From: mk_thisisit
Complex numbers are a fundamental concept in mathematics, introduced to enable operations like multiplication on points in a plane, which was not previously possible through traditional arithmetic. They are considered an elegant solution that broadens the scope of mathematical operations and provides deep insights into various fields of science [00:00:26].
The Origin of Complex Numbers
Historically, mathematicians initially considered the idea of adding points on a plane to be nonsensical [00:10:39]. However, the introduction of the coordinate system by Descartes enabled the addition of points by summing their respective coordinates [00:10:46]. For example, two points (a, b) and (c, d) could be added to yield (a+c, b+d) [00:10:57].
The desire to extend this capability to multiplication led to a new notation for points: (a, b) was re-expressed as a + ib (or a + *i*b using the speaker’s notation a + and b) [00:11:52]. When attempting to multiply two such expressions, for example, (a + ib) * (c + id), an i² term naturally arose [00:13:14]. This i² initially had no meaning [00:13:17].
The Imaginary Unit ‘i’
To define multiplication for points on a plane, mathematicians introduced the “imaginary unit,” denoted as i [00:13:53]. This unit is defined by the property that i² = -1 [00:13:39]. While there is no real number that, when squared, results in -1, mathematicians “invented” such a number to solve this problem, thereby enabling the multiplication of points on the plane [00:13:43].
Role in Euler’s Formula
Euler’s formula, e^(ix) = cos(x) + i sin(x), is considered one of the most elegant and beautiful formulas in mathematics [00:03:42]. It uniquely combines five fundamental mathematical constants: 0, 1, π, e, and i [00:05:16]. These numbers originate from diverse areas of mathematics, including trigonometry, complex numbers, and mathematical analysis (differential calculus) [00:05:32].
The formula demonstrates how the imaginary unit i links the exponential function e^x with trigonometric functions (sine and cosine) [00:19:54]. Specifically, it shows that raising e to the power of ix (where x is a real number) results in a complex number whose real part is cos(x) and imaginary part is sin(x). This can be visualized as a point on a unit circle in the complex plane, where x represents the arc length from the positive x-axis [00:08:05].
For example, when x = π, Euler’s formula becomes e^(iπ) = cos(π) + i sin(π) [00:20:08]. Since cos(π) = -1 and sin(π) = 0 [00:09:53], this simplifies to e^(iπ) = -1 + i(0), or e^(iπ) + 1 = 0 [00:04:04], a concise expression highlighting the deep connections between these fundamental constants [00:04:25].