-1 (number)

In mathematics, −1 is the negative integer greater than negative two (−2) and less than 0. It is the additive inverse of 1, that is, the number that when added to 1 gives 0.

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:

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 is a sensible definition to make since it preserves the exponential law x^ax^b = x^{(a + b)} in the case when a or b is negative (i.e. in the case when a and b are not both nonnegative).

Negative one bears relation to Euler's identity since .

Negative one is one of three possible return values of the Möbius function. When passed a square-free integer with an odd number of distinct prime factors, the Möbius function returns negative one.

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

[edit] Why is −1 times −1 equal to 1?

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Why is −1 multiplied by −1 equal to 1? More generally, why is a negative times a negative a positive? There are two ways to answer this question. The first is intuitive and conceptual; the second is formal and algebraic.

[edit] Intuitive explanation

There are many ways to conceptualize multiplication.

Imagine, for a moment, a hot-air balloon in the sky. The flame is going, so the balloon is rising. Let's say that the balloon is rising at a steady pace of 2 feet every second. Let's also say that we'll consider up to be a "positive" direction, and down to be a "negative" direction. Two questions shall be asked about the balloons height at different points in time:

Question: Compared to where the balloon is now, where will it be in 5 seconds?
Answer: The answer is found by multiplying the number of seconds by the speed. (5 seconds from now)(2 feet higher every second) = 10 feet higher. 5 x 2 = 10 -- a positive result.

Question: Compared to where the balloon is now, where was it 5 seconds ago?
Answer: Let's think of going back in time as a "negative time" direction. (5 seconds ago)(2 feet higher every second) = 10 feet lower. (-5) x 2 = -10.

Now, imagine a change in the situation. The flame isn't on and there's a small hole in the balloon, so its falling at a rate of 2 feet every second. The same two questions shall be asked with the new direction.

Question: Compared to where the balloon is now, where will it be in 5 seconds?
Answer: (5 seconds from now)(2 feet lower every second) = 10 feet lower. 5 x (-2) = -10 -- a negative result.

Question: Compared to where the balloon is now, where was it 5 seconds ago?
Answer: (5 seconds ago)(2 feet lower every second) = 10 feet higher. (-5) x (-2) = +10.

And it has been shown that a negative times a negative comes out to be a positive.

[edit] Algebraic explanation

The algebraic explanation is essentially a formalisation of the above intuitive explanation. Start with the equation

The first equality follows from the fact that "anything times zero is zero". 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 and simple addition. But now we add 1 to both sides of this last equation to see that

The above argument holds in any ring. It has a flavour common to some of the basic results in abstract algebra.

[edit] 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 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.

[edit] The square root of -1

The square root of -1 is i, the imaginary unit.