Lectures for Mathematics 112, Fall 2025-26
- M 9/1: Labor Day Holiday
- T 9/2: Overview
- W 9/3: Skim through Ch1
- F 9/5: Start 2.1: Binary operators, identity, inverses, associativity
- M 9/8: Polygon triangulations and product parenthesizations in
connection with 2.30 and 2.31, noncommutative computer "and",
nonassociative computer "*", (Z_n,+,*)
questions about homework
- T 9/9: Continue through 2.3 field axioms, 2.4 some of their consequences
- W 9/10: Finish 2.4, go over homework
- F 9/12: [Add/section-change deadline]
2.5 Subtraction and division
- M 9/15: Finish 2.5, start order
- T 9/16: Go over homework, start 2.6
through inequalities
- W 9/17: Finish inequalities, start absolute value
- F 9/19: Finish absolute value; inductive sets, the natural numbers
of a field
- M 9/22: Induction theorem, addition and multiplication in the
integers of a field, no natural number between 0 and 1
- T 9/23: Least Integer Principle; no square root of 2 in the
rational field
- W 9/24: Recursive definitions, a^{m+n} = a^m a^n for all natural
n,m,
go over homework
- F 9/26: The integers and rational numbers of a field; finish 3.3
- M 9/29: Discuss first exam; numerically unstable sequence; 4.1 the
complexification of a field with no square root of -1
- T 9/30: Skim 4.2 complex conjugate, start 5.1 sequences and binary
search sequences
- W 10/1: Go over homework, start binary search
- F 10/3: More binary search, pth roots
- M 10/6: [No-W drop deadline]
Review the binary searches and their consequences, rational
exponents, begin into chapter 6
- T 10/7: More chapter 6: triangle inequality, geometry of complex
addition, geometry of complex multiplication, roots of unity
- W 10/8: Addition laws for cosine and sine, polar decomposition of
nonzero complex numbers, finding the nth roots of a nonzero complex number
- F 10/10: Complex mappings, work on exercises 6.35, 6.40a as
revised on the assignments page for this course
- M 10/13: Full triangle inequality writeup, first few pages
of chapter 7 and then start complex sequences writeup: complex
sequences form a ring, bounded complex sequences a subring
- T 10/14: The null sequences form an ideal of the bounded sequences,
convergence in terms of nullness, convergent sequences are bounded,
sum rule and product rule
- W 10/15: Review sum rule and product rule, (r^n), (b^{1/n}),
(a_n^\alpha), preview Friday's homework problems from chapter 7
- F 10/17: Finish r^n rule, b^{1/n} rule, 1/n^\alpha rule,
x_n^\alpha rule, reciprocal rule
- Fall break week
- M 10/27: Interesting case of the geometric sequence theorem via
the fact that a convergent sequence is Cauchy; geometric series;
relation between area and circumference of the unit disk
- T 10/28: Limit of nonnegative sequence is nonnegative,
index-translate of a sequence has the same convergence, Fibonacci
quotient example
- W 10/29: Monotonic sequence theorem, point to various formulations
of completeness
- F 10/31: Binomial theorem and {n^(1/n)}, review chapter 7
- M 11/3: Continuity: examples from early chapter 8, continuity of
the rational power function
- T 11/4: Progenitive continuity results, start the limit of a function
- W 11/5: Every f:Z->C continuous? f(p/q)=1 but f(x)=0 for irrational x?
- F 11/7: Function limit as common sequence limit, derivative of the
squaring function as a function limit, three subtle function limits
- M 11/10: [Withdraw/leave deadline]
Review continuity and function limits
- T 11/11: Intermediate Value Theorem; Extreme Value Theorem
- W 11/12: Work on chapter 9 exercises
- F 11/14: Start derivatives, derivative of the rational power function
- M 11/17: Generative derivative rules, including chain rule
- T 11/18: Mean value theorem and its consequences, x/2+x^2 sin(1/x) example
- W 11/19: Start series: harmonic series diverges, nth term test,
comparison test
- F 11/21: p-test, {|z|t^n/n!} summable, absolute convergence test
- M 11/24: Limit-comparison test, ratio test, alternating series test, start
power series
- T 11/25: Power series radius of convergence
- W 11/26: Differentiation theorem
- F 11/28: Thanksgiving Holiday
- M 12/1: Start specific power series examples
- T 12/2: More power series examples
- W 12/3: More power series examples
- F 12/5: Values of zeta
- M 12/8: Values of zeta
- T 12/9: No meeting (college on Thursday schedule)
- W 12/10: Loose ends
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