What is the modulus of a complex number?

The modulus of a complex number is indeed defined as the distance from the number to the origin in the complex plane. A complex number is typically expressed in the form ( z = a + bi ), where ( a ) is the real part and ( b ) is the imaginary part. The modulus is calculated using the formula ( |z| = \sqrt{a^2 + b^2} ).

This formula derives from the Pythagorean theorem, where ( a ) and ( b ) can be seen as the two perpendicular sides of a right triangle, and the modulus represents the length of the hypotenuse, which extends from the origin (0,0) to the point (a,b) in the complex plane. Thus, the modulus provides a quantitative measure of how far the complex number is from the origin, giving it geometric significance.

The other options refer to properties or aspects of complex numbers but do not correctly define the modulus. For instance, while the angle of a complex number in polar form is important, it relates to its argument, not the modulus. The scaling factor for the imaginary part does not align with the definition of modulus, and stating that it is the maximum of the real parts mis