What does 2/3ds the US Senate mean?

Question

For normal bills to pass the US Senate, all it takes is a simple majority. If the Senate is evenly divided (50/50), the vice president votes as to break the tie with a 51/50 or 50/51.

But for constitutional amendments and convicting an impeached President, the constitution puts the Senates bar at 2/3rds of the senators. The Senate is composed of 100 Senators and 2/3rds of 100 persons is more than 66.666 persons, so does 2/3rds of the Senate mean 67 Senators, or 66 Senators, or is 66 Senators a tie and requires the vice president to break the tie?

Answer 1

67 senators is the majority if all 100 are present. There's no notion of tie in such 2/3-vote circumstances. (Also, for presidential impeachment at least the VP does not participate at all; he is replaced by the Chief Justice).

Wikipedia has a complete list of tie breaking votes (since 1945); none involve a [near] 2/3 majority case. Nor does it mention that ever having been possible. The relevant wording in Article One of the Constitution is:

The Vice President of the United States shall be President of the Senate, but shall have no Vote, unless they be equally divided.

The other relevant bits of the Constitution don't ever mention the VP in the context of such 2/3 votes, but just Senators, E.g. the Treaty Clause speaks of two-thirds of Senators present:

[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...

The amendment procedure is similar: only two-thirds of those Senators present. And even for impeachment, as Wikipedia notes:

If not all Senators were present, a quorum would be needed for a vote. In the Senate a quorum is half plus one, or 51 out of 100. If only the minimum for a quorum were present, two-thirds would mean 34 voting in favor of impeachment.

Answer 2

The key factor in determining the minimum number of votes required is that the comparison operation for a simple majority vote is different from that for a supermajority vote.

A simple majority means "greater than half of the votes." Equivalently, it means "more votes for than against." In mathematical terms, this is the greater than comparison. It may be described using either of these expressions:

votes_in_favor > total_votes / 2
votes_in_favor > votes_opposed

The vice president can break ties even if the vote isn't 50/50; if ten senators are absent, and 45 vote for and 45 vote against a proposal, then the vice president can also vote to make the result 46/45 or 45/46.

Suppose the vice president is absent, and an even number of senators votes. The votes are evenly divided, with the same number being for and against. In that case, the measure does not pass, because there was no majority in favor of its passage.

A two thirds majority means "at least two thirds of the votes cast are in favor." Equivalently, it means "at least twice as many votes for as against." In mathematical terms, this is the greater than or equal to comparison. It could be described with either of these expressions:

votes_in_favor >= total_votes * 2 / 3
votes_in_favor >= 2 * votes_opposed

Therefore, if there were 102 senators, and all of them cast a vote, the narrowest possible vote for a two thirds majority would be 68/34, which is exactly equal to two thirds in favor and one third opposed.

Under senate rules, questions of cloture are decided by a three fifths majority. In common with the two thirds majority, the comparison operation is greater than or equal to, so in these cases, with 100 votes, the narrowest possible vote in favor is 60/40. However, the total against which the number of votes is compared is not the total number of votes, but the total number of senators whether present or not (vacant seats are not counted, however). Therefore, the mathematical expressions analogous to those above would have to use different variables:

votes_in_favor >= total_senators * 3 / 5
votes_in_favor >= 1.5 * (votes_opposed + non_voting_senators)

(The second expression is much less useful here than are the corresponding expressions above.)

As Fizz notes in another answer, there's no possibility of a tie in a supermajority vote. More precisely, a tie vote does not come close to the threshold for deciding the question at hand. Therefore, the vice president never gets to vote in these cases (or, if it were for some reason decided that the vice president could vote when the the votes were evenly divided even when the vote threshold were two thirds or three fifths, such a vote would have no impact on the outcome).