# How to Solve Congress's Fraction Problem

Historians/History

George Szpiro holds an MA in mathematics and physics from the Swiss Federal Institute of Technology, an MBA from Stanford University, and a PhD in finance and mathematical economics from the Hebrew University of Jerusalem. His latest book is "Numbers Rule: The Vexing Mathematics of Democracy, from Plato to the Present" (Princeton, 2010).

Each decade, on April 1 of the year that ends in a zero, the United States takes a census of its population.  The results affect the allocation of more than \$400 billion in federal funding per yearto local, state and tribal governments.  But a more important reason for the census is its political dimension.  It determines how many congresspersons each state sends to the House of Representatives.

The U.S. Constitution states that "the number of representatives shall not exceed one for every thirty thousand, but each state shall have at least one representative."  In the early days, the number of representatives was increased as the population grew.  With a population of over 300 million, and the so-called divisor set at 30,000, the House of Representatives would now number about 10,000 members.  But in 1929 the size of the House was fixed at 435.  Accordingly the divisor has been adjusted and now stands at one representative for about 700,000 citizens.

The apportionment of seats has, however, presented a problem that has plagued the United States since its inception.  Invariably, the number of representatives that a state would be due according to the census, is a fractional number.  So, should a state that is due, say, 7.38 representatives, send 7 or 8 congresspersons to the Capitol?  Rounding the numbers up or down, according to whether the fractions are above or below 0.5, won't do since, depending on how many states get rounded up or down, the total could be 434, 436 or some other number.  In 1792, Alexander Hamilton, the first secretary of the treasury, suggested a two-stage process.  First, each state would be given the integer part of its initial allocation.  Next, the states would be ranked according to the leftover fractions, and those with the largest fractions would receive an additional representative until all available seats of the House have been allocated.  But Thomas Jefferson, the astute secretary of state, pointed to a troubling aspect of this allocation method.  He noticed that his native Virginia (also, of course, President Washington's home soil), would lose a seat under Hamilton's allocation scheme.  At Jefferson's instigation, the president cast his veto against the method.  Of course, the reason given was a better one:  Hamilton's method would have awarded the rounder-upper states more than one representative per 30,000 citizens and this would have been unconstitutional.  As a way out of the dilemma, Jefferson suggested finding a divisor so that the seat allocations, when rounded down, would give exactly the required size of the House.  Of course, this method had its own problem:  it disenfranchised the citizens who were rounded down.  Their votes were, in effect, lost.  But by now nobody could be bothered, and Jefferson's "divisor method" remained in force until 1830.

After a while, however, small states began to feel cheated, one reason being—there are others—that being rounded down from 3.5 hurts much more than being rounded down from 30.5 to 30.  The former president and elder statesman John Quincy Adams weighed in, suggesting—surprise, surprise—Jefferson's method with a twist:  round up instead of down.  His rationale was that this would correspond more closely to the spirit of the Constitution.  By rounding the fractional parts of a seat up to a full seat, every citizen would have his say, and then some.  Of course, this did not sit well with the large states (rounding up 3.5 to 4 gives a bigger boost than does rounding up from 30.5 to 31).  At this point, Secretary of State Daniel Webster had a good idea.  Rather than round up or down, why not round to the nearest number?  The problem would then be reduced to seeking an appropriate divisor such that the result, when rounded up or down, would just give the desired number of seats.  The "method of major fractions," as it was henceforth called, would not avoid inequities but at least it was unbiased.  Sometimes it would favor large states, at other times small states.  In 1842 Congress adopted Webster's method.  But it may have been too reasonable, because at the very next census squabbles started anew.  Senator Samuel Vinton re-discovered Hamilton's round-down method and re-packaged it as the Vinton method.  It remained in force for three decades.  In addition, the size of the House was increased after every census by as many seats as were needed to make as many people as possible happy.  The fact that the inflation in the size of the House in effect diluted each congressperson's voting power did not seem to bother anybody.

Then something extraordinary happened.  After the results of the census of 1880 became known, everybody expected the House to grow again.  In order to give the congressmen the necessary ammunition for the infighting that would undoubtedly precede the next apportionment of the House, the chief clerk of the Census Office did some computations.  Using the census results of 1880, he worked out the apportionments according to Vinton’s method for all House sizes between 275 and 350.  Starting with 275 representatives everything worked out just fine, all the way up to 299.  Whenever he added an additional seat it was picked up by some lucky state.  But when he reached 300 seats, a bombshell fell in his lap.  The delegation from the state of Alabama decreased by one representative, from 8 to 7.  In its stead, two states, Illinois and Texas, each got an additional seat.  Seaton was dumbfounded.  The congressmen were flabbergasted.  How could such a thing happen?  The phenomenon became known as the Alabama Paradox.

The reason for this paradox becomes apparent when we delve a little deeper into the numbers.  When the total number of seats increases from 299 to 300, the states’ “raw” numbers of seats grow on average by about one-third of 1 percent.  But Texas and Illinois start out with larger populations and therefore gain more in absolute numbers.  Thus the number of “raw” seats grows by only 0.025 in Alabama (from 7.646 to 7.671), by 0.032 in Texas (from 9.640 to 9.672), and by 0.061 in Illinois (from 18.640 to 18.701).  As a consequence, the larger states creep past Alabama.

299 seats are to be allocated.  Total population is 49,713,370 and the appropriate divisor is 165,120.

 Alabama Texas Illinois Population 1,262,505 1,591,749 3,077,871 “Raw” allocation 7.646 9.640 18.640 Seats in first round 7 9 18 Fractional part 0.646 0.640 0.640 Additional seats 1 0 0 Total seats 8 9 18

Now 300 seats are to be allocated.  The appropriate divisor is 164,580.

 Alabama Texas Illinois Population 1,262,505 1,591,749 3,077,871 “Raw” allocation 7.671 9.672 18.701 Seats in first round 7 9 18 Fractional part 0.671 0.672 0.701 Additional seats 0 1 1 Total seats 7 10 19

Alabama loses one seat; Texas and Illinois each gain a seat.

Once again, Congress got around that problem by increasing the size of the House to 325 seats.  This made everybody happy.  In 1890, for the same reason, the number of seats was increased to 356 seats.  But ten years later, in 1900, no such luck.  When tables on the apportionment were prepared in 1901 for sizes of the House between 350 and 400, Maine’s apportionment oscillated between 3 and 4 seats and Colorado would receive 3 seats for every size of the House, except at 357 seats where it would be allocated just 2.  Of course, the chairman of the Select Committee on the Twelfth Census, no friend of Colorado’s and Maine’s, suggested fixing the size of the House precisely at 357.  Tempers rose and the atmosphere again became ugly.  The suggestion was raised to keep the size of the House constant once and for all so that the Alabama Paradox, as it was henceforth known as, would never again rear its ugly head.

But more bad news was on the way.  The congressmen and the pencil pushers had failed to notice that problems could arise even with a House of constant size.  Let me illustrate by example.  In 1900 the populations of Virginia and Maine stood at 1,854,184 and 694,466 citizens, respectively.  During the following year, Virginia’s population grew by 19,767 citizens (+1.06 percent), while Maine’s increased by 4,648 (+0.7 percent).  If an additional seat were to be allocated to one of the two states, one would think that it would have to go to Virginia.  Far from it.  The surprising fact is that by Hamilton’s method of allocating leftover seats to the states with the largest remainders, it would have been Maine that would have received an additional seat, while Virginia would have lost one.  Let us look at the numbers:

1900                                   1901
Population      Seats                  Population           Seats
raw    rounded                               raw  rounded
Virginia    1,854,184   9.599*    10       1,873,951     9.509    9
Maine          694,466   3.595      3         699,114     3.548*   4

Total       74,562,608            386      76,069,522            386

*rounded up

Total population grew from 74,562,608 to 76,069,522 and the appropriate divisors were 193,167 in 1900 and 197,071 in 1901. Virginia’s population grew by 19,767, while Maine’s grew by only 4,648. Nevertheless, if a new House had been appointed in1901, Virginia would have lost a seat to Maine. (The numerical values for 1901 are inferred from the population growth between 1900 and 1910. There was no separate census in 1901.)

The reason for this strange situation, which would henceforth be known as the Population Paradox, is that the remainders traded places.  In 1900, Maine had the smaller remainder behind the decimal point (0.595) and failed to make the mark.  A year later—because the nation as a whole grew faster than either of the two states—it would have been Virginia that would have had the smaller remainder (0.509) and Maine would have picked up the extra seat.

The Population Paradox was even more of a threat to Hamilton’s method of apportionment than the Alabama Paradox.  The latter, which appears when the size of the House increases, has been avoided since Congress decided in 1929 to fix the number of its members at 435.  But population increase cannot be stopped and so this paradox is here to stay.

The above problem and paradoxes, which have kept politicians and courts busy for over two hundred years, stem from the annoying fact that rational numbers (such as when a state's population is divided by the total population, and the result is multiplied by 435) are nearly never integers (whole numbers).  So let us think outside of the box!  My suggestion:  send fractional congresspersons to the Capitol.

How can that be done?  If the raw census data shows that a state has, say, 3.7356 percent of the U.S.'s population, and therefore is due 16.25 congresspersons (3.7356% of 435), that state should send 17 people to Washington DC.  Sixteen representatives will have a full vote whenever a vote is taken, one will count for 0.25 of one vote.  Since about half of the states are already rounded up under the current system the new system would result, on average, in twenty-five additional representatives.

Who will the fractional representative be?  When the process of redrawing congressional districts takes place following the decennial U.S. Census, sixteen districts will have populations as equal to each other as possible, while one district, the seventeenth, will only be 0.25 as large as the others.  In Congress the representative of this seventeenth district, representing 0.25 as many citizens as his colleagues, will have a quarter of a vote.

In February 1964, in Wesberry v. Sanders, the Supreme Court held that the Constitution requires that "as nearly as is practicable, one person's vote in a congressional election is to be worth as much as another's."  Four months later, in June of the same year, the Supreme Court held in Reynolds v. Simms that the constitutional requirement is that "as nearly as practicable, districts be of equal population."

Well, in June the honorable judges erred.  "One person's vote to be worth as much as another's" is not equivalent to "districts must be of equal population," except if fractional congresspersons are disallowed.  If a congressional district has only one quarter as many citizens as the other districts, and sends a quarter congressperson to Washington, then the constitutional requirement is satisfied even if districts are not of equal population.  Citizens in the fractional district will share their congressperson with only a quarter as many citizens as those in the other districts.  All citizens, regardless of where they reside in a state, will have equal legislative representation.

### Lawrence Brooks Hughes - 7/23/2010

Ideally, I think we would raise the number of members to 2.5x or 3x the present 435, and continue to apportion them among the states as at present. (You can't jump state lines with House districts without fouling the electoral college).

Our government could handle this expansion quite easily, thanks to the Rules Committee stranglehold. And such large numbers have worked in other representative assemblies.

One of the chief benefits would be that each member would know more of his constituents personally, as each did in the 19th century,when contact was much easier. Another benefit would make membership easier to attain, with less competition and less money chasing after every seat. Another would be to encourage more specialists in the House, i.e., deeper instutional knowledge on many complex subjects.

The last enlargement was in 1912 or 1913, following the census of 1910. That's when it went to 435. The U.S. populaton was then 100 million. We need to triple the present membership to have the same advantages as before 1913.

To do this would increase the liberal or democratic-left proportion of the House membership, so you might think the heavy Democratic majority today would jump to it. Wrong, though all the Republican members would oppose it for that reason. The inner city members today enjoy 80% Democratic voters who provide them with super-safe seats for life, and they have no desire to see their advantage changed, even if it would increase Democratic membership in the House overall. (Also, injecting a modicum of competition into those inner city seats would sharply reduce the number of crooked legislators).

### Bill Glowacki - 7/21/2010

What if instead we just let the states send however many representatives they want and each representative would get to cast as many votes in congress as there were people who voted for him at the last election. So for example in a typical district you might have 250000 people vote. Say 150000 vote Democrat and 100000 Republican. Why not send both politicians to congress and let the Dem cast 150000 votes for every bill, amendment, or whatever and let the Repub cast 100000 votes. To pass a bill you'd need ~50000000 votes (since roughly a third of the country votes in most elections) and they'd probably have to use a spreadsheet to figure out how many votes they have in favor of any bill at any time since all the congressmen would have different weights but that's not too much to ask.

Such a system would have numerous benefits. First, the actual shape and size of the districts would be meaningless (read: no need to gerrymander) since nobody would really lose an election, they'd just lose voting power by moving voters around. Second, it would encourage people to vote since the voting power of their district would be directly related to the number of votes cast. Third, it would give a voice (though a small one) to third parties, though we'd probably have to put some minimal number of votes in place for a person to go to Washington purely for practical purposes.

### Jonathan Dresner - 7/18/2010

I like this better, yeah. Nobody's going to want to live in a fractional district: mathematical equality in that fashion will not, I think, stand the Constitutional test of representational equality. Fractional Representatives would have to form a caucus all their own to have any real influence.

Representation using the Marthaler method, ignoring state boundaries, would be particularly useful in cases (St. Louis and Kansas City come to mind, the 4-cities region on the Iowa border, etc.) where significant urban concentrations span state boundaries, as well as drawing together large rural areas.

### Dan E Marthaler - 7/15/2010

I think the better method would be to just allocate districts according to population. That is, make 435 contiguous districts of equal population. If the district overlaps a state boundary, so be it. The representation isn't to be by state but by *citizen*! You can still have your 2 senators per state.

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