change**. Physical processes that control Earth’s atmosphere, ocean, and climate. Quantitative methods for constructing energy and water budgets. Topics include clouds, rain , server storms, regional climate, the ozone layer, air pollution, ocean currents and productivity, the seasons, El Nino, the history of Earth’s climate, global warming, energy, and water resources.

**LING 224a: Formal Foundations of Linguistic Theories.**Study of formal systems that play an important role in the scientific study of language. Exploration of a range of mathematical structures and techniques; demonstrations of their application in theories of grammatical competence and performance including set theory, graphs and discrete structures, algebras, formal language and automata theory. Evaluation of strengths and weaknesses of existing formal theories of linguistic knowledge.

**LING 263a: Semantics.**Introduction to truth-conditional compositional semantics. Set theory, first- and higher-order logic, and the lambda calculus as they relate to the study of natural language meaning. Some attention to analyzing the meanings of tense/aspect markers, adverbs, and modals.

**MATH 101: Geometry of Nature**. Geometric patterns in nature, including classical models of spirals in seashells and sunflowers, symmetry of honeycombs and snowflakes, and the curvature of soap films; the shape of the universe; ways to visualize the fourth dimension; and a brief introduction to fractal geometry.

*“One word: excellent. I highly recommend this class and Professor Frame is an incredible teacher. Help is always there when needed and this has been one of my favorite classes at Yale. I really enjoyed it!”*

**MATH 107a: Mathematics in the Real World. **The use of mathematics to address real-world problems. Applications of exponential functions to population growth and radiocarbon dating; geometric series in mortgage payments, amortization of loans, future value of money; applications of basic probability theory and Bayes’s law in disease detection and drug testing; elements of logic; elements of cryptography. No knowledge of calculus required. Enrollment limited to students who have not previously taken a high school or college calculus or statistics course.

**MATH 112/115: Calculus of Functions of One Variable I & II**. Applications of integration, with some formal techniques and numerical methods. Improper integrals, approximation of functions by polynomials, infinite series.

**MATH 190a: Fractal Geometry. **A visual introduction to the geometry of fractals and the dynamics of chaos, accessible to students not majoring in science. Study of mathematical patterns repeating on many levels and expressions of these patterns in nature, art, music, and literature.

**MUSI 175b: The Mathematics of Music.**An introduction to applied mathematics in the context of music theory and analysis. Concepts from algebra, modular arithmetic, set theory, geometry, and elementary topology are applied to the study of musical rhythms, melodies, and chords across a wide repertoire of classical, atonal, and popular musics. Prerequisite: ability to read music.

**PHIL 115a: First Order Logic.**An introduction to formal logic. Study of the formal deductive systems and semantics for both propositional and predicate logic. Some discussion of metatheory.

**PHIL 281b: Infinity.**The idea of infinity. Traditional and contemporary versions of the paradoxes of space, time, and motion, as well as the paradoxes of classes, chances, and truth. Some elementary arithmetic, geometry, probability theory, and set theory.

**PHYS 060a: Energy Technology and Society.**The technology and use of energy. Impacts on the environment, climate, security, and economy. Application of scientific reasoning and quantitative analysis. Intended for students not committed to a major in science or engineering; no college-level science or mathematics required. Enrollment limited to freshmen. Preregistration required; see under Freshman Seminar Program.

**PHYS 101a or b: Movie Physics.**A critical evaluation of Hollywood action movies using the laws of physics and back of the envelope estimates to distinguish between fictional and real movie physics. Enrollment limited to freshmen and sophomores. Intended for students with little or no prior exposure to calculus and statistics.

**PHYS 115a: The Physics of Dance.**Critical investigation of introductory concepts in physics through the lens of dance. Topics in physics include the normal force, friction, Newton’s laws, projectile motion, potential and kinetic energy, and conservation of energy. Topics in dance include aspects of dance history, contemporary artists who engage with science, and the development of movement studies. Class meetings include movement exercises. Prerequisite: basic trigonometry and algebra. Prior dance experience is not required.

**PHYS 295a:**

**Research Methods in Astrophysics.**The acquisition and analysis of astrophysical data, including the design and use of ground- and space-based telescopes, computational manipulation of digitized images and spectra, and confrontation of data with theoretical models. Examples taken from current research at Yale and elsewhere. Use of the Python programming language. Includes an optional field trip during October recess to the Arecibo 300-meter radio telescope. No previous programming experience required.

**PLSC 452a:**

**Introduction to Statistics: Political Science.**Statistical analysis of politics, elections, and political psychology. Problems presented with reference to a wide array of examples: public opinion, campaign finance, racially motivated crime, and public policy.

**PLSC 453a: Introduction to Statistics: Social Sciences.**Descriptive and inferential statistics applied to analysis of data from the social sciences. Introduction of concepts and skills for understanding and conducting quantitative research.

**PSYC 200b: Statistics.**Measures of central tendency, variability, association, and the application of probability concepts in determining the significance of research findings.

**SOCY 162a: Methods in Quantitative Sociology.**Introduction to methods for reading and conducting quantitative sociological research. Data description and graphical approaches to data analysis; elementary probability theory; assumptions and properties of bivariate and multivariate linear regression; regression diagnostics.

**STAT 100b: Introductory Statistics.**An introduction to statistical reasoning. Topics include numerical and graphical summaries of data, data acquisition and experimental design, probability, hypothesis testing, confidence intervals, correlation and regression. Application of statistical concepts to data; analysis of real-world problems.

**STAT 101a: Introduction to Statistics: Life Sciences.**Statistical and probabilistic analysis of biological problems presented with a unified foundation in basic statistical theory. Problems are drawn from genetics, ecology, epidemiology, and bioinformatics.

**STAT 102a: Introduction to Statistics: Political Science.**Statistical analysis of politics, elections, and political psychology. Problems presented with reference to a wide array of examples: public opinion, campaign finance, racially motivated crime, and public policy.

**STAT 103a: Introduction to Statistics: Social Sciences.**Descriptive and inferential statistics applied to analysis of data from the social sciences. Introduction of concepts and skills for understanding and conducting quantitative research.

**STAT 105a: Introduction to Statistics: Medicine.**Statistical methods relied upon in medicine and medical research. Practice in reading medical literature competently and critically, as well as practical experience performing statistical analysis of medical data.

**STAT 230b: Introductory Data Analysis.**Survey of statistical methods: plots, transformations, regression, analysis of variance, clustering, principal components, contingency tables, and time series analysis. The R computing language and Web data sources are used. EP&E: Intro Statistics