Learn the math behind determining the number of congressional seats for each state in the United States from the American Mathematical Society, publisher of Mathematical Moments, a program that promotes appreciation and understanding of the role mathematics plays in science, nature, technology, and human culture.

As difficult as it is to conduct the census, the ensuing process of determining the number of congressional seats for each state can be even harder. The basic premise, that the proportion of each state’s delegation in the United States House of Representatives should match its proportion of the U.S. population, is simple enough. The difficulty arises when deciding what to do with the fractions that inevitably arise (e.g., New York can’t have 28.7 seats). Over the past 200 years, several methods of apportioning seats have been used. Many sound good but can lead to paradoxes, such as an increase in the total number of House seats actually resulting in a reduction of a state’s delegation. The method used since the 1940s, whose leading proponent was a mathematician, is one that avoids such paradoxes.

A natural question is, “Why 435 seats?” Nothing in the U.S. Constitution mandates this number, although there is a prohibition against having more than one seat per 30,000 people. One model, based on the need for legislators to communicate with their constituents and with each other, uses algebra and calculus to show that the ideal assembly size is the cube root of the population it represents. Remarkably, the size of the House mirrored this rule until the early 1900s. To obey the rule now would require an increase to 670, which would presumably both better represent the population and increase the chances that the audience in the seats for those late speeches would outnumber the speaker.

"One model, based on the need for legislators to

communicate with their constituents and with each

other, uses algeba and calculus to show that the ideal

assembly size is the cube root of the population it represents."

For more information: “E pluribus confusion”, Barry Cipra, American Scientist, July-August 2010.

“Assigning Seats” was first published in Mathematical Moments, a program by the American Mathematical Society that promotes appreciation and understanding of the role that mathematics plays in science, nature, technology, and human culture. The Mathematical Moments site features downloadable posters for classroom or office use, as well as a recurring podcast.

MathSciNet is published by the American Mathematical Society and available on the EBSCOhost and EBSCO Discovery Service, which provides the ability to search indexed fields such as author, institution and MSC Primary and Secondary Classifications, and allows mathematics librarians, students and faculty the opportunity to easily link to related full text in the library’s collection.

Mathematical Moments “Assigning Seats” from the American Mathematical Society

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