−2 - Wikiwand

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]

Cardinal−2, minus two, negative two

Ordinal−2nd (negative second)

Divisors1, 2

Arabic−٢

Quick facts ← −3 −2 −1 →, Cardinal ...

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−2

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Cardinal

−2, minus two, negative two

Ordinal

−2nd (negative second)

1, 2

−٢

负二，负弍，负貳

−২

Binary (byte)

S&M:	100000010₂
2sC:	11111110₂

Hex (byte)

S&M:	0x102₁₆
2sC:	0xFE₁₆

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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]

Properties

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 $\mathbb {Q} [{\sqrt {d}}]$ $\mathbb {Q} [{\sqrt {d}}]$ 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 $\mathbb {Q} [{\sqrt {d}}]$ 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., $H_{2}=H_{2}(0)=-2$ .^[26]^[27]
Negative two makes $k(k+1)(k+2)$ 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]

Divisors of negative two

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 $-1$ can be factored out as $-1\times 2$ ,^[30] making 2 a prime factor but not the result of prime factorization. As a Gaussian integer, negative two can be factored as $i\times (1+i)^{2}$ , where $1+i$ is a Gaussian prime and $i$ is the imaginary unit.^[31]

Powers of negative two

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 $2i$ can represent complex numbers.^[38]

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

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

Using the geometric series formula,^[41]

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

with the first term $a=1$ and common ratio $r=-2$ , the result is ${\frac {1}{3}}$ . 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 ${\frac {1}{3}}$ to this series,^[43] known as Euler summation.^[44]

Negative second power

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 $n$ , its square reciprocal $n^{-2}$ 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 ${\frac {\pi ^{2}}{6}}$ :^[49]^[48]

\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 ${\frac {1}{4}}$ . The square reciprocals of the first few natural numbers are:

More information

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

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Square root of negative two

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

Representation

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

Plus or minus two ( $\pm 2$ ) 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 $x^{2}=4$ , i.e., ${\sqrt {4}}=\pm {2}$ . 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]

Notes

[1]
When $d<0$ , if the ring of integers of $\mathbb {Q} [{\sqrt {d}}]$ is a unique factorization domain, all numbers in $\mathbb {Q} [{\sqrt {d}}]$ have a unique factorization. For example, $\mathbb {Q} [{\sqrt {-5}}]$ is not, as 6 can be factored in two ways in $\mathbb {Z} [{\sqrt {-5}}]$ : $2\times 3$ and $(1+{\sqrt {-5}})(1-{\sqrt {-5}})$ .
[2]
The repeating decimal for 7's square reciprocal has a period of 42: 0.0204081632 6530612244 8979591836 7346938775 51... See reciprocal of 49.
[3]
In the bi-imaginary number system $\left\langle {\sqrt {R}},Z_{R}\right\rangle$ , where $R$ is negative two and $Z_{R}$ is two, it is $\left\langle \pm \mathrm {i} {\sqrt {2}},Z_{2}\right\rangle$ . Donald E. Knuth (April 1, 1960). "A imaginary number system" [A imaginary number system]. Communications of the ACM. 3 (4): 245–247. doi:10.1145/367177.367233. Retrieved August 21, 2025.
[4]
The principal value of the square root of –2 is ${\sqrt {-x}}=\pm i{\sqrt {x}}$ taking the positive value; with respect to –2, that is ${\sqrt {-2}}=i{\sqrt {2}}$ .

−2