Question

Now examine ∣a+bi∣ and complete the definition below. The absolute value of any complex number a+bi is the distance from ________.

Real (a)
Imaginary (i)

Answer 1

The absolute value of any complex number a+biis the distance from the origin (0,0) to the point a+bi in the complex plane.

Step-by-step explanation:

The absolute value or modulus of a complex number a+bi is denoted as |a+bi|. Geometrically, in the complex plane, a complex number a+bi represents a point with coordinates (a,b). The absolute value of this complex number is the distance from the origin (0,0) to the point (a,b). Mathematically, the absolute value of a complex number |a+bi| is calculated using the formula
`\(|a+bi| = √(a^2 + b^2)\)`, which represents the distance formula in the Cartesian plane. This formula determines the magnitude or distance of the complex number from the origin, irrespective of the real (a) or imaginary (bi) components.

Understanding the concept of absolute value for complex numbers is crucial in comprehending their geometric interpretation. In the context of the complex plane, the modulus of a complex number represents its distance from the origin, analogous to how the absolute value of a real number represents its distance from zero on the number line. For any complex number a+bi the modulus |a+bi| equals the square root of the sum of the squares of its real (a) and imaginary (b) components, thus indicating the length of the line segment connecting the origin to the complex number's point in the complex plane. This geometric interpretation aids in visualizing and analyzing complex numbers in various mathematical contexts and applications.