1. **Problem Statement:** Define the modulus of a complex number and explain its geometric interpretation with an example. 2. **Definition:** The modulus of a complex number $z = a + bi$ (where $a$ and $b$ are real numbers, and $i$ is the imaginary unit) is the distance of the point $(a,b)$ from the origin in the complex plane. 3. **Formula:** The modulus is given by $$|z| = \sqrt{a^2 + b^2}$$ This formula comes from the Pythagorean theorem applied to the right triangle formed by the point $(a,b)$ and the origin. 4. **Geometric Interpretation:** Geometrically, the complex number $z = a + bi$ can be represented as a point or vector in the 2D plane with coordinates $(a,b)$. The modulus $|z|$ is the length of the vector from the origin $(0,0)$ to the point $(a,b)$. 5. **Example:** Consider the complex number $z = 3 + 4i$. - Here, $a = 3$ and $b = 4$. - Calculate the modulus: $$|z| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5$$ - So, the modulus of $3 + 4i$ is 5, which means the point $(3,4)$ is 5 units away from the origin in the complex plane. 6. **Summary:** The modulus measures the size or magnitude of a complex number and is always a non-negative real number. It is fundamental in understanding the geometric representation of complex numbers.