Math 105 – Introduction to Mathematical Ideas

Course Coordinator

Christian Roettger

Catalog Description

MATH 104. Introduction to Mathematical Ideas

(3-0) Cr. 3. F.S.SS.

Prereq: Satisfactory performance on placement assessment, 2 years of high school algebra, 1 year of high school geometry.
Introduction to the use of basic mathematics to solve real-world problems in the areas of voting issues, measuring power in situations where people have different numbers of votes, apportionment, fair division, and elementary game theory. No prior background in politics or history is necessary for this course.


Math 105


ISU Custom Edition

ISBN: 9781269121453


  • The mathematics of voting; the various ways to determine a winner in preference ballot voting situations, and why none is totally satisfactory. (Chapter 1)
  • Measurement of power in yes/no voting situations: Banzhaf and Shapley-Shubik power indices. (Chapter 2)
  • The mathematics of fair division. (Chapter 3)
  • Apportionment problems. (Chapter 4)
  • Introduction to game theory. (Chapter 5)


Understanding the basic methods and limitations of preference voting methods

  • To be able to understand what the idea of a “preference ballot” is when voters choose among several alternatives
  • To be able to understand how to tabulate a preference ballot and select a winner according to a variety of different voting methods: Borda Count, plurality, Hare, Coombs, Copeland, etc.
  • To be able to understand the various criteria that have been developed to judge the adequacy of the voting methods discussed above
  • To be able to understand and construct simple arguments or counter-examples illustrating that a various method does or does not violate a given criteria
  • To be able to understand the statement of Arrow’s Theorem, which (loosely speaking) asserts that no “perfect” voting system exists

Understanding the basic methods of yes-no voting and the assessment of power in such a voting system

  • To be able to understand and analyze simple yes-no voting systems where different people may have a different number of votes
  • To be able, in such systems, to determine who (if anybody) is a dictator, has veto power, etc.
  • To understand the Banzhaf and Shapley-Shubik methods for computing power in such a voting system, and to be able to perform simple computations of voting power

Understanding the basic ideas of apportionment and fair division

  • To be able to understand what an apportionment problem is, and to understand the basic terminology concerning such problems
  • To understand the various apportionment methods (Hamilton, Jefferson, Adams, Webster,Huntington-Hill) and perform simple calculations
  • To understand the various methods that have been developed (Lone Divider, Divide andChoose, Method of Markers, Adjusted Winner, etc.) for solving problems in which an object or objects must be divided among several people

Understanding the basic ideas of game theory

  • To understand the meaning of a mathematical “game” and understand several classical examples: Chicken, Battle of the Bismarck Sea, Prisoner’s Dilemma, etc.
  • To understand the difference between a zero-sum and nonzero-sum game
  • To understand what a “saddle point” is in a zero-sum game and to be able to determine whether, for a given game, a saddle point exists
  • To understand the concept of “equilibrium” for a simple zero-sum game
  • To understand the mathematical techniques for determining optimal strategies for simple zero-sum games given by, at least, 2 x 2 matrices

Free Expression Statement

Iowa State University supports and upholds the First Amendment protection of freedom of speech and the principle of academic freedom in order to foster a learning environment where open inquiry and the vigorous debate of a diversity of ideas are encouraged. Students will not be penalized for the content or viewpoints of their speech as long as student expression in a class context is germane to the subject matter of the class and conveyed in an appropriate manner.

Students With Disabilities

Iowa State University is committed to assuring that all educational activities are free from discrimination and harassment based on disability status. Students requesting accommodations for a documented disability are required to work directly with staff in Student Accessibility Services (SAS) to establish eligibility and learn about related processes before accommodations will be identified. After eligibility is established, SAS staff will create and issue a Notification Letter for each course listing approved reasonable accommodations. This document will be made available to the student and instructor either electronically or in hard-copy every semester. Students and instructors are encouraged to review contents of the Notification Letters as early in the semester as possible to identify a specific, timely plan to deliver/receive the indicated accommodations. Reasonable accommodations are not retroactive in nature and are not intended to be an unfair advantage. Additional information or assistance is available online at, by contacting SAS staff by email at, or by calling 515-294-7220. Student Accessibility Services is a unit in the Dean of Students Office located at 1076 Student Services Building.

More information about disability resources in the Mathematics Department can be found at