Euler's formula is widely regarded as one of the most beautiful and profound equations in all of mathematics. The special case e to the i pi plus one equals zero elegantly connects five of the most fundamental constants in mathematics: e, the base of natural logarithms; i, the imaginary unit; pi, the ratio of a circle's circumference to its diameter; one, the multiplicative identity; and zero, the additive identity. The more general form, e to the i x equals cosine x plus i sine x, reveals a deep and surprising connection between exponential functions and trigonometric functions, bridging seemingly unrelated areas of mathematics.
To understand Euler's formula, we first need to understand complex numbers. Complex numbers extend the real number system by introducing the imaginary unit i, which has the fundamental property that i squared equals negative one. A complex number is written in the form a plus b i, where a is called the real part and b is called the imaginary part. Complex numbers can be visualized geometrically in the complex plane, where the horizontal axis represents the real part and the vertical axis represents the imaginary part. For example, the complex number 3 plus 4 i corresponds to the point with coordinates 3 comma 4 in this plane, and can also be represented as a vector from the origin to this point.
Before diving into Euler's formula, we need to understand the key functions involved. The exponential function e to the x has the remarkable property that it equals its own derivative, making it fundamental in calculus and differential equations. The trigonometric functions cosine and sine describe circular motion and periodic phenomena. On the unit circle, cosine of x gives the x-coordinate of a point, while sine of x gives the y-coordinate. As the angle x changes, the point traces out the circle, with cosine and sine values oscillating between negative one and positive one. These functions are periodic with period two pi, meaning they repeat their values every two pi units.
Now let's see how Euler's formula emerges naturally from Taylor series. The exponential function e to the x can be expressed as an infinite series: one plus x plus x squared over two factorial plus x cubed over three factorial, and so on. Similarly, cosine and sine have their own Taylor series expansions. When we substitute i x into the exponential series, we get e to the i x equals one plus i x plus i x squared over two factorial, and so on. Using the powers of i - where i squared equals negative one, i cubed equals negative i, and i to the fourth equals one - we can expand and rearrange terms. Separating the real and imaginary parts, we discover that the real part matches the cosine series and the imaginary part matches the sine series, giving us Euler's remarkable formula: e to the i x equals cosine x plus i sine x.
The geometric interpretation of Euler's formula provides profound insight into its meaning. In the complex plane, the expression e to the i x represents a point on the unit circle, where x is the angle measured counterclockwise from the positive real axis. As we vary x from zero to two pi, the point e to the i x traces out the entire unit circle exactly once. This beautiful visualization shows that complex exponentials are fundamentally connected to circular motion and rotation. The real part, cosine of x, gives the horizontal coordinate of the point, while the imaginary part, sine of x, gives the vertical coordinate. This geometric perspective reveals why Euler's formula is so powerful - it provides a unified way to describe rotation and oscillation using the elegant language of complex exponentials.