Those geometric assumptions in turn may be justified starting with a more detailed discussion of geometry. So the quickest way to understand and explain complex numbers and enjoy the consequences is to assume the distributive law. Chains of reason leading to it may be studied later. The Rectangular Way to Multiply . This is first consequence of the key properties.
Many further consequences follow from it.
Complex Conjugates and Reciprocals f complex numbers are illustrated here. The Key Properties lessons covered the existence of reciprocals or multiplicative inverses of complex numbers without the diagrams. Pythagorean Theorem . Here is the complex number based proof, another consequence of assuming the distributive law and obtaining two different ways to compute the product of a complex number with its complex conjugate. Multiplication of an point by its reflection across the horizontal axis can be done in two different ways with the aid of polar coordinates and with the aid of real and imaginary.