From: 3blue1brown
Introduction to Complex Numbers
Complex numbers, which the speaker considers a fundamental piece of mathematics for engineering, mathematics, and quantum mechanics, are unfortunately given a “terrible, terrible name” [00:00:04]. The components that lead to complex numbers are called “imaginary numbers” [00:00:17]. The speaker aims to demonstrate the usefulness of imaginary and complex numbers to imbue them with more “reality” [00:04:51].
The “Reality” of Numbers
A poll was conducted asking the audience which numbers they considered to “really exist” among values like 2, the square root of 2, the square root of negative 1, and infinity [00:01:01]. The speaker’s personal view is that any numerical construct helpful in the real world should be considered “real” [00:04:37].
Poll results showed a significant portion of the audience was comfortable with the square root of negative 1, but many rejected infinity as being “real” [00:14:29]. Historically, René Descartes coined the term “imaginary” for these numbers derisively, implying they were not serious mathematics, yet the term stuck [00:10:29].
Defining Imaginary Numbers and Their Geometric Home
The starting point for complex numbers is to assume the existence of a number i such that i² = -1 [00:06:43]. This concept initially seems strange, as squaring any real number (positive or negative) results in a positive number [00:08:46].
To give i a “home”, complex numbers are proposed to be two-dimensional [00:08:37]. Instead of just the real number line, i lives in a “different dimension,” perpendicularly to the real axis [00:10:53]. This extends the number system into a plane, where i is one unit perpendicularly above the real number line [00:11:19]. This geometric representation of complex numbers is similar to vectors, where numbers can be scaled (e.g., negative 2i) [00:08:33].
Operations with Complex Numbers
Addition
Addition of complex numbers is straightforward, similar to adding vectors [00:12:02]. For example, adding two complex numbers involves summing their real parts and their imaginary parts separately [00:17:58]. If 4 (real) and 2i (imaginary) are added to -2 (real) and i (imaginary), the sum is 2 + 3i [00:17:58].
Multiplication and Rotation
Where things become interesting is in the multiplication of complex numbers [00:18:32]. Unlike vectors in a 2D plane, complex numbers have a well-defined multiplication operation [00:18:32].
Multiplying by i has the action of rotating things by 90 degrees [00:21:53]. For a point (3, 2) (representing 3 + 2i), a 90-degree counterclockwise rotation results in (-2, 3) (representing -2 + 3i) [00:19:59].
Mechanistically, when multiplying i by 3 + 2i:
i * (3 + 2i) = 3i + 2i² [00:20:49]
Since i² = -1, the expression becomes:
3i + 2(-1) = -2 + 3i [00:20:55]
This algebraic result corresponds precisely to a 90-degree rotation of the point (3, 2) [00:21:04]. The coordinates are swapped (3, 2) to (2, 3), and the first coordinate is multiplied by -1 [00:21:21].
This ability of complex numbers to encode rotations provides a computational mechanism for various types of rotations, not just 90 degrees [00:22:15]. This property is crucial for understanding how complex numbers transform a 2D plane, keeping lines parallel, evenly spaced, and perpendicular [00:26:08].
Connection to Trigonometry
The lesson will eventually build towards understanding two identities from trigonometry using complex numbers [00:05:10]. While trigonometric identities like the cosine of the sum of two angles are “hard to remember” and seem unrelated to the square root of negative one [00:06:00], complex numbers can simplify them [00:06:59]. This demonstrates how complex numbers can make other pieces of math more understandable and useful [00:07:22].