Operations in the Complex Plane

To visualize a complex number $z = a + bi$, we use the complex plane. The horizontal axis represents the Real part ($a$), and the vertical axis represents the Imaginary part ($b$). Instead of just a dot, we often view a complex number as a vector pointing from the origin $(0,0)$ to the point $(a, b)$. The 'length' of this vector is called the modulus (or absolute value), denoted as $|z|$. Using the Pythagorean theorem, we calculate it as $|z| = \sqrt{a^2 + b^2}$. This value represents the distance of the complex number from the origin, providing a bridge between algebra and geometry.

Multiplying complex numbers follows the distributive property (often called FOIL), but with a twist: whenever you encounter $i^2$, you must replace it with $-1$. Algebraically, $(a + bi)(c + di) = ac + adi + bci + bdi^2$. Since $bdi^2 = -bd$, the result is $(ac - bd) + (ad + bc)i$. Geometrically, multiplying by a complex number doesn't just scale the vector; it rotates it. For instance, multiplying any number by $i$ results in a counter-clockwise rotation of $90^\circ$ in the complex plane.

Division in the complex plane is tricky because we cannot easily divide by an imaginary unit. To solve this, we use the complex conjugate. For any number $z = a + bi$, its conjugate is $\bar{z} = a - bi$. The magic happens when you multiply a number by its conjugate: $(a + bi)(a - bi) = a^2 + b^2$. This always results in a real number. To divide two complex numbers, multiply both the numerator and the denominator by the conjugate of the denominator. This 'rationalizes' the bottom, allowing you to split the result into distinct real and imaginary parts.