absolute value of a real number is that number's distance from zero along the real number line, and more generally the absolute value of the difference of two real numbers (their absolute difference) is the distance between them. The notion of an abstract distance function in mathematics can be seen to be a generalisation of the absolute value of the difference (see "Distance" below). Since the square root symbol represents the unique positive square root, when applied to a positive number, it follows that This is equivalent to the definition above, and may be used as an alternative definition of the absolute value of real numbers. The absolute value has the following four fundamental properties (a, b are real numbers), that are used for generalization of this notion to other domains: Non-negativity, positive definiteness, and multiplicativity are readily apparent from the definition. To see that subadditivity holds, first note that with its sign chosen to make the result positive. Now, since it follows that, whichever of is the value one has for all Consequently, , as desired. Some additional useful properties are given below. These are either immediate consequences of the definition or implied by the four fundamental properties above. Two other useful properties concerning inequalities are: These relations may be used to solve inequalities involving absolute values. For example: The absolute value, as "distance from zero", is used to define the absolute difference between arbitrary real numbers, the standard metric on the real numbers. Complex numbers Since the complex numbers are not ordered, the definition given at the top for the real absolute value cannot be directly applied to complex numbers. However, the geometric interpretation of the absolute value of a real number as its distance from 0 can be generalised. The absolute value of a complex number is defined by the Euclidean distance of its corresponding point in the complex plane from the origin. This can be computed using the Pythagorean theorem: for any complex number where and are real numbers, the absolute value or modulus is and is defined by the Pythagorean addition of and , where and denote the real and imaginary parts respectively. When the is zero, this coincides with the definition of the absolute value of the When a complex number is expressed in its polar form its absolute value Since the product of any complex number and its with the same absolute value,