1. **Problem:** Find the modulus and amplitude (argument) of the complex number $1 + 3i$. 2. **Formula:** - Modulus $|z| = \sqrt{x^2 + y^2}$ where $z = x + iy$. - Amplitude (argument) $\theta = \tan^{-1}\left(\frac{y}{x}\right)$. 3. **Calculation:** - For $z = 1 + 3i$, $x=1$, $y=3$. - Modulus: $$|z| = \sqrt{1^2 + 3^2} = \sqrt{1 + 9} = \sqrt{10}.$$ - Amplitude: $$\theta = \tan^{-1}\left(\frac{3}{1}\right) = \tan^{-1}(3).$$ 4. **Explanation:** - The modulus is the distance from the origin to the point $(1,3)$ in the complex plane. - The amplitude is the angle the line from the origin to $(1,3)$ makes with the positive real axis. 5. **Final answer:** - Modulus: $\sqrt{10}$. - Amplitude: $\tan^{-1}(3)$ radians (approximately 1.249 radians or 71.57 degrees).