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−1 (number)
−1  

List of numbers  Integers (whole numbers)
 
Cardinal  −1, minus one, negative one  
Arabic  −١  
Chinese numeral  负一，负弌，负壹  
Bengali  ô১  
Binary (byte) 
 
Hex (byte) 

In mathematics, −1 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 negative two (−2) and less than 0.
Negative one has some similar properties as positive one. But some are different.^{[1]}
Negative one is related to Euler's identity. This is because the identity states [math]e^{i \pi} = 1.\![/math]
In computer science, −1 is a common initial value for integers. It is also used to show that a variable has no useful information.
Contents
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
 [math]x+(1)\cdot x=1\cdot x+(1)\cdot x=(1+(1))\cdot x=0 \cdot x=0[/math]
where we used the fact that 0 multiplied by any real number x equals 0, shown by cancellation from the equation
 [math]0\cdot x=(0+0)\cdot x=0\cdot x+0\cdot x \, [/math]
In other words,
 [math]x+(1)\cdot x=0 \, [/math]
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
 [math]0 =1\cdot 0 =1\cdot [1+(1)][/math]
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
 [math]0 =1\cdot [1+(1)]=1\cdot1+(1)\cdot(1)=1+(1)\cdot(1)[/math]
The second equality follows from the fact that 1 is a multiplicative identity, that is : [math]x\cdot 1=x \, [/math]. But now adding 1 to both sides of this last equation means
 [math](1) \cdot (1) = 1[/math]
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 i^{2} = −1. So it is a square root of −1. The only other complex number x for which the equation x^{2} = −1 holds is −i. In the algebra of quaternions, which has the complex plane, the equation x^{2} = −1 has an infinity of solutions.^{[2]}
Exponentiation to negative integers
A nonzero 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^{a}x^{b} = x^{(a + b)} for a,b nonzero 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
 ↑ Deshpande, J.V. (2004). Mathematical Analysis and Applications: An Introduction. Alpha Science Int'l Ltd.. . https://books.google.com/?id=Xrh89dLWZqEC.
 ↑ "Math Forum  Ask Dr. Math". http://mathforum.org/library/drmath/view/58251.html.