Welcome to one of mathematics' most elegant discoveries: Euler's Identity. This remarkable equation, e to the i pi plus one equals zero, is often called the most beautiful formula in mathematics. It was discovered by the Swiss mathematician Leonhard Euler in the 18th century. What makes this identity so special is how it connects five of the most important 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. These constants come from completely different areas of mathematics, yet Euler's identity shows they are intimately connected through the fundamental operations of exponentiation, multiplication, and addition.
To understand Euler's identity, we first need to explore complex numbers. Complex numbers extend our number system beyond real numbers by introducing the imaginary unit i, defined as the square root of negative one, or i squared equals negative one. A complex number has the form a plus b i, where a is the real part and b is the imaginary part. We can visualize complex numbers in the complex plane, where the horizontal axis represents real numbers and the vertical axis represents imaginary numbers. Each complex number corresponds to a point in this plane, and can also be represented as a vector from the origin to that point. For example, 3 plus 4 i is located at coordinates 3, 4 in the complex plane. Complex numbers follow specific rules for addition and multiplication that preserve their geometric interpretation.
Now let's explore Euler's number e, approximately 2.718, which is the base of natural logarithms. The exponential function e to the x has a remarkable property: its derivative equals the function itself. This makes e special among all possible bases for exponential functions. We can express e to the x using its Taylor series expansion: 1 plus x plus x squared over 2 factorial plus x cubed over 3 factorial, and so on. This infinite series converges for all real numbers, and remarkably, it also works for complex arguments. When we substitute i x into this series, we get e to the i x, which will be crucial for understanding Euler's identity. The graph shows how e to the x grows exponentially, and at the point where x equals 1, the function value is e and the slope is also e.
Complex exponentials have a beautiful geometric interpretation. When we write e to the i x, we're describing rotation in the complex plane. The magnitude of e to the i x is always 1, which means all these points lie on the unit circle. As x varies, e to the i x traces out points on this circle. When x equals 0, we get 1. When x equals pi over 2, we get i. When x equals pi, we get negative 1. When x equals 3 pi over 2, we get negative i. And when x equals 2 pi, we return to 1, completing a full rotation. This shows that e to the i x represents rotation by angle x from the positive real axis.
Now we'll derive Euler's famous formula using Taylor series. We start with the series for e to the x, cosine x, and sine x. When we substitute i x into the exponential series, we get terms with powers of i. Since i squared equals negative 1, i cubed equals negative i, and i to the fourth equals 1, the pattern repeats. By separating the real and imaginary parts, we find that the real part gives us the cosine series, while the imaginary part gives us the sine series. This leads to Euler's formula: e to the i x equals cosine x plus i sine x. On the unit circle, we can see this relationship visually - the x-coordinate represents cosine x, and the y-coordinate represents sine x.