Negative integers

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The negative integers are real integers that are less than 0. For example, −147 and −4 are negative integers, but −0.4181554... and 10 are not (the former is a negative number but not an integer, the latter is a positive integer). The negative integers are listed in A001478, which can be thought of as the "negative equivalent" of A000027.

The absolute value of a negative integer is the number multiplied by −1. The set of all negative integers may be denoted Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathbb{Z}^-} . And so we may write for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle n \in \mathbb{Z}^-} that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle |n| \in \mathbb{Z}^+} .

The study of prime numbers is generally unconcerned with negative integers. But the question does occasionally arise: how do you write the factorization of a negative integer? A number of different solutions suggest themselves, for example, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle -48 = -(2^4) \times 3 = (-2)^4 \times -3 = 2^4 \times -3 = \ldots} , etc., but these are unsatisfactory on account of the seemingly arbitrary sign choices. To insure uniformity, the factorization of a negative integer could be expressed as −1 followed by the factorization of the absolute value, e.g., Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle -48 = (-1) \times 2^4 \times 3} .

In the OEIS, if a sequence includes negative numbers it gets the keyword "sign" in the Keywords field (this is mutually exclusive with keyword:nonn). Note that in such OEIS sequence entries, the short dash "-" is used rather than "−"; but since a monospace font is used, this difference becomes relevant only when copying and pasting.