Discovering Imaginary Numbers in Everyday Life: The Popularization of Complex Numbers

The book is inspired by social media users' exceptional laudation for the author's work

on redeveloping the foundation of complex numbers from the ground up through first

principles. Nowadays complex numbers formed from imaginary numbers have

widespread applications in mathematics, physics and engineering. But many learners

still have difficulties accepting the existence of the imaginary unit 𝑖 because the

property 𝑖² = -1 is used to define 𝑖 while 𝑖 itself is also used to define the property

with 𝑖 remaining undefined despite the abstract representation in the complex plane.

Complex numbers were first introduced so that all polynomials would have solutions.

Traditionally, a top-down approach has been taken in the development of complex

number theory based on 𝑖² = -1. By contrast, the author's work uses a bottom-up

approach in which rotational numbers are introduced to rotate physical vectors through

multiplication. As a result, the re-creation of the imaginary unit, complex numbers and

Euler's formula as well as the discovery of a rotational number set can then be achieved

in the reality without relying on 𝑖² = -1. This removes learners' skepticism about

complex numbers by redefining the imaginary unit as a self-evident presence in daily

life, and makes the complex plane interchangeable with the familiar x-y plane, which

has a direct connection to the real world. The work opens up the opportunity to explore,

extend and share the newly gained comprehensive understanding about complex

numbers for the thorough demystification, popularization and empowerment to broader

educational levels including elementary 6th grade and up.

Like the real number set for 1D (one-dimensional) arithmetic, the existence of the

rotational number set gives rise to 2D arithmetic that greatly simplifies the learning by

allowing elementary students to use the familiar position and rotation concepts of the

physical world without the deep abstract knowledge behind complex numbers. The

contemporary ubiquitous use of touch-screens on mobile phones indicates that selecting

a point on a surface can actually be more primitive and intuitive than selecting a point

on a line, making the 2D arithmetic more appealing. The goal of the book is to provide

an essential read for anyone wanting to keep abreast of the latest understanding of

complex numbers for inspirational learning and teaching.