|Cardinal||−1, minus one, negative one|
−1 has some similar properties as positive one. But some are different.
−1 is related to Euler's identity. This is because the identity states
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
where we used the fact that 0 multiplied by any real number x equals 0, shown by cancellation from the equation
In other words,
so (−1) · x or −x is the arithmetic inverse of x.
Square of −1Edit
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
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
The second equality follows from the fact that 1 is a multiplicative identity, that is : . But now adding 1 to both sides of this last equation means
Square roots of −1Edit
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.
Exponentiation to negative integersEdit
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 xaxb = x(a + b) for a,b non-zero real numbers is true even if a or b is negative.
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.
- Deshpande, J.V. (2004). Mathematical Analysis and Applications: An Introduction. Alpha Science Int'l Ltd. ISBN 978-1-84265-189-6.
- "Math Forum - Ask Dr. Math". mathforum.org.