-1 (number)

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-1 (number)

In mathematics

Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, −1 is the additive inverse

Additive inverse

In mathematics, the additive inverse, or opposite, of a number n is the number that, when addition to n, yields 0 .The additive inverse of F is denoted −F....
of 1

1 (number)

1 is a number, number names, and the name of the glyph representing that number.It represents a single entity, the unit of counting or measurement....
, that is, the number that when added

Addition

Addition is the mathematics process of putting things together. The plus sign "+" means that numbers are added together. For example, in the picture on the right, there are 3 + 2 apples?meaning three apples and two other apples?which is the same as five apples, since 3 + 2 = 5....
to 1 gives 0.

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Binary Binary numeral system The binary numeral system, or notation with a radix of 2. Owing to its straightforward implementation in digital electronic circuitry using logic gates, the binary system is used internally by all modern computers....	−1
Octal Octal The octal numeral system, or oct for short, is the radix-8 number system, and uses the digits 0 to 7. Numerals can be made from Binary numeral system numerals by grouping consecutive digits into groups of three ....	−1
Duodecimal Duodecimal The duodecimal system is a numeral system using 12 as its radix. In this system, the number 10 may be written as 'A', and the number 11 as 'B' ....	−1
Hexadecimal Hexadecimal In mathematics and computer science, hexadecimal is a numeral system with a radix, or base, of 16. It uses sixteen distinct symbols, most often the symbols 0–9 to represent values zero to nine, and A, B, C, D, E, F to represent values ten to fifteen....	−1
Two's complement Two's complement The two's complement of a binary number is defined as the value obtained by subtracting the number from a large power of two .A two's-complement system or two's-complement arithmetic is a system in which negative numbers are represented by the two's complement of the absolute value; this system is the most common Signed number r... signed Signed number representations In mathematics, negative numbers in any base are represented in the usual way, by prefixing them with a "−" sign. However, on a computer, there are various ways of representing a number's sign.... byte Byte A byte is a basic unit of measurement of Computer storage in computer science. In many computer architectures it is a Byte addressing memory address space....	11111111 (0xFF)

In mathematics

Mathematics

Additive inverse

In mathematics, the additive inverse, or opposite, of a number n is the number that, when addition to n, yields 0 .The additive inverse of F is denoted −F....
of 1

1 (number)

1 is a number, number names, and the name of the glyph representing that number.It represents a single entity, the unit of counting or measurement....
, that is, the number that when added

Addition

Negative and non-negative numbers

A negative number is a real number that is inequality 0 , such as -3. A positive number is a real number that is greater than zero, such as 2....
integer

Integer

The integers are natural numbers including 0 and their negative and non-negative numberss . They are numbers that can be written without a fractional or decimal component, and fall within the set ....
greater than negative two (−2) and less than 0

0 (number)

0 is both a number and the numerical digit used to represent that number in numeral system. It plays a central role in mathematics as the additive identity of the integers, real numbers, and many other algebraic structures....
.

Negative one has some similar but slightly different properties to positive one. Negative one would be a multiplicative identity if it were not for the sign change:

Negative one bears relation to Euler's identity

Euler's identity

In mathematical analysis, Euler's identity, named after Leonhard Euler, is the equationwhere is E , the base of the natural logarithm, is the imaginary unit, one of the two complex numbers whose square is negative one , and...
since .

In computer science

Computer science

Computer science is the study of the theoretical foundations of information and computation, and of practical techniques for their implementation and application in computer systems....
, −1 is a common initial value for integers and is also used to show that a variable contains no useful information.

Algebraic properties

The sign change can be proved using the distributive law and the fact 1 is a multiplicative identity: for x real

Real number

In mathematics, the real numbers may be described informally in several different ways. The real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as pi and the square root of two; or, a real number can be given by an infinite decimal representation, such as 2.4871773339...., where the digits co...
, we have:

where we used the fact that any real x times 0 equals 0, implied by cancellation

Cancellation property

In mathematics, the notion of cancellative is a generalization of the notion of invertible.An element a in a magma has the left cancellation property if for all b and c in M, a * b = a * c always implies b = c....
from the equation

Square of -1

The square

Square (algebra)

In algebra, the square of a number is that number multiplication by itself. To square a quantity is to multiply it by itself.Its notation is a superscripted "2"; a number x squared is written as x?....
of −1, i.e. −1 multiplied by −1, equals 1. As a consequence, a product of negative real numbers is positive.

For an algebraic proof of this result, start with the equation

The first equality follows from the above result. The second follows from the definition of −1 as additive inverse of 1: it is precisely that number that when added to 1 gives 0. Now, using the distributive law, we see that

The second equality follows from the fact that 1 is a multiplicative identity. But now adding 1 to both sides of this last equation implies

The above arguments hold in any ring

Ring (mathematics)

In mathematics, a ring is a type of algebraic structure. There is some variation among mathematicians as to exactly what properties a ring is required to have, as described in detail below....
, a concept of abstract algebra

Abstract algebra

Abstract algebra is the subject area of mathematics that studies algebraic structures, such as group , ring , field , module , vector spaces, and algebra over a field....
generalizing integers and real numbers.

Square roots of -1

The complex number

Complex number

In mathematics, the complex numbers are an extension of the real numbers obtained by adjoining an imaginary unit, denoted i, which satisfies:...
i
Imaginary unit

In mathematics, physics, and engineering, the imaginary unit is denoted by or the Latin or the Greek iota . It allows the real number system, to be extended to the complex number system, Its precise definition is dependent upon the particular method of extension....
satisfies , and as such can be considered as a square root

Square root

In mathematics, a square root of a number x is a number r such that r2 = x, or, in other words, a number r whose square is x....
of -1. The only other complex number x satisfying the equation is -i. In the algebra of quaternion

Quaternion

Quaternions, in mathematics, are a non-commutative number system that extends the complex numbers. The quaternions were first described by Irish mathematician Sir William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space....
s, containing the complex plane, the equation has an infinity of solutions.

Exponentiation to negative integers

Exponentiation

Exponentiation is a mathematics operation , written 'an', involving two numbers, the base a and the exponent n....
of a non-zero real number can be extended to negative integers

Exponentiation

Exponentiation is a mathematics operation , written 'an', involving two numbers, the base a and the exponent n....
. We make the definition that x⁻¹ = 1/x, meaning that we define raising a number to the power −1 to have the same effect as taking its reciprocal. This definition then extended to negative integers preserves the exponential law x^ax^b = x^{(a + b)} for a,b non-zero real numbers.

Exponentiation to negative integers can be extended to invertible elements of a ring, by defining x⁻¹ as the multiplicative inverse of x.

Computer representation

There are a variety of ways that −1 (and negative numbers in general) can be represented in computer systems, the most common being as two's complement

Two's complement

The two's complement of a binary number is defined as the value obtained by subtracting the number from a large power of two .A two's-complement system or two's-complement arithmetic is a system in which negative numbers are represented by the two's complement of the absolute value; this system is the most common Signed number r...
of their positive form. Since this representation could also represent a positive integer in standard binary representation, a programmer must be careful not to confuse the two. Negative one in two's complement could be mistaken for the positive integer 2ⁿ − 1, where n is the number of digits in the representation (that is, the number of bits in the data type). For example, 11111111₂ (binary) and FF₁₆ (hex) each represents −1 in two's complement, but represents 255 in standard numeric representation.