Euler's formula states that e to the power i theta equals cosine theta plus i sine theta. Its most famous case, e to the i pi plus 1 equals zero, connects five fundamental constants: e, i, pi, 1, and 0. It is often called the most beautiful equation in mathematics.
A complex number a plus b i is represented by a point on the complex plane, with the horizontal axis for the real part and the vertical axis for the imaginary part. In polar form, it is written as r e to the i theta, linking to rotation and magnitude.
In the unit circle, e to the i theta corresponds to the point with coordinates cosine theta and sine theta. As theta increases, the point moves counterclockwise, demonstrating rotation.
Starting from the series of e to the x, replace x with i theta. Group the even powers to recover cosine, and the odd powers to recover sine, resulting in cosine theta plus i sine theta.
For example, e to the i pi over two equals i, e to the i pi over four equals root two over two plus i times root two over two, and e to the i two pi equals one. De Moivre's theorem generalizes to any integer power.
Setting theta to pi, e to the i pi equals cosine pi plus i sine pi, which equals negative one. Adding one gives zero, uniting e, i, pi, one, and zero, spanning analysis, algebra, and geometry.