Did You Know: The History of Complex Numbers
Did You Know: The History of Complex Numbers

Today I am gonna write about History of Imaginary Number System and Complex numbers. At the ancient time it was very difficult to imagine a number which is in fact imaginary, but when time grows, mathematicians felt a need to incorporate an imaginary number system for the sake of solving many problems. How how it was started – I am gonna write about this.

As we all probably know, (I am quoting Wikipedia here) A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit, which satisfies the equation i2 = −1. In this expression, a is the real part and b is the imaginary part of the complex number. Complex numbers extend the concept of the one-dimensional number line to the two-dimensional complex plane by using the horizontal axis for the real part and the vertical axis for the imaginary part. The complex number a + bi can be identified with the point (a, b) in the complex plane.

Now, it is of course that the creation of i was not instant or accidental. Rather it took centuries to convince some mathematicians to accept certain number system. i was created due to the fact that mathematicians simply needed it. While solving some problems of 2nd, 3rd, 4th degree polynomials often needed such numbers which does not exist in real numbers. Problems such as solving x2+1=0 or finding square roots of negative numbers were the source of problem and were thought to be impossible. However some people thought that a new number should be created to solve such kind of problems and hence the result is ‘i‘.

Today, i is very useful to the world. Engineers use it to study stresses on beams and to study resonance. Complex numbers help us study the flow of fluid around objects, such as water around a pipe. They are used in electric circuits, and help in transmitting radio waves. We often use imaginary numbers to solve polynomials.

The very first mention of people trying to use imaginary numbers dates all the way back to the 1st century. In 50 A.D., Heron of Alexandria studied the volume of an impossible section of a pyramid. What made it impossible was when he had to take√81-114. However, he deemed this impossible, and soon gave up. For a very long time, thoughts of solving such problems were silent. Once negative number system was arrived into the picture, mathematicians tried to find a number that, when squared, could equal a negative one. Not finding an answer, they gave up. In the 1500’s, some speculation about square roots of negative numbers was brought back. Formulas for solving 3rd and 4th degree polynomial equations were discovered, and people realized that some work with square roots of negative numbers would occasionally be required. Naturally, they didn’t want to work with that, so they usually didn’t. Finally, in 1545, the first major work with imaginary numbers took place.

In 1545, Gerolamo Cardano wrote a book titled Ars Magna. He solved the equation x(10-x)=40, finding the answer to be 5 ± √-15. Although he found that this was the answer, he himself greatly disliked imaginary numbers. He said that work with them would be, “as subtle as it would be useless”, and referred to working with them as “mental torture.” For a while, most people agreed with him. Later, in 1637, Rene Descartes came up with the standard form for complex numbers, which is a+bi. However, he didn’t like complex numbers either. He assumed that if they were involved, you couldn’t solve the problem. Lastly, he came up with the term “imaginary”, although he meant it to be negative. Issac Newton agreed with Descartes, and Albert Girad even went as far as to call these, “solutions impossible”. Although these people didn’t enjoy the thought of imaginary numbers, they couldn’t stop other mathematicians from believing that i might exist.

Rafael Bombelli was a firm believer in complex numbers. He helped introduce them, but since he didn’t really know what to do with them, he mostly wasn’t believed. He did understand that i × i should equal -1, and that –i times i should equal one. Most people did not believe this fact either. Lastly, he did have what people called a “wild idea”- the idea that you could use imaginary numbers to get the real answers. Today, this is known as conjugation. Although Bombelli himself did not have much of an impact at the time, he helped lead the way for imaginary numbers.

The first person who considered to put imaginary number system into graph was John Wallis. In 1685, he said that a complex number was just a point on a plane, but he was ignored. More than a century later, Caspar Wessel published a paper showing how to represent complex numbers in a plane, but was also ignored. In 1777, Euler made the symbol i stand for √-1, which made it a little easier to understand. In 1804, Abbe Buee thought about John Wallis’s idea about graphing imaginary numbers, and agreed with him. In 1806, Jean Robert Argand wrote how to plot them in a plane, and today the plane is called the Argand diagram. In 1831, Carl Friedrich Gauss made Argand’s idea popular, and introduced it to many people. In addition, Gauss took Descartes’ a+bi notation, and called this a complex number. It took all these people working together to get the world, for the most part, to accept complex numbers.

Mathematicians kept working to make sure that imaginary and complex numbers were understood. In 1833, William Rowan Hamilton expressed complex numbers as pairs of real numbers (such as 4+3i being expresses as (4,3)), making them less confusing and even more believable. After this, many people, such as Karl Weierstrass, Hermann Schwarz, Richard Dedekind, Otto Holder, Henri Poincare, Eduard Study, and Sir Frank Macfarlane Burnet all studied the general theory of complex numbers. Augustin Louis Cauchy and Niels Henrik Able made a general theory about complex numbers accepted. August Mobius made many notes about how to apply complex numbers in geometry. All of these mathematicians helped the world better understand complex numbers, and how they are useful.

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Scientific History Blog Writer • Art enthusiast and Illustrator • Amateur Photographer • Biker and Hiker • Beer Enthusiast • Electrical Engineer • Chicago

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