−1 (number)

In mathematics, −1 (negative one, minus one or one below) is the additive inverse of 1. That is, if it is added to 1, the result is 0. It is the negative integer greater than −2 and less than 0.

−1
−1 0 1 2 3 4 5 6 7 8 9 → List of numbers -- Integers (whole numbers) 0 10 20 30 40 50 60 70 80 90 >>
Cardinal	minus one, negative one, one below
Arabic	−١
Chinese numeral	负一，负弌，负壹
Bengali	ô১
Binary (byte)	S&M: 2sC: 1000000012 111111112
Hex (byte)	S&M: 2sC: 0x1018 0xFF8

−1 has some similar properties as positive one. But some are different.^[1]

−1 is related to Euler's identity. This is because the identity states ${\displaystyle e^{i\pi }=-1.\!}$

In computer science, −1 is a common initial value for integers. It is also used to show that a variable has no useful information.

Algebraic properties

Multiplying a number by −1 is the same as changing the sign on the number. This can be proved using the distributive law and the axiom at 1 is the multiplicative identity, that is, a number multiplied by 1 is the number itself. So, for x real, we have

${\displaystyle x+(-1)\cdot x=1\cdot x+(-1)\cdot x=(1+(-1))\cdot x=0\cdot x=0}$

where we used the fact that 0 multiplied by any real number x equals 0, shown by cancellation from the equation

${\displaystyle 0\cdot x=(0+0)\cdot x=0\cdot x+0\cdot x\,}$

0, 1, −1, i, and −i in the complex or cartesian plane

In other words,

${\displaystyle x+(-1)\cdot x=0\,}$

so (−1) · x or −x is the arithmetic inverse of x.

Square of −1

The square of −1, i.e. −1 multiplied by −1, equals 1. So, a square of negative real numbers is positive.

To prove this with algebra, start with the equation

${\displaystyle 0=-1\cdot 0=-1\cdot [1+(-1)]}$

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

${\displaystyle 0=-1\cdot [1+(-1)]=-1\cdot 1+(-1)\cdot (-1)=-1+(-1)\cdot (-1)}$

The second equality follows from the fact that 1 is a multiplicative identity, that is : ${\displaystyle x\cdot 1=x\,}$. But now adding 1 to both sides of this last equation means

${\displaystyle (-1)\cdot (-1)=1}$

The above arguments hold in any ring. Ring is a concept of abstract algebra generalizing integers and real numbers.

Square roots of −1

The complex number i satisfies i² = −1. So it is a square root of −1. The only other complex number x for which the equation x² = −1 holds is −i. In the algebra of quaternions, which has the complex plane, the equation x² = −1 has an infinity of solutions.^[2]

Exponentiation to negative integers

A non-zero real number can have a negative number as its power. We define that x⁻¹ = 1/x. This means a number raised to a power of −1 is equal to the reciprocal of that number. The exponential law x^ax^b = x^{(a + b)} for a,b non-zero real numbers is true even if a or b is negative.

Computer representation

There are many ways that −1 (and negative numbers in general) can be represented in computer systems. The most common is as two's complement of their positive form. In standard binary representation, this can also represent a positive integer.

References

1. Deshpande, J.V. (2004). Mathematical Analysis and Applications: An Introduction. Alpha Science Int'l Ltd. ISBN 978-1-84265-189-6.
2. "Math Forum - Ask Dr. Math". mathforum.org.