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Mathematics Department - University of Michigan




Content: Math 486 can be described as the rudiments of analysis and algebra underlying theorems used in secondary mathematics. Math486 examines the principles of analysis and algebra underlying theorems concerning fields -- especially the rationals, reals, and complex numbers; and concerning functions -- especially polynomials, exponential functions, and logarithmic functions. Mathematical underpinnings of these ideas can serve as intellectual resources for secondary teachers. Major topics covered by Math486-W11 vary from year to year, but have included:

  • Properties of fields, including the parallels between Q[√p] and C=R[i], for positive integers p such that √p is irrational.
  • Properties of the rational numbers, including density.
  • A rigorous description of the long division algorithm for integers and for polynomials; invariance of the sequence of remainder terms for a particular integer dividend, assuming an integer divisor.
  • The proof that a degree n polynomial must have exactly n complex roots (counting with multiplicity); the proof establishing an equivalence between factors of a polynomial and roots of a polynomial.
  • Limits, convergence, and divergence of sequences of real and complex numbers; accumulation points of sets of real and complex numbers.
  • Limits, convergence, and divergence of sequences of real functions; uniform convergence, pointwise convergence.
  • Properties of functions including injectivity, surjectivity, invertibility, continuity, and periodicity.
  • Definition of the exponential function and logarithmic function on a complex domain.
The students are engaged via constructing collective explanations of key mathematical topics. Mathematical practices emphasized by Math486 include:
  • Assessing the completeness and soundness of explanations and proofs of mathematical ideas.
  • Reading, explaining, and writing conjectures and proofs.
  • Alertness to mathematical language and precision; how small differences in phrasing may have significant mathematical implications (e.g., the phrase "the solution" versus "a solution").
  • Giving mathematical motivations for different given formulations of equivalent mathematical ideas; using different ways of representing the same mathematical idea, and giving explanations of why they are equivalent.