The following is the Introduction section of the book I have authored. The book is based on my work in Complex Numbers published in “International Journal of Scientific and Engineering Research” in Issue 12 of Volume 3, December 2012.
‘2’ happens to be a very special number for mankind. It is evident in both mythology and philosophy that duality has always been a part of mankind. Good and evil, yin and yang, dark and light, high and low, hot and cold and so on. The dualisms of physical – spiritual states and life-death are probably the most debated of all time. Such a topic isn’t new to science either, the wave-particle duality being one of the most famous and intriguing concept. Although mythological duality and scientific duality aren’t quite the same, the point to be taken is that things present in two’s aren’t a wholly novel idea.
Duality is known to exist in physics but such a concept has never been theorized in mathematics. According to the basic language of mathematics, a variable or any entity can correspond to only one value or one value among two or more possible values. Mathematics has no provision to incorporate the idea of a variable corresponding to more than one value. This book is targeted entirely on this – establishing the fact that duality exists in mathematics. But what does that have to do with complex numbers? Continue reading, it will soon make sense. The terminology, duality, is rather misleading. It is first better to tell what the duality, dealt in this book, is not. Duality, in mathematics also refers to the transformation of concepts and theorems into another concept or theorem. This book doesn’t deal with those kinds of duality neither does it deal with the square roots of positive numbers (± root). The book deals with special variables that can correspond to two values, which is what is mean by ‘duality’ or ‘dual nature’. Those two words “two values” are almost forbidden in the world of mathematics. To excavate this concept of duality, we will try to find the value of a few numbers that technically (up to current mathematical standards) have no real value. Those special numbers are known better by the name – complex numbers.
Complex numbers have no real value at all. The rest of the book deals with the very same – trying to give a complex number a real value and thereby understand duality. If none of this makes any sense carry on, it will all fit in, at the end.
The quest to find a value for the imaginary numbers has a bit of history behind it but it is a fact that, since its inception, complex numbers remain without a proper mathematical value i.e. they cannot be designated a position on the number line. No better why, are they termed imaginary numbers. A complex number is a number which can be put in the form a + i b, where a and b are real numbers and i is called the imaginary unit. Mathematically i = √−1. Italian mathematician Gerolamo Cardano, arguably, introduced complex numbers and he called them “fictitious”, during his attempts to find solutions to cubic equations in the 16th century. Complex numbers are still imaginary numbers, despite the huge advancements in mathematics. Many mathematicians in the past years have tried to give a quantitative value to complex numbers but none could give ‘i’ a comprehensive value. Complex numbers just don’t have any real value no matter what we try.
For e.g. we consider the number ‘pi’ which has a value 3.14… and the decimals keep going. Though this number’s decimals aren’t going to stop, we have a specific idea about where it would be on the number line. But if we consider a complex number 3 + i 5, there is absolutely no way to have any idea of where that number would be on the number line.
A solution for the real value nature of complex numbers has been found in the current research work and a way is there, by which we can assign real values to a complex number.
“All such expressions as -1, -2, etc., are consequently impossible or imaginary numbers, since they represent roots of negative quantities; and of such numbers we may truly assert that they are neither nothing, nor greater than nothing, nor less than nothing, which necessarily constitutes them imaginary or impossible.” – Leonhard Euler, 1770
The bottom-line is, the book’s central core is a hypothesis that attempts to give complex numbers a real (mathematical) value, real in a sense of not only comprehension but also quantization. The current mathematical work investigates how a complex number would behave in terms of real numbers thereby finding a way to give complex numbers a real value. By doing so, it unexpectedly brings the property of duality to mathematics.
The hypothesis may seem unacceptable but its mathematical and physical significances, discussed in the book, are vindictive of such an answer. The very fact that a complex number can be given a real value can prove to be useful especially in the field of complex analysis. Complex numbers, if expressed in terms of real values, could prove to be advantageous for the future of mathematics.
To catch the complete book, check the ISBN: 978-3-659-37757-0 titled “Complex Numbers: A Real Approach”