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.
At the end of a leg, since one needs to double out to win, the 2-adic absolute value becomes the relevant measure.
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.
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.