How do I find the modulus and argument of the complex number 1 + i?

Write the complex number as z = x + iy, so here x = 1 and y = 1. The modulus |z| is the distance from the origin, given by the square root of (x² + y²). That is the square root of (1 + 1), which equals the square root of 2. The argument is the angle the line to z makes with the positive real axis, found from tan θ = y/x = 1. Since both x and y are positive, z lies in the first quadrant, so θ = π/4 (45°). Therefore 1 + i has modulus root 2 and argument π/4. In polar form it is root 2 times (cos π/4 + i sin π/4). Always check the quadrant before fixing the argument, because tan alone is ambiguous.