The Mathematical Limits of Representation

By Muriel Bristol | June 1, 2018

Many have spoken, over eons, of the practical, logical, and philosophical limits of political representation. Here we will consider only some of its mathematical limits.

The U.S. Constitution provided that

The number of Representatives shall not exceed one for every thirty Thousand, but each State shall have at Least one Representative; and until such enumeration shall be made, the State of New Hampshire shall be entitled to chuse [Sic] three, Massachusetts eight, Rhode-Island and Providence Plantations one, Connecticut five, New-York six, New Jersey four, Pennsylvania eight, Delaware one, Maryland six, Virginia ten, North Carolina five, South Carolina five, and Georgia three.

That made for a total of 65 House Representatives originally. This was only an estimate with which to start. The number of Representatives expanded to 105 after the first census provided actual population data in 1790. That number of Representatives continued to grow as the population increased to maintain the desired ratio of 1 Representative for roughly 30,000 people. It grew to 142 Representatives after the 1800 census, 182 after 1810, 213 after 1820, and 240 Representatives after the 1830 census, which recorded a population of 12,855,020 people. Representation began to lose ground after that.

There were only 223 Representatives after the 1840 census, 234 after 1850, 241 after 1860, 292 after 1870, 325 after 1880, 356 after 1890, and 386 Representatives after the 1900 census. This process continued until Congress passed the Apportionment Act of 1911, which capped the number of increasingly less representative Representatives at 435 after the 1910 census.

Each U.S. House member represented about 212,000 people in 1920, 280,675 in 1930, 301,164 in 1940, 334,587 in 1950, 410,481 in 1960, 469,088 in 1970, 510,818 in 1980, 571,477 in 1990, 646,946 in 2000, and 709,760 people in 2010.

The U.S. Census Bureau projects a population of 314,500,000 people by 2020, which would be about 723,000 people per Representative, or 1/24th of the representation originally intended. (It would take a House of 10,434 Representatives to provide the original degree of representation).

A mathematical limit is the value that an equation, function, or sequence “approaches” as its input or index approaches some value. The function or f(x) of House representation can be represented as f(x) = 435/x, where x is the size of the population. When x = 435, the function f(x) = 1, i.e., everyone represents themselves, and when x = 13,050,000 or less, the level of representation would be about as the framers intended – 30,000 people per Representative. However, as x grows larger, the degree of representation falls increasingly below their intent.

When the U.S. House is capped at 435 (or any other number), the degree of representation must shrink thereafter as population grows. For our House representation function f(x) = 435/x, when x grows larger and larger and finally approaches infinity, the function f(x) approaches its limit of 0. That is to say, the degree to which anyone is “represented” must shrink increasingly until it ceases finally to have any meaning at all.

The NH House was capped at 400 members in 1942. The same mathematics of representation applies to that institution as well.


Baker, Peter (NYT). (2009, September 17). Expand the House? Retrieved from

Bartlett, Bruce (NYT). (2014, January 7). Enlarging the House of Representatives. Retrieved from

Colby, Sandra L. and Ortman, Jennifer M. (2015, March). Projections of the Size and Composition of the U.S. Population: 2014 to 2060. Retrieved from

Election Data Services. (2017, December 26). Some Change in Apportionment Allocations With New 2017 Census Estimates; But Greater Change Likely by 2020. Retrieved from

NH House of Representatives. (2006). NH House Facts. Retrieved from

US House of Representatives. (2018, May 8). Proportional Representation. Retrieved from

Wikipedia. (2018, May 2). Limit (Mathematics). Retrieved from

Wikipedia. (2018, May 27). Limit of a Function. Retrieved from


Author: Muriel Bristol

"Lady drinking tea"

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