One of the most well-known clauses in the original Constitution was known as the “Three-Fifths Clause.” This name comes from a fraction in the clause, which is no longer in effect. But there are other fractions in the Constitution which are still in effect.
For example, you can find the fractions “one-fifth,” “one-third,” “two-thirds,” and “three-fourths” in various parts of the Constitution that are still in effect. Even the concept of a “majority” is something of a fractional concept – that is, more than one-half.
With 50 states, 100 Senators, and 435 Representatives …
Thus, if you really want to understand the Constitution, you have to know your fractions. Some examples are available below, along with sources to back them up:
United States Capitol
One-fifth required to have votes recorded (usually 20 Senators or 87 Representatives)
“Each House shall keep a journal of its proceedings, and from time to time publish the same, excepting such parts as may in their judgement require secrecy; and the yeas and nays of the members of either House on any question shall, at the desire of one-fifth of those present, be entered on the journal.” (Source: Article 1, Section 5, Paragraph 3) This means that more than four-fifths must vote for secrecy before it can be allowed under the Constitution (which is generally 81 Senators or 349 Representatives, depending on which house we're talking about).
Capitol Dome
One-third of the Senators chosen every second year (which is either 33 Senators or 34 Senators)
“Immediately after they [the Senators] shall be assembled in consequence of the first election, they shall be divided as equally as may be into three classes. The seats of the Senators of the first class shall be vacated at the expiration of the second year, of the second class at the expiration of the fourth year, and of the third class at the expiration of the sixth year, so that one-third may be chosen every second year” (Source: Article 1, Section 3, Paragraph 2)
Great Seal of the United States
Bare majority of each house shall constitute a quorum to do business (51 Senators and 218 Representatives)
The word “majority” means more than one-half (which is a fraction). Thus, it is a fractional concept. With this in mind, the Constitution says that “Each House shall be the judge of the elections, returns and qualifications of its own members, and a majority of each shall constitute a quorum to do business” (Source: Article 1, Section 5, Paragraph 1)
Capitol Dome
Bare majority of the Senate needed to confirm the president's nominees (generally 51 Senators)
The Constitution does not say explicitly how many Senators are needed to confirm a nominee. Thus, the default number required here is a bare majority. In the words of the Constitution, the president “shall nominate, and by and with the advice and consent of the Senate, shall appoint ambassadors, other public ministers and consuls, judges of the Supreme Court, and all other officers of the United States, whose appointments are not herein otherwise provided for, and which shall be established by law: but the Congress may by law vest the appointment of such inferior officers, as they think proper, in the President alone, in the courts of law, or in the heads of departments.” (Source: Article 2, Section 2, Paragraph 2)
Supreme Court of the United States
Two-thirds of both houses needed to override a presidential veto (67 Senators and 290 Representatives)
“Every bill which shall have passed the House of Representatives and the Senate, shall, before it become a law, be presented to the president of the United States; if he approve, he shall sign it, but if not, he shall return it, with his objections, to that house in which it shall have originated, who shall enter the objections at large on their journal, and proceed to reconsider it. If after such reconsideration, two thirds of that house shall agree to pass the bill, it shall be sent, together with the objections, to the other house, by which it shall likewise be reconsidered, and if approved by two-thirds of that house, it shall become a law. But in all such cases the votes of both houses shall be determined by yeas and nays, and the names of the persons voting for and against the bill shall be entered on the journal of each house respectively. If any bill shall not be returned by the president within ten days (Sundays excepted) after it shall have been presented to him, the same shall be a law, in like manner as if he had signed it, unless the Congress by their adjournment prevent its return, in which case it shall not be a law.” (Source: Article 1, Section 7, Paragraph 2)
The White House, where the President of the United States lives
(continued)
“Every order, resolution, or vote to which the concurrence of the Senate and House of Representatives may be necessary (except on a question of adjournment) shall be presented to the president of the United States; and before the same shall take effect, shall be approved by him, or, being disapproved by him, shall be re-passed by two-thirds of the Senate and House of Representatives, according to the rules and limitations prescribed in the case of a bill.” (Source: Article 1, Section 7, Paragraph 3)
Thus, the president can increase the number of votes required to pass a bill by roughly one-sixth of each house (16 Senators and 72 Representatives)
This means that a veto can increase the number of votes required to pass a bill by roughly one-sixth of each house (a calculation that involves finding a least common denominator between one-half and two-thirds). A president can thus typically increase the number of votes needed to pass a bill by 16 Senators and 72 Representatives.
The White House
Two-thirds of the Senate required to ratify treaties (generally 67 Senators)
The Constitution says that the president “shall have power, by and with the advice and consent of the Senate, to make treaties, provided two-thirds of the Senators present concur” (Source: Article 2, Section 2, Paragraph 2)
U. S. State Department, the part of the government which handles foreign diplomacy
Bare majority of the House of Representatives needed to impeach (generally 218 Representatives)
The Constitution does not say explicitly how many Representatives are needed to impeach someone. Thus, the default number required here is a bare majority. In the words of the relevant clause, “The House of Representatives shall choose their Speaker and other officers; and shall have the sole power of impeachment.” (Source: Article 1, Section 2, Paragraph 5)
Andrew Johnson, who was impeached but not removed
Bill Clinton, who was also impeached but was not removed
Two-thirds of the Senate needed to try the impeachments (generally 67 Senators)
“The Senate shall have the sole power to try all impeachments. When sitting for that purpose, they shall be on oath or affirmation. When the President of the United States is tried, the Chief Justice shall preside: and no person shall be convicted without the concurrence of two-thirds of the members present.” (Source: Article 1, Section 3, Paragraph 6)
Richard Nixon, who resigned under the threat of certain impeachment and removal
Two-thirds of either house needed to expel a member of that house (67 Senators or 290 Representatives, depending on which house)
“Each House may determine the rules of its proceedings, punish its members for disorderly behavior, and, with the concurrence of two-thirds, expel a member.” (Source: Article 1, Section 5, Paragraph 2)
Senator William Blount, who was expelled from the Senate via the use of this power
Two-thirds of both houses required to make the Vice President into the Acting President against the president's wishes (67 Senators and 290 Representatives)
“Whenever the Vice President and a majority of either the principal officers of the executive departments or of such other body as Congress may by law provide, transmits to the President Pro tempore of the Senate and the Speaker of the House of Representatives their written declaration that the President is unable to discharge the powers and duties of his office, the Vice President shall immediately assume the powers and duties of the office as Acting President.” (Source: Twenty-Fifth Amendment, Section 4)
George H. W. Bush, who almost became Acting President via this section after an assassination attempt on Ronald Reagan
(continued)
“Thereafter, when the President transmits to the President Pro tempore of the Senate and the Speaker of the House of Representatives his written declaration that no inability exists, he shall resume the powers and duties of his office unless the Vice President and a majority of either the principal officers of the executive departments or of such other body as Congress may by law provide, transmits within four days to the President Pro tempore of the Senate and the Speaker of the House of Representatives their written declaration that the President is unable to discharge the powers and duties of his office. Thereupon Congress shall decide the issue, assembling within forty-eight hours for that purpose if not in session. If the Congress, within twenty-one days after receipt of the latter written declaration, or, if Congress is not in session within twenty-one days after Congress is required to assemble, determines by two-thirds vote of both houses that the President is unable to discharge the powers and duties of his office, the Vice President shall continue to discharge the same as Acting President; otherwise, the President shall resume the powers and duties of his office.” (Source: Twenty-Fifth Amendment, Section 5)
President Ronald Reagan, who resumed his office after the assassination attempt against him
Two-thirds of the states needed for a quorum in the backup plans for presidential elections (34 states)
A majority of the electoral votes is needed to become president in the traditional way. With 538 electoral votes, a majority is thus 270 electoral votes. A later amendment clarifies that “The person having the greatest number of votes for President, shall be the President, if such number be a majority of the whole number of Electors appointed; and if no person have such majority, then from the persons having the highest numbers not exceeding three on the list of those voted for as President, the House of Representatives shall choose immediately, by ballot, the President. But in choosing the President, the votes shall be taken by States, the representation from each State having one vote; a quorum for this purpose shall consist of a member or members from two-thirds of the States, and a majority of all the States shall be necessary to a choice.” (Source: Twelfth Amendment, Section 1)
John Quincy Adams, the only person to become president via the backup plans in this section
Two-thirds of the Senators needed for a quorum in the backup plans for vice presidential elections (67 Senators)
“The person having the greatest number of votes as Vice-President, shall be the Vice-President, if such numbers be a majority of the whole number of electors appointed, and if no person have a majority, then from the two highest numbers on the list, the Senate shall choose the Vice-President; a quorum for the purpose shall consist of two-thirds of the whole number of Senators, and a majority of the whole number shall be necessary to a choice.” (Source: Twelfth Amendment, Section 2)
Richard Mentor Johnson, the only man to become vice president via the backup plans in this section
Two-thirds and three-fourths both mentioned in the constitutional amendment process (67 Senators and 290 Representatives to begin, or 34 state legislatures to begin; and 38 state legislatures or 38 state ratifying conventions to ratify)
“The Congress, whenever two-thirds of both houses shall deem it necessary, shall propose amendments to this constitution, or on the application of the legislatures of two-thirds of the several states, shall call a convention for proposing amendments, which, in either case, shall be valid to all intents and purposes, as part of this constitution, when ratified by the legislatures of three-fourths of the several states, or by conventions in three-fourths thereof, as the one or the other mode of ratification may be proposed by the Congress: Provided … that no state, without its consent, shall be deprived of its equal suffrage in the Senate.” (Source: Article 5)
The United States Bill of Rights, passed in accordance with this process
Representatives shall be “apportioned” among the several states according to population (a fractional concept itself)
Apportionment is itself a fractional concept. With this in mind, the Fourteenth Amendment says that “Representatives shall be apportioned among the several States according to their respective numbers, counting the whole number of persons in each State, excluding Indians not taxed.” (Source: Section 2 of the amendment) This replaced the Three-Fifths Clause that I mentioned earlier.
Census Bureau headquarters in Suitland, Maryland
Conclusion: Fractions in the Constitution are important
Thus, if you want to truly understand the Constitution, then you have to know your fractions.
Footnote to this blog post:
Before the advent of calculators, direct fractions were more convenient to use than decimals or percentages; although I should acknowledge that decimals are indirect ways of doing fractions, and so are percentages (specifically, hundredths).
If you liked this post, you might also like:
Some thoughts on math education
The difference between skepticism and closed-mindedness
How do the economic laws of supply and demand work?
Why is my stats class so focused on bell curves?
A review of the BBC's “The Story of Maths” (by Marcus du Sautoy)
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