The reason for i (imaginary unit)


Since my first met with imaginary unit have past about 3 years. I’ve been thinking about it many times. First step forward in understanding i
was made when I acquainted with Schrödinger equation, it gave me some kind of experience with complex numbers.

Meantime I’ve heard a lot of voices like “What is it for if it doesn’t exist?!” or “Who needs this kind of abstraction?”, so I asked myself, is there a reason for i
to take place in Mathematics? The obvious answer could be “of course! It is used in solving cubic equations!”, but I think that, this problem has a depth Philosophical aspect.

Let’s start from the beginning. Mathematics (in spite of many opinions) was born from recognition the TRUE from FALSE, numbers weren’t before! That’s the beauty of Mathematics.

First there were natural numbers, I mean 1,2,3… and so up. Please notice than even those simple numbers are abstract! There are no physical things such as numbers. Although numbers were necessary in our physical world (it’s a nice exercise for a reader that have enough courage to invent something instead them). Then it was time for integers, rational number and finally real numbers (The last ones are extending of the others). Integers were defined to complete the operation of subtraction for example without them a question 3 - 5= comes without answer. Integers gave us -2 .
Would somebody say that it was a bad decision? Maybe only the ancient Greek. Rational numbers were defined to fulfill the operation of division, example: 6/2= have an obvious answer in integers 3 , but what with quotations like 6/5= ? Finally – real numbers, the product of extending the power operation, not every number could be represented as a rational number as we may think, example: \sqrt{2} Let’s suppose that there exist two integers p, q , such that p/q = \sqrt{2} if we now square both sides and multiply we will receive 2*q^2 =p^2 , but it’s contradiction, because p cannot have odd number of 2’s in her decomposition into prime numbers. That’s why \sqrt{2} \not \in Q , so we need real numbers to define them. If our intuition failed us once there’s no reason for her to stop doing it again. It was hard for mathematicians to break through with idea of i , they hadn’t seen a reason to define \sqrt{-1} , it wasn’t OK with their concept of placing all the numbers on an single axis and before Girolamo Cardano nobody dared to do that. It’s hard now to judge properly how fruity in new ideas was that step. If we can’t see or imagine something, it doesn’t mean that it does not exist! It all can look like wandering about in a dark room, but Mathematics is our great candle by which we explore the nature of things that surely were not created by human hand. Although one might assume that a further set of imaginary numbers need to be invented to account for the square root of i . However this is not necessary as it can be expressed as either of two complex numbers \sqrt{i} = \frac{1}{\sqrt{2}} (1 + i) . Everything pure and simple. For more information look at an article on Wikipedia

If anyone of you finds a bug in this article please let me know about it by leaving a comment or writing an e-mail.


3 Responses to “The reason for i (imaginary unit)”

  1. In my opinion, one of the most trivial yet amazing facts form the theory of complex numbers is the following

    \bf{Theorem}. The field \mathbb{C} cannot be given a simple order. (Under simple ordering we understand that if a\ne b, then either $latex ab$).

    \bf{Proof}. Suppose that such an ordering exists, then for i\in \mathbb{C} we should have either i>0 or i0 and (-i) * (-i) =-1>0 both false.

    If i < z_2 or z_1 < z_2 are nonsense unless z_1 and z_2 are real.
    Let’s think about this fact as a would-be philosopher. Can you really compare two human beings? Well, a person can be more attractive than others, someone is smarter, taller, kinder, etc. But that’s like comparing magnitudes of two complex numbers (certainly you can compare their moduli). So it seems that we can more or less liken people according to their income, education, occupation etc. but trying to simply order them have no meaning as totally ordering the field of complex numbers is impossible.

  2. 2 masteranza

    for some reasons LateX doesn’t work, even when I tried to correct it… but I hope we all got the author intentions. if i>0 and if i<0 we’ve got false…
    However I don’t find that fact so significant, maybe because I always have on mind their interpretation on a complex plane…

  3. 3 Rohedi

    Mr.Masteranza, recently Rohedi read at many math blogs that discussing about the proof by contradiction for example to prove that the square root of two is irrational number. My question, how to prove that 0.999…is not equal to 1. Thx for your help. Oh yeah don’t forget to click my address that inform smart technique for formulating the circumference and area of a circle.

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