 Jean Le Ronde d'Alembert was a French philosopher and major contributor to Diderot's Encyclopédie, the first modern encyclopedia. He's best known in mathematics for being wrong. However, he was wrong in a way that made others think more clearly about mathematics. D'Alembert was wrong in several areas, but one of the more important was his incorrect conclusions regarding the log of negative 1. D'Alembert claimed that the log of negative number was the same as the log of the corresponding positive number. Now, he gave several arguments for why this had to be the case. One of them is the following. Suppose x is the log of negative 1. We have the log of negative 1 cubed, while our rules of logarithms say that that's the same as 3 log negative 1. But negative 1 cubed itself is negative 1, and so we have the equation x equals 3x, so x must be 0. Consequently, if a is a positive number, the log of negative a, well, that's log of negative 1 times a, and our rules of logarithms say that's log negative 1 plus the log of a, but since log of negative 1 is 0, that's just the log of a. And this led to a great controversy in mathematics. To resolve that controversy, let's go back to De Wauff's theorem. De Wauff's theorem allows us to evaluate algebraic functions at any complex value. But what about transcendental functions? Well, let's see what happens. In 1748, Euler made an argument we can summarize as follows. From our Taylor series for e to the x, which is true for all x, we can substitute in a complex number e to bi and get. Now, remember the even power of an imaginary number is a real number, and an odd power can be reduced to something multiplied by i. And we can separate our real and imaginary terms. And notice that this first series is the same as our Taylor series for cosine, and the second series is the same as the Taylor series for sine, and so e to a pure imaginary power cosine b plus i sine b. In other words, e to the i theta is equal to sys theta. And this leads to the exponential form of a complex number. A complex number x plus i y in rectangular form can be written in polar form as r sys theta. But since e to the bi is sys b, we can turn the polar form into the exponential form r e to the i theta. So let's find the log of negative one. So suppose the log of negative one is a plus bi. Now by definition, e to power a plus bi must be negative one. And we can split that exponent into the real and complex portions e to the a times e to the bi. And e to the bi is sys b. Meanwhile, negative one is sys pi. And if we compare the two sides in order for them to be equal, e to the a must be one and b must be pi. And consequently, a must be zero, p must be pi, and log of negative one is pi i. The fact that log of negative one equals pi i also means that e to the pi i is negative one, and so e to the pi i plus one is equal to zero, which unites the five basic mathematical symbols and is known as Euler's identity. One other important point is that since the trigonometric form of a complex number is not unique, the exponential form is also not unique. This means that functions like log z can output different values for the same number. We might call them multi-valued functions. But we like our functions to have a single output. So instead, we call them multi-valued functions. We'll come back to this topic. But as a preview of things to come, if you've been wondering about the image that opens these lectures, it has to do with that.