Through elementary school and usually most if not all of middle school, mathematics in America consists of real numbers and the basic mathematical functions we can apply to them (addition, subtraction, etc.). Then, algebra steps on stage, bringing with it the familiar numbers and symbols of elementary mathematics… and the alphabet.

In most cases in algebra, the letters stand for variables (e.g. the *x* in *y** = 2x*).* *However, two noteworthy letters do not – *e* and *i*. These letters serve not as place-holders for numbers, but as numbers themselves! For the present, we will concern ourselves with *i*, also called the imaginary number, and, by extension, complex numbers.

First, for those unfamiliar with the number *i* and complex numbers, definitions are in order.

Basically, *i* is the square root of -1, or the radical of -1.

*i* = √-1

Paired with a real number in the form *a* + *b*i, we have a complex number, where *a* is the real part and *b* is the complex part.

This post has two parts. In this first part, we will explore the origin of imaginary and complex numbers.

Mathematics begins with the geometric constructions of the ancient Greeks, the ever-familiar length, width, area, and volume the enabled construction. As a negative measurement could not possibly exist, none meddled in the idea of negative numbers except to deem their existence absurd and, according to particularly vehement opponents of the concept, a stain upon mathematics.

Mathematicians in the East had toyed for centuries with these funny negatives though, like European scholars, did not regard them as numbers in themselves. As late the 18th and even 19th centuries, mathematicians held this ancient opinion of negative numbers. Still, in the 15th century they came more into use, and calculations for 3rd- and 4th- degree polynomials revealed a need for negative radicals.

Thus the imaginary number, named as an insult to its ludicrousy, emerged at the the same time that mathematicians were just acclimating to negative numbers.

Italian intellectual (and gambler – mathematicians are human, too) Gerolamo Cardano, a Renaissance thinker who contributed to the emerging study of probability, published his *Ars Magna*, which included solutions to the cubic and quartic equations. One calculation of his yielded a negative radical which, though acknowledged as the answer, he did not accept as useful or meaningful mathematical constructs.

Some brave hearts after Cardano promoted the legitimacy of imaginary numbers. In his 1573 treatise *L’Algebra, *Rafael Bombelli (1526-72) established rules for the properties of imaginary numbers, namely the multiplication of them, such as that *i* squared equals -1. However, not until Leonhard Euler (1707-83), who dabbled much in complex numbers, and Carl Friedrich Gauss (1777-1855), who presented a clear investigation of complex numbers and of the connection between imaginary numbers and real numbers, did mathematicians really start to accept the concept.

Gauss popularized the graphical explanation of complex numbers that others before him, Jean Robert Argand in particular, discovered. Now call an Argand diagram, this graph plots complex numbers as points on a complex plane, where the x-axis represents real numbers and the y-axis represents imaginary numbers.

In Part 2, we will try to understand how the complex plane works and learn about the importance and application of complex numbers.

In China, negative numbers have been recognized almost two thousand years ago. I have personally seen old Chinese documents dated 2nd century to 3rd century where simultaneous equations with negative number solutions are solved.

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Fascinating! The Chinese did exceed Europeans in many scientific pursuits, at least for the first several centuries, so I’m not surprised that they accepted negative nunbers sooner and more readily.

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Do you know why?

The Chinese are good at coming up with new ideas. However, they don’t like to share them. A master may teach his student 70 percent (or even lower) of what he knows. When the student becomes a master, he’ll do the same, or even worse, to his student. This is the reason most of the ancient skills in China have dwindled over the years like martial arts, cooking, and fine arts.

Do you know about Sichuan’s face mask changing act (bian lian) used in Chinese opera? I think that this is one of the most secretive art forms of all time. Usually, only people native to Sichuan are taught. Even if they are taught, the student will never surpass the master.

I’ve read from old books that many centuries ago, some people could change their masks 40 to 50 times per minute and even 60 times per minute if they waned to showoff. However, the best masters today can only do 15 to 20 per minute, and those are rare.

Nowadays, China is slowly returning to the scientific scene. I’m not sure what would happen in the future but we’ll just have to find out I suppose…

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Now, the Chinese wants to lead even in the area of quantum computing.

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This is the first time I’m hearing of this innovation. I researched it a little this morning after reading your comment, SoundEagle, and am both surprised and not surprised that China has advanced ahead of Western countries. Plenty of funding and few government limitations (e.g. Congressional oversight) certainly helps.

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Hi Abigail,

Since you like mathematics, here are the weblinks to some excellent and thought-provoking articles for you to find in one of my detailed comments located in a special post published at https://soundeagle.wordpress.com/2016/08/17/do-plants-and-insects-coevolve/comment-page-1/#comment-5582

Please enjoy!

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Thank you, SoundEagle, for directing me to those links. I look forward to reading through them. It sounds like you’re a bit of a math/science enthusiast too!

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Not just a bit, but in a BIG way, as you may gauged from many of my posts.

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Oops! Please pardon my typo, as I meant “… may gauge …”.

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Please be informed that you might need to use a desktop or laptop computer with a large screen to view the rich multimedia contents available for heightening your multisensory enjoyment at SoundEagle’s websites, some of which could be too powerful and feature-rich for iPad, iPhone, tablet or other portable devices to handle properly or adequately.

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Advanced mathematics began appearing far earlier than most people realized. Luke Hodgkin’s History of Mathematics: From Mesopotamia to Modern Times is a good starting point for learning the history of mathematics.

Morris Kline’s Mathematical Thought from Ancient to Modern Times is also good.

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Thank you, Edmark, for the direction to the math resources! I’ve been wanting to find a good survey of mathematical history for a couple of months, but haven’t had the opportunity to search for one. I’ll be on the lookout for the two you mentioned.

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No problem.

Thos3 two books are just good introductory books. If you want suggestions for more specific references, like math history during the 18th century or history of topology during the 20th century, then you can ask me. I have done quite a few readings and research regarding this subject.

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