Integers

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"God made the integers, all else is the work of man."—Leopold Kronecker

The number set of integers $\scriptstyle \mathbb {Z} \,$ (from german Zahlen, plural of Zahl, meaning number) is the not well-ordered (having no smallest term) set, which given in increasing order gives the doubly infinite sequence (not in the OEIS, which only contains sequences with an initial term)

{..., –20, –19, –18, –17, –16, –15, –14, –13, –12, –11, –10, –9, –8, –7, –6, –5, –4, –3, –2, –1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, ...}

A permutation of the integers, using the formula

$a(n)=(-1)^{n}\left\lfloor {\frac {n+1}{2}}\right\rfloor ,\quad n\geq 0,\,$

gives the following ordering of integers (A130472)

{0, –1, 1, –2, 2, –3, 3, –4, 4, –5, 5, –6, 6, –7, 7, –8, 8, –9, 9, –10, 10, –11, 11, –12, 12, –13, 13, –14, 14, –15, 15, –16, 16, –17, 17, –18, 18, –19, 19, –20, 20, –21, 21, –22, 22, ...}

The set of integers $\scriptstyle \mathbb {Z} \,$ together with the operations of addition and multiplication constitutes a ring.

Zero, units, primes and composites

Zero is divisible by all (infinite number of) nonzero integers (thus 0 is neither prime nor composite,) and it is also not the product of nonzero integers. Zero is also non-invertible (thus 0 is not a unit.)

A unit (i.e. invertible integer) is neither prime nor composite since it is divisible by no nonunit whatsoever, thus the units −1 and 1 of $\scriptstyle \mathbb {Z} \,$ are neither prime nor composite.

The integers are either:

• Negative composite numbers: {−4, −6, −8, −9, −10, −12, −14, −15, −16, −18, −20, −21, −22, −24, −25, −26, −27, −28, ...} (−1 × A002808)
• Negative prime numbers: {−2, −3, −5, −7, −11, −13, −17, −19, −23, −29, −31, −37, −41, −43, −47, -53, -59, ...} (−1 × A000040)
• Negative unit: {−1}
• Zero: {0}
• Positive unit: {1}
• Positive primes numbers: {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, ...} (A000040)
• Positive composite numbers: {4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, ...} (A002808)

Even and odd integers

A005843 The even [nonnegative integer] numbers: $\scriptstyle a(n)\,=\,2n\,$.

{0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, ...}

A005408 The odd [nonnegative integer] numbers: $\scriptstyle a(n)\,=\,2n+1\,$.

{1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, ...}

See also

• Orderings of integers