The cause for i (imaginary unit)

02Nov07

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 numbers 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 http://en.wikipedia.org/wiki/Imaginary_unit

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 cause for i (imaginary unit)”

1. Well, 0 is a natural number now, isn’t it?

Yes, you’re right because you wrote “FIRST there were natural numbers, I mean 1,2,3… and so up”

2. 2 masteranza

If think that someday we will figure it out.

3. A little bug – typing mismatch:
Then it was time for integers, rational number and finally real numbers
(numbers need to be in plural).

If talking about the essence, it is very nicely explained about that i thing. Enjoyable article.