−2

In mathematics, negative two or minus two is an integer two units from the origin,^[1] denoted as −2^[2] or ⁻2.^[3] It is the additive inverse of 2, following −3 and preceding −1, and is the largest negative even integer. Except in rare cases exploring integral ring prime elements,^[4] negative two is generally not considered a prime number.^[5]

Negative two is sometimes used to denote the square reciprocal in the notation of SI base units, such as m·s⁻².^[6] Additionally, in fields like software design, −1 is often used as an invalid return value for functions,^[7] and similarly, negative two may indicate other invalid conditions beyond negative one.^[8] For example, in the On-Line Encyclopedia of Integer Sequences, negative one denotes non-existence, while negative two indicates an infinite solution.^[9][10]

• Negative two is a complementary Bell number (also known as Rao Uppuluri-Carpenter numbers),^[11][12] and a Hermite number.
• Negative two makes the class number of the quadratic field ${\displaystyle \mathbb {Q} [{\sqrt {d}}]}$ equal to 1, meaning its ring of integers is a unique factorization domain.^{[Note 1][13]} According to the Stark–Heegner theorem, only nine negative numbers have this property,^[14][15][16] corresponding to Heegner numbers.^[17]
  • Negative two also makes the quadratic field ${\displaystyle \mathbb {Q} [{\sqrt {d}}]}$ a simple Euclidean field (or norm-Euclidean field).^[18] Only five negative numbers have this property: −11, −7, −3, −2, −1.^[19][20] If relaxed, −15 is included.^[21][22]
• Negative two is the largest negative number unreachable from 1 in two steps using addition, subtraction, or multiplication.^[23] The largest in one step is −1, and in three steps, −4.^[23] This relates to the straight-line program combined with addition, subtraction, and multiplication,^[24] exploring the algebraic complexity of integers in relation to NP = P.^[25]
• Negative two is the second-order i.e., ${\displaystyle H_{2}=H_{2}(0)=-2}$.^[26][27]
• Negative two makes ${\displaystyle k(k+1)(k+2)}$ a triangular number.^[28] Only nine integers have this property, with negative two being the smallest: −2, −1, 0, 1, 4, 5, 9, 56, and 636.^[29]

The divisors of negative two, including negative divisors, are identical to those of two: −2, −1, 1, 2. By definition, negative numbers are not typically subjected to prime factorization, though ${\displaystyle -1}$ can be factored out as ${\displaystyle -1\times 2}$,^[30] making 2 a prime factor but not the result of prime factorization. As a Gaussian integer, negative two can be factored as ${\displaystyle i\times (1+i)^{2}}$, where ${\displaystyle 1+i}$ is a Gaussian prime and ${\displaystyle i}$ is the imaginary unit.^[31]

The first few powers of negative two are −2, 4, −8, 16, −32, 64, −128, oscillating between positive and negative.^[32] The positive terms are powers of four, and the negative terms differ from powers of four by a factor of negative two.^[33] This property makes negative two the largest negative number that can represent all real numbers as a base without using a negative sign or two's complement.^{[32][34][35][36]} In 1957, some computers used a base-negative-two numeral system for calculations.^[37] Similarly, using ${\displaystyle 2i}$ can represent complex numbers.^[38]

The sum of the powers of negative two is a divergent geometric series. Although divergent, its generalized sum is ${\displaystyle {\frac {1}{3}}}$.^[39][40]

${\displaystyle \sum _{k=0}^{n}(-2)^{k}=1-2+4-8+\dots }$

Using the geometric series formula,^[41]

${\displaystyle \sum _{k=0}^{\infty }ar^{k}={\frac {a}{1-r}}}$

with the first term ${\displaystyle a=1}$ and common ratio ${\displaystyle r=-2}$, the result is ${\displaystyle {\frac {1}{3}}}$. However, the series is divergent, with partial sums:^[42]

1, −1, 3, −5, 11, −21, 43, −85, 171, −341...^[42]

Though divergent, Euler assigned the value ${\displaystyle {\frac {1}{3}}}$ to this series,^[43] known as Euler summation.^[44]

The negative second power of a number is its square reciprocal, applicable to functions as well.^[45] In daily life, it is occasionally used to denote units without a division sign, such as acceleration, typically written as m/s² but also as m·s⁻² in SI notation.^[6]

Common topics related to the square reciprocal include: for any real number ${\displaystyle n}$, its square reciprocal ${\displaystyle n^{-2}}$ is always positive; the inverse-square law;^[46] grid turbulence decay;^[47] and the Basel problem.^[48] The Basel problem states that the sum of the square reciprocals of natural numbers converges to ${\displaystyle {\frac {\pi ^{2}}{6}}}$:^[49][48]

${\displaystyle \sum _{n=1}^{\infty }{n^{-2}}={1^{-2}}+{2^{-2}}+{3^{-2}}+\cdots ={1 \over 1^{2}}+{1 \over 2^{2}}+{1 \over 3^{2}}+\cdots ={\pi ^{2} \over 6}}$

This value equals the Riemann zeta function at 2.^[50][51]

For any real number, the square reciprocal is positive. For negative two, it is ${\displaystyle {\frac {1}{4}}}$. The square reciprocals of the first few natural numbers are:

Square reciprocal	1	2	3	4	5	6	7	8	9	10
${\displaystyle x^{-2}}$	1	${\displaystyle {\frac {1}{4}}}$	${\displaystyle {\frac {1}{9}}}$	${\displaystyle {\frac {1}{16}}}$	${\displaystyle {\frac {1}{25}}}$	${\displaystyle {\frac {1}{36}}}$	${\displaystyle {\frac {1}{49}}}$	${\displaystyle {\frac {1}{64}}}$	${\displaystyle {\frac {1}{81}}}$	${\displaystyle {\frac {1}{100}}}$
	1	0.25	${\displaystyle 0.{\overline {1}}}$	0.0625	0.04	${\displaystyle 0.0{\overline {27}}}$	0.0204081632...[Note 2]	0.015625	${\displaystyle 0.{\overline {012345679}}}$	0.01

The square root of negative two, defined with the imaginary unit ${\displaystyle i}$ satisfying ${\displaystyle i^{2}=-1}$, is derived from ${\displaystyle {\sqrt {-x}}=\pm i{\sqrt {x}}}$. For negative two, it is ${\displaystyle {\sqrt {-2}}=\pm i{\sqrt {2}}}$.^{[Note 3][52][53][54][55]} The principal value is ${\displaystyle {\sqrt {-2}}=i{\sqrt {2}}}$.^{[Note 4][52][53][54][55]}

Negative two is typically represented by adding a negative sign before 2,^[56] commonly called "negative two" or "minus two" in English.^[57]

In binary, especially in computing, negative numbers are often represented using two's complement.^[58] Negative two is represented as "...11111110₍₂₎", specifically "1110₍₂₎" for 4-bit, "11111110₍₂₎" for 8-bit, and "1111111111111110₍₂₎" for 16-bit integers.^[59] In signed notation, it is "−10₍₂₎".^[60]

Plus or minus two (${\displaystyle \pm 2}$) is expressed using the plus-minus sign, representing both positive and negative two. It denotes the square roots of 4 or the solutions to the quadratic equation ${\displaystyle x^{2}=4}$, i.e., ${\displaystyle {\sqrt {4}}=\pm {2}}$. Plus or minus two appears more frequently in cultural contexts, such as music compositions^[61] or the documentary ±2 °C, which explores the environmental impact of a global temperature change of two degrees.^[62][63]

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