nLab even number



An even number is that which is divisible into two equal parts.

An odd number is that which is not divisible into two equal parts, or that which differs by a monad [=unit] from an even number.

[Euclid, Def. 6 & 7 of Elements Book VII (~ 400-300 BC), see here]


An even number is an integer nn \in \mathbb{Z} that is a multiple of 2, hence such that n=2kn = 2 k for kk \in \mathbb{Z}.

An odd number is an integer that is not an even number.



(adjoint modality of even and odd integers)

Regard the integers as a preordered set (,)(\mathbb{Z}, \leq) in the canonical way, and thus as a thin category.

Consider the full subcategory inclusions

(,) even (,) n 2nAAAAA(,) odd (,) n 2n+1 \array{ (\mathbb{Z}, \leq ) & \overset{even}{\hookrightarrow}& (\mathbb{Z},\leq) \\ n &\mapsto & 2 n } \phantom{AAAAA} \array{ (\mathbb{Z}, \leq ) & \overset{odd}{\hookrightarrow}& (\mathbb{Z},\leq) \\ n &\mapsto & 2 n + 1 }

of the even and the odd integers, as well as the functor

(,) /2 (,) n n/2 \array{ (\mathbb{Z}, \leq ) & \overset{\lfloor-/2\rfloor}{\longrightarrow}& (\mathbb{Z},\leq) \\ n &\mapsto& \lfloor n/2 \rfloor }

which sends any nn to the floor n/2\lfloor n/2 \rfloor of n/2n/2, hence to the largest integer which is smaller or equal to the rational number n/2n/2.

These functors form an adjoint triple

even/2odd even \;\dashv\; \lfloor -/2 \rfloor \;\dashv\; odd

and hence induce an adjoint modality

EvenOdd Even \;\dashv\; Odd

on (,)(\mathbb{Z}, \leq) with

  1. Even2/2Even \coloneqq 2 \lfloor -/2 \rfloor sending any integer to its “even floor value”

  2. Odd2/2+1Odd \coloneqq 2 \lfloor -/2 \rfloor + 1 sending any integer to its “odd ceiling value”.


Observe that for all nn \in \mathbb{Z} we have

2n/2ϵ nnη n2n/2+1, 2 \lfloor n/2 \rfloor \overset{ \epsilon_n }{\leq} n \overset{ \eta_n }{\leq} 2 \lfloor n/2 \rfloor + 1 \,,

where the first inequality is an equality precisely if nn is even, while the second is an equality precisely if nn is odd. Hence this provides candidate unit η\eta and counit.

Hence by this characterization of adjoint functors

  1. the adjunction /2odd\lfloor -/2 \rfloor \dashv odd is equivalent to the condition that

    for every n2k+1n \leq 2 k + 1 we have 2n/2+12k+12 \lfloor n/2 \rfloor + 1 \leq 2 k + 1;

  2. the adjunction even/2even \dashv \lfloor -/2 \rfloor is equivalent to the condition that

    for every 2kn2k \leq n we have 2k2n/22k \leq 2 \lfloor n/2 \rfloor ,

which is readily seen to be the case.

Last revised on August 19, 2023 at 11:38:19. See the history of this page for a list of all contributions to it.