1. **State the problem:** Express the complex number $1 + i$ in polar form. 2. **Recall the polar form:** A complex number $z = x + yi$ can be represented in polar form as $$z = r(\cos \theta + i \sin \theta)$$ where $r$ is the magnitude and $\theta$ is the argument. 3. **Calculate the magnitude:** The magnitude $r$ is given by $$r = \sqrt{x^2 + y^2}$$ For $1 + i$, $x = 1$, $y = 1$, so $$r = \sqrt{1^2 + 1^2} = \sqrt{2}$$ 4. **Calculate the argument:** The argument $\theta$ is the angle formed with the positive real axis, given by $$\theta = \tan^{-1}\left(\frac{y}{x}\right) = \tan^{-1}\left(\frac{1}{1}\right) = \frac{\pi}{4}$$ radians. 5. **Write the polar form:** Substitute $r$ and $\theta$ into the formula: $$1 + i = \sqrt{2}\left(\cos \frac{\pi}{4} + i \sin \frac{\pi}{4}\right)$$ **Final answer:** The polar form of $1 + i$ is $$\boxed{\sqrt{2}\left(\cos \frac{\pi}{4} + i \sin \frac{\pi}{4}\right)}$$