Question

Compare and contrast the absolute value of a real number to that of a complex number.

Answer 1

The definition of complex, real and pure imaginary number is as follows:

$A \ \mathbf{complex \ number} \ is \ written \ in \ \mathbf{standard \ form} \ as:\\ \\ \ (a+bi) \\ \\ where \ a \ and \ b \ are \ real \ numbers. \ If \ b=0, \ the \ number \ a+bi=a \\ is \ a \ \mathbf{real \ number}. \ If \ b\neq 0, \ the \ number \ (a+bi) \ is \ called \ an \\ \mathbf{imaginary \ number}. \ A \ number \ of \ the \ form \ bi, \ where \ b\neq 0, \\ is \ called \ a \ \mathbf{pure \ imaginary \ number}$

The absolute value of this number is given by:

$|a+bi|=\sqrt{a^{2}+b^{2}}$

So, the absolute value of a complex number represents the distance between the origin and the point in the complex plane. On the other hand, the absolute value of a real number means only how far a number is from zero without considering any direction.

Answer 2

The absolute value of a real number is a positive value of the number. Which means that the absolute value is the distance from zero of the number line. However, that of the complex numbers is the distance from the origin to the point in a complex plane.