The former names are traditional ones, derived from ancient Greek mathematics; the latter have become common in recent decades.
For an integer n, the 2-order of n (also called valuation) is the largest natural number ν such that 2ν divides n. This definition applies to positive and negative numbers n, although some authors restrict it to positive n; and one may define the 2-order of 0 to be infinity (see also parity of zero).
At the beginning of a leg, "smaller" has the usual meaning of absolute value, and the basic strategy is to aim at high-value areas on the dartboard and score as many points as possible.
With any odd score no matter how small in absolute value, it takes at least two darts to win.
A common miss when aiming at the double ring is to hit a single instead and accidentally halve one's score.
Given a score of 22 — a singly even number — one has a game shot for double 11.
By contrast, when shooting for double 12, one may make the same mistake but still have 3 game shots in a row: D12, D6, and D3.
[4][5] The classic proof that the square root of 2 is irrational operates by infinite descent.
Usually, the descent part of the proof is abstracted away by assuming (or proving) the existence of irreducible representations of rational numbers.
, yielding a direct proof of irrationality not relying on the law of excluded middle.
[6] In geometric topology, many properties of manifolds depend only on their dimension mod 4 or mod 8; thus one often studies manifolds of singly even and doubly even dimension (4k+2 and 4k) as classes.
For example, doubly even-dimensional manifolds have a symmetric nondegenerate bilinear form on their middle-dimension cohomology group, which thus has an integer-valued signature.
Conversely, singly even-dimensional manifolds have a skew-symmetric nondegenerate bilinear form on their middle dimension; if one defines a quadratic refinement of this to a quadratic form (as on a framed manifold), one obtains the Arf invariant as a mod 2 invariant.
If a compact oriented smooth spin manifold has dimension n ≡ 4 mod 8, or ν2(n) = 2 exactly, then its signature is an integer multiple of 16.
Lambert's continued fraction for the tangent function gives the following simple continued fraction involving the positive singly even numbers:[10] This expression leads to similar representations of e.[11] In organic chemistry, Hückel's rule, also known as the 4n + 2 rule, predicts that a cyclic π-bond system containing a singly even number of p electrons will be aromatic.