# B.A./B.S. in Mathematics

The information and requirements given here apply to the 2017–2018 catalog. For other catalog years, please consult the archive.

Upon declaring a major in mathematics, students must select a degree in which the major will appear. We offer your choice of a Bachelor of Arts (B.A.) or Bachelor of Science (B.S.). These degrees have many common requirements, including minimum units, English composition, and many of the general education courses. There are some differences, however.

## Comparing the B.A. and the B.S.

As far as the mathematics coursework is concerned, the B.A. and B.S. are identical: they each require the core courses, a supporting computer science course, and one of seven math major emphases. The requirements for the two degrees differ only in the following ways:

- Second language requirement (details below).
- Science/application course requirements (details below).

## The B.A. in Mathematics

**Language Requirement:**Students must demonstrate*fourth semester proficiency*in a second language.- The degree is not science-intensive. Students need to complete only the two Tier One NATS (170) and one Tier Two Natural Science courses, as specified in the University General Education requirements.

## The B.S. in Mathematics

**Language Requirement:**Students must demonstrate*second semester proficiency*in a second language.**Laboratory Science Courses Requirement:**The degree is science-intensive and requires one of the following sequences of laboratory science courses:- PHYS 141& either PHYS 142 or PHYS 241
- PHYS 161H& either PHYS 162H or PHYS 261H
- CHEM 151& CHEM 152
- CHEM 105A& CHEM 106A& CHEM 105B& CHEM 106B
- MCB 181R& MCB 181L& ECOL 182R& ECOL 182L
- PSIO 201& PSIO 202
- GEOS 251& GEOS 302
- GEOS 251& GEOS 304

**Application Courses Requirement:**(This requirement does not apply to the Mathematics Education emphasis.) Students must complete at least six units of non-math coursework with a prerequisite or corequisite of at least Calculus I (MATH 122B or higher). Choose from the following:

Agric & Biosystems Eng | ABE 201 , ABE 284, ABE 428, ABE 481A, ABE 481B; |

Astronomy | ASTR 250, ASTR 302; |

Atmospheric Sciences | ATMO 421C (no longer offered), ATMO 436A, ATMO 469A, ATMO 469B; |

Biochemistry | BIOC 462A, BIOC 462B, BIOC 463A, BIOC 466; |

Biomedical Engineering | BME 481B; |

Civil Engineering | CE 214; |

Chemical & Environmental Engr | CHEE 201, CHEE 201L, CHEE 202, CHEE 301A, CHEE 481A, CHEE 481B; |

Chemistry | CHEM 105A, CHEM 105B, CHEM 404A, CHEM 480A, CHEM 480B, CHEM 481; |

Computer Science | CSC 345, CSC 422, CSC 433, CSC 436, CSC 437 (not offered recently), CSC 445, CSC 453, CSC 460, CSC 477; |

Computer Science UA South | CSCV 345; |

Electrical & Computer Engr | ECE 201R, ECE 381A, ECE 404, ECE 429; |

Ecology & Evolutionary Biology | ECOL 302, ECOL 447, ECOL 496N (not offered recently); |

Economics | ECON 332,ECON 361; |

Engineering | ENGR 211C, ENGR 211P, ENGR 265; |

Environmental Science | ENVS 420, ENVS 470; |

Geography & Development | GEOG 463; |

Geosciences | GEOS 322, GEOS 356, GEOS 419, GEOS 432, GEOS 434A, GEOS 440 (not offered recently), GEOS 469, GEOS 479; |

Hydrology and Water Resources | HWRS 421 (not offered recently), HWRS 431, HWRS 460A; |

Information Sci, Tech & Arts | ISTA 352, ISTA 421; |

Molecular & Cellular Biology | MCB 303, MCB 305, MCB 480; |

Materials Science & Engr | MSE 345, MSE 404, MSE 415; |

Optical Sciences | OPTI 201R, OPTI 404; |

Physics | PHYS 140, PHYS 141, PHYS 142, PHYS 143, PHYS 151 (not offered recently), PHYS 152 (not offered recently), PHYS 161H, PHYS 162H, PHYS 240, PHYS 241, PHYS 251 (not offered recently), PHYS 261H, PHYS 440; |

Physiology | PSIO 303A, PSIO 303B, PSIO 304, PSIO 472; |

Planetary Sciences | PTYS 407, PTYS 411; |

Systems & Industrial Engr | SIE 250, SIE 265; |

Watershed Management | WSM 402, WSM 460A; |

or courses approved by your academic advisor. |

**Note:** B.S. students may satisfy both the Laboratory Science Courses Requirement and the Application Courses Requirement by completing one of the Physics sequences.

## Double Degrees

Students pursuing a second major that has a degree title different from that of the first major are technically double degree students. Additional requirements may apply to double degree students that do not pertain to double major students (students with two majors that have exactly the same degree title). See an advisor for details.

## Math Major Emphases

Both the Bachelor of Arts (B.A.) and the Bachelor of Science (B.S.) in Mathematics require a core of basic courses followed by additional courses specific to one of seven possible emphases:

**Comprehensive emphasis:**This emphasis prepares students for graduate study in mathematics, applied mathematics, or most other scientific fields.**Computer Science emphasis:**This emphasis is for students interested in applications of computers to mathematical problems, including math majors who plan to attend graduate school in computer science.**Economics or Business emphasis:**This emphasis is for students with a particular interest in business applications of mathematics, especially those preparing for graduate school in economics or finance.**Applied Mathematics emphasis:**This emphasis is for students who intend to enter the job market upon graduation, but may also be appropriate for students who plan to go on to graduate school in a field of science or engineering.**Probability and Statistics emphasis:**This emphasis is for students considering a career as an actuary or statistician, as well as for students wanting to attend graduate school in statistics.**Life Sciences emphasis:**This emphasis is for students considering a career in medicine, as well as for students wanting to attend graduate school in the biological sciences.**Mathematics Education emphasis:**This emphasis prepares students for teaching mathematics at the secondary school level. The emphasis has two main components: a set of courses in mathematics, and a set of courses in teaching and learning mathematics in secondary schools.

Each emphasis requires at least 15 units of 400-level mathematics course work, and each emphasis, except Mathematics Education, requires the student to complete a minor. Courses in minors must be distinct from those in the major.

The complete official requirements for each emphasis are given in the *University Catalog* in the form of an *Academic Advisement Report (ADVIP)*. Below we cover the portion of the requirements specific to the mathematics major. It is important for students to consult with their academic advisor about their choice and order of courses, as well as which additional courses would strengthen their degree program.

The information and requirements given here apply to the 2017–2018 catalog. For other catalog years, please consult the archive.

## Supporting Computer Science Requirement

All math majors are required to complete a computer programming course, regardless of the degree and emphasis selected. Choose one course:^{(2)}

- CSC 110— Introduction to Computer Programming I
- ISTA 130— Computational Thinking and Doing

## Core Courses

All seven emphases for the B.A./B.S. require the following core courses, which should ideally be completed by the end of the sophomore year.

- MATH 122A AND MATH 122B
^{(2)}or MATH 125— Calculus I - MATH 129— Calculus II
- MATH 223— Vector Calculus
- MATH 313— Introduction to Linear Algebra
^{(3)} - MATH 323— Formal Mathematical Reasoning and Writing
^{(4)} - MATH 355— Analysis of Ordinary Differential Equations
^{(5)}

^{(1)}*Either CSC 110 or ISTA 130 is recommended for most students. Other courses that can be used to satisfy the programming requirement are: CSC 127A, CSC 227, ECE 175,MIS 301, MSE 350, and PHYS 305. These latter courses may require prior programming experience, additional prerequisites and/or a major or minor declared in another subject.*

^{(2)}*MATH 122A and MATH 122B are a single-semester sequence of courses that cover Calculus I. They are equivalent to the old MATH 124 course, which is no longer offered.*

^{(3) }*MATH 313 replaces MATH 215, which is no longer offered. Students who completed MATH 215 prior to fall 2015 or who have transfer credit equivalent to MATH 215 will still fulfill this requirement, though they will not earn upper-division credit for the course.*

^{(4)}*MATH 323 is a writing-emphasis course that is the foundation for many of the advanced courses taken by math majors. It is prerequisite for more theoretical 400-level courses in the major. Students who do not do well in MATH 323 should speak to an advisor about their choice of major/emphasis. MATH 396L, the Wildcat Proofs Workshop, is a 1 unit supplemental instruction course intended to give students additional practice and guidance in learning proof writing techniques. The course is optional, but recommended!*

^{(5)}*MATH 355 is the differential equations course that all mathematics majors are expected to take. This course emphasizes the ideas of dynamical systems and makes use of a more sophisticated approach to differential equations. MATH 254 is a differential equations course that is aimed at engineering and science majors. Students with an additional major in engineering or science may ask their math faculty advisors for permission to substitute MATH 254 course for MATH 355 if a scheduling issue arises.*

## Comprehensive emphasis

The information and requirements given here apply to the 2017–2018 catalog. For other catalog years, please consult the archive.

The Academic Advising Report (ADVIP) gives complete degree requirements:

This emphasis covers the minimum requirements for admission to most graduate programs in mathematics or applied mathematics. We have separate information on selecting the appropriate emphasis and courses to prepare for various types of graduate programs. Students selecting the comprehensive emphasis should consult with a Mathematics Department faculty advisor in choosing additional course work to ensure that they are prepared for the graduate school of their choice. A minor in any subject is required with this emphasis.

Major requirements:

- Core Courses
- MATH 413— Linear Algebra
- MATH 424— Theory of Complex Variables
- MATH 425A— Real Analysis of One Variable
- MATH 425B— Real Analysis of Several Variables
- One of the following two pairs:
- MATH 415A— Introduction to Abstract Algebra
- MATH 415B— Second Course in Abstract Algebra

- MATH 454— Ordinary Differential Equations and Stability Theory
- MATH 456— Applied Partial Differential Equations

Students completing this emphasis are expected to be proficient in proof writing before beginning most of the 400-level courses above. It is recommended that students who do not earn an A or B in MATH 323 speak with an advisor about their selected math major emphasis before enrolling in proof-intensive 400-level courses.

For an example of how to order your coursework to heed prerequisites and meet all degree requirements, please consult the sample 4-year plans. You will develop an individualized plan in consultation with your faculty advisor. Students planning to enroll in three MATH courses in a single semester are advised to talk to their math faculty advisor before finalizing their schedule. The Math Department requires that students enrolling in four or more MATH courses for a single term obtain permission from their math faculty advisor. Note that special courses like Supplemental Instruction, Workshops, Teaching Assistantship enrollment, and Pedagogy courses for the Secondary Math Education Program are not counted toward this limit.

## Computer Science emphasis

The Academic Advising Report (ADVIP) gives complete degree requirements:

A minor in computer science is required with this emphasis.

Major requirements:

- Core Courses
- One of the following three sequences:
- MATH 415A— Introduction to Abstract Algebra
- MATH 415B— Second Course in Abstract Algebra

- MATH 464— Theory of Probability
- MATH 466— Theory of Statistics

- MATH 475A— Mathematical Principles of Numerical Analysis
- MATH 475B— Mathematical Principles of Numerical Analysis

- Two of the following six courses:
- MATH 413— Linear Algebra
- MATH 443— Theory of Graphs and Networks
- MATH 445— Introduction to Cryptography
- MATH 446— Theory of Numbers
- MATH 447— Combinatorial Mathematics
- CSC 473— Automata, Grammars and Languages (*)

- A 5
^{th}400-level MATH course, to be selected with approval from your math faculty advisor. You may choose from the pre-approved list: MATH 401A, MATH 401B, MATH 402, MATH 413, MATH 415A, MATH 443, MATH 445, MATH 446, MATH 447, MATH 464, MATH 468, MATH 475A, MATH 485, CSC 473. If you prefer to select another course, you may do so ONLY if approved by your math faculty advisor.

(*) *Students minoring in Computer Science may not double-dip courses with their math major. Students with a double major in Computer Science and Mathematics may be able to double-dip additional courses between the two majors, within reasonable limits. It is important to consult with both major advisors when planning your coursework to avoid graduation delays.*

For an example of how to order your coursework to heed prerequisites and meet all degree requirements, please consult the sample 4-year plans. You will develop an individualized plan in consultation with your faculty advisor. Students planning to enroll in three MATH courses in a single semester are advised to talk to their math faculty advisor before finalizing their schedule. The Math Department requires that students enrolling in four or more MATH courses for a single term obtain permission from their math faculty advisor. Note that special courses like Supplemental Instruction, Workshops, Teaching Assistantship enrollment, and Pedagogy courses for the Secondary Math Education Program are not counted toward this limit.

## Economics or Business emphasis

The Academic Advising Report (ADVIP) gives complete degree requirements:

This emphasis requires either

- an Economics Minor, or
- a Business Administration Minor, or
- a Finance Minor, or
- a Thematic Minor emphasizing courses from two or more disciplines/subject areas such as Economics, Business, and Finance.

The minor should be chosen in consultation with an advisor.

Major requirements:

- Core Courses
- MATH 425A— Real Analysis of One Variable
- MATH 464— Theory of Probability
- Two of the following four courses:
- MATH 413— Linear Algebra
- MATH 425B— Real Analysis of Several Variables
- MATH 466— Theory of Statistics
- MATH 468— Applied Stochastic Processes

- A 5
^{th}400-level MATH course, to be selected with approval from your math faculty advisor. You may choose from the pre-approved list: MATH 413, MATH 422,MATH 425B, MATH 432, MATH 454, MATH 456, MATH 462 (new), MATH 466, MATH 468, MATH 485. If you prefer to select another course, you may do so ONLY if approved by your math faculty advisor.

Students completing this emphasis are expected to be proficient in proof writing before beginning their 400-level course work. It is recommended that students who do not earn an A or B in MATH 323 speak with an advisor about their selected math major emphasis before enrolling in 400-level courses.

For an example of how to order your coursework to heed prerequisites and meet all degree requirements, please consult the sample 4-year plans. You will develop an individualized plan in consultation with your faculty advisor. Students planning to enroll in three MATH courses in a single semester are advised to talk to their math faculty advisor before finalizing their schedule. The Math Department requires that students enrolling in four or more MATH courses for a single term obtain permission from their math faculty advisor. Note that special courses like Supplemental Instruction, Workshops, Teaching Assistantship enrollment, and Pedagogy courses for the Secondary Math Education Program are not counted toward this limit.

## Applied Mathematics emphasis

The Academic Advising Report (ADVIP) gives complete degree requirements:

A minor in any subject is required with this emphasis.

Major requirements:

- Core Courses
- MATH 422— Advanced Applied Analysis
- One of the following three sequences:
- MATH 454— Ordinary Differential Equations and Stability Theory
- MATH 456— Applied Partial Differential Equations

- MATH 464— Theory of Probability
- MATH 466— Theory of Statistics

- MATH 475A— Mathematical Principles of Numerical Analysis
- MATH 475B— Mathematical Principles of Numerical Analysis

- MATH 485— Mathematical Modeling (*)
- A 5
^{th}400-level MATH course, to be selected with approval from your math faculty advisor. You may choose from the pre-approved list: MATH 413, MATH 424, MATH 454, MATH 456, MATH 464, MATH 468, MATH 475A, MATH 488. If you prefer to select another course, you may do so ONLY if approved by your math faculty advisor.

(*) *Students should plan to take MATH 485 in their final spring semester (it is only offered in spring semesters). In addition to more typical prerequisites, MATH 485 requires completion of at least one 400-level MATH course (422, 454, 456, 464, or 475A). Prerequisites also include linear algebra (313), differential equations (254 or 355), and a programming course.*

## Probability and Statistics emphasis

The Academic Advising Report (ADVIP) gives complete degree requirements:

A minor in any subject is required with this emphasis. Studies in statistics can equip students for careers working with data in a wide variety of fields, from biology to economics. By discussing with an advisor the choice of the minor, the student can greatly enhance his/her career opportunities. Students who are interested in becoming actuaries will find additional information and suggestions on our Careers in Actuarial Science page.

Major requirements:

- Core Courses
- MATH 425A— Real Analysis of One Variable
- MATH 464— Theory of Probability
- MATH 466— Theory of Statistics
- One of the following two courses:
- MATH 468— Applied Stochastic Processes
- MATH 413— Linear Algebra

- A 5
^{th}400-level MATH course, to be selected with approval from your math faculty advisor. You may choose from the pre-approved list: MATH 413, MATH 422,MATH 425B, MATH 454, MATH 456, MATH 462 (new), MATH 468, MATH 485. If you prefer to select another course, you may do so ONLY if approved by your math faculty advisor.

Students completing this emphasis are expected to be proficient in proof writing before beginning their 400-level course work. It is recommended that students who do not earn an A or B in MATH 323 speak with an advisor about their selected math major emphasis before enrolling in 400-level courses.

## Life Sciences emphasis

The Academic Advising Report (ADVIP) gives complete degree requirements:

A minor in the biological sciences is required for this emphasis. Possibilities include Astrobiology, Biochemistry, Biology, Biosystems Engineering, Ecology & Evolutionary Biology, Environmental Sciences, Molecular & Cellular Biology, Physiology, Physiological Sciences, and Pre-Health Professions, or a thematic minor with a life sciences emphasis. A second major in a life science area may also be used to fulfill the minor requirement. Courses which may be of interest for your minor include: ECOL 320, ECOL 340, ECOL 380, ECOL 426, ECOL 496J; MCB 315, MCB 410, MCB 411, MCB 422.

Major requirements:

- Core Courses
- MATH 422— Advanced Applied Analysis
- One of the following two courses:
- MATH 454— Ordinary Differential Equations and Stability Theory
- MATH 456— Applied Partial Differential Equations

- MATH 464— Theory of Probability
- One of the following two courses:
- MATH 466— Theory of Statistics
- MATH 468— Applied Stochastic Processes

- MATH 485— Mathematical Modeling (*)

(*) *Students should plan to take MATH 485 in their final spring semester (it is only offered in spring semesters). In addition to more typical prerequisites, MATH 485 requires completion of at least one 400-level MATH course (422, 454, 456, 464, or 475A). Prerequisites also include linear algebra (313), differential equations (254 or 355), and a programming course.*

## Mathematics Education emphasis

The Academic Advising Report (ADVIP) gives complete degree requirements:

This emphasis is for students preparing to teach mathematics at the secondary school level. The Education emphasis has two main components, a set of courses in mathematics, and a set of pedagogy courses. This emphasis does not require a minor. See the Secondary Math Education Program (SMEP) website for additional information.

The Noyce Interns and Noyce Scholars programs offer experience to students interested in learning more about the teaching profession; see the SMEP website for details and to apply.

Major requirements:

- Core Courses
- Courses in Mathematics:
- MATH 315— Introduction to Number Theory and Modern Algebra
- MATH 330— Topics in Geometry
- MATH 361— Elements of Statistics Using Calculus
- MATH 404— History of Mathematics
- MATH 407— Synthesis of Mathematical Concepts

- Courses in Teaching and Learning Mathematics (Pedagogy):
- MATH 205— Teaching Secondary Mathematics
- EDP 301— Educational Psychology and Child and Adolescent Development
- SERP 400— Survey of Exceptional Students
- TLS 416— An Introduction to Structured English Immersion
- TLS 435— Content Area Literacy in a Multicultural School
- MATH 406A— Curriculum and Assessment in Secondary School Mathematics
- MATH 406B— Methods of Teaching Mathematics in Secondary Schools
- MATH 494C— Student Teaching

- Constitution Requirement for Arizona State certification.
- This requirement is fulfilled by completing one of the following: POL 210 or equivalent; a passing score on the Arizona and US Constitution exams.

- GPA requirements:
- GPA ≥ 2.5 in MATH 122A&B/125, 129, 223
- Pedagogy GPA ≥ 2.5
- Major GPA ≥ 2.0
- Cumulative UA GPA ≥ 2.0