ABSTRACT
Complex numbers were first studied in the 1500s but were only fully accepted in the 1800s. Although a complex number can be regarded as labelling a point in the complex plane, it can also be regarded as labelling the directed line segment from the origin to the point, and this turns out to be more fruitful. Thus, the complex numbers enable us to find the hidden connections between the three most important functions of calculus: the exponential function, the sine function and the cosine function. The word isomorphism means the ‘same shape’. Informally, two algebraic systems are isomorphic if they have all the same properties even though they may look different. The concept of isomorphism is vitally important in mathematics. This is because mathematics deals with the properties that an object possesses, that is the rules that it obeys, and not with what it looks like.
