−2

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

Properties

• Negative two is a complementary Bell number (also known as Rao Uppuluri-Carpenter numbers), 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. According to the Stark–Heegner theorem, only nine negative numbers have this property, corresponding to Heegner numbers.
  • Negative two also makes the quadratic field ${\displaystyle \mathbb{Q}[\sqrt{d}]}$ a simple Euclidean field (or norm-Euclidean field). Only five negative numbers have this property: −11, −7, −3, −2, −1. If relaxed, −15 is included.
• Negative two is the largest negative number unreachable from 1 in two steps using addition, subtraction, or multiplication. The largest in one step is −1, and in three steps, −4. This relates to the straight-line program combined with addition, subtraction, and multiplication, exploring the algebraic complexity of integers in relation to NP = P.
• Negative two is the second-order i.e., ${\displaystyle H_2=H_2(0)=-2}$.
• Negative two makes ${\displaystyle k(k+1)(k+2)}$ a triangular number. Only nine integers have this property, with negative two being the smallest: −2, −1, 0, 1, 4, 5, 9, 56, and 636.

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 ${\displaystyle -1}$ can be factored out as ${\displaystyle -1\times 2}$, 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.

Powers of negative two

The first few powers of negative two are −2, 4, −8, 16, −32, 64, −128, oscillating between positive and negative. The positive terms are powers of four, and the negative terms differ from powers of four by a factor of negative two. 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. In 1957, some computers used a base-negative-two numeral system for calculations. Similarly, using ${\displaystyle 2i}$ can represent complex numbers.

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

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

Using the geometric series formula,

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

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:

1, −1, 3, −5, 11, −21, 43, −85, 171, −341...

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

Negative second power

The negative second power of a number is its square reciprocal, applicable to functions as well. 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.

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; grid turbulence decay; and the Basel problem. The Basel problem states that the sum of the square reciprocals of natural numbers converges to ${\displaystyle \frac{{\pi }^2}{6}}$:

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

This value equals the Riemann zeta function at 2.

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 root of negative two

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 ${\displaystyle \sqrt{-2}=\pm i\sqrt{2}}$. The principal value is ${\displaystyle \sqrt{-2}=i\sqrt{2}}$.