Introduction To IOTA

Go back to  'Complex Numbers'

The concept of complex numbers is fundamentally important to many sciences and engineering branches and is a powerful tool to solve a diverse array of problems. However, this concept is a radical departure from the concept of numbers that students generally have in mind before they first encounter complex numbers. Therefore, this section is devoted to an intuitive introduction to complex numbers, why they are required at all, and what their actual significance is. For this section in particular, you are urged to really think and reflect deeply on the statements and concepts that will be presented here.

You must all be familiar by now with the various number systems, starting from the natural numbers, to the real numbers.

\[\begin{align}\text{Natural Numbers} \qquad  \mathbb{N} \quad &:  \qquad  1, 2, 3 \dots \dots \infty\\\text{Whole numbers} \quad   \;\rm{W}\;\;\; &: \qquad     0, 1, 2, 3 \dots \dots \infty\\
\text{Integers}   \qquad\mathbb{Z} \quad &: \qquad   – 3, –2, –1, 0, 1, 2, 3 \dots \dots\\  
\text{Rational numbers }\; \;Q\quad &: \qquad    \text{Numbers of the form} \frac{p}{q},\,\,q \ne 0,\,\,\,\,\,p,q \in \mathbb{Z}\\
\text{Real numbers}      \quad\mathbb{R}  \quad &:   \qquad  \text {{Rational numbers}}  \cup   \text{{irrational numbers}}\end{align}\]

You also know that \(\mathbb{N} \subset W \subset \mathbb{Z} \subset Q\,\, \subset \,\,\mathbb{R}\) .

For each set in this sequence (except \(\mathbb{R}\) ), there are mathematical equations which have no solution in that set, but do have a solution in the next higher set. For example

\[\begin{align}&\Rightarrow \quad \,\,x + 3 = 3             \qquad\qquad\text{Has no solution in } \mathbb{N}\\
 &\quad\qquad \qquad\qquad\qquad\qquad\text{Has a solution in W }{x = 0}\\\\
&\Rightarrow \qquad x + 3 = 2  \!\!\!\!\qquad\qquad\text{Has no solution in W}\\
&\quad\qquad \qquad\qquad\qquad\qquad\text{Has a solution in }\mathbb{Z}   {x = –1}\\\\
&\Rightarrow \qquad 3x = 2\qquad\qquad \;\;\text{Has no solution in  }\mathbb{Z}\\
&\quad\qquad \qquad\qquad\qquad\qquad\text{Has a solution in Q} \left\{ {x = \frac{2}{3}} \right\}\\\\
&\Rightarrow\qquad {x^2} = 2                \;\;  \qquad\qquad\text{Has no solution in Q}\\
&\quad\qquad \qquad\qquad\qquad\qquad\text{Has a solution in } \mathbb{R}\left\{ {x = \sqrt 2 } \right\}\end{align}\]

Therefore, we see that each successive set in this sequence is an ‘improvement’ or extension over the previous set, because it has more scope than its predecessor in terms of solvability of equations.

The question that now arises is, can \(\mathbb{R}\)  be ‘improved’ or extended? That is, are there equations which have no solutions in \(\mathbb{R}\) ? Yes; consider the following equation:

\[{x^2} + 1 = 0\]

This has obviously no solution in \(\mathbb{R}\)  since the LHS is always greater than or equal to 1. This means that \(\mathbb{R}\) is ‘insufficient’, for there do exist equations having no solution in \(\mathbb{R}\) . So, for example, from the equation above, there is no real number whose square is –1.

However, being mathematicians, nothing stops us from defining a number whose square is –1. Though such a number ‘does not exist’ according to us, let us still go ahead and define such a number; we will (due to the convention followed) call this self-defined number as ‘iota’, written as i.  And since i ‘does not exist’, let us call it an imaginary number.

\[{i^2} + 1 = 0\\ \text{or}\qquad {i^2} =  - 1\]

We can now say with satisfaction that even the equation \({x^2} + 1 = 0\) has a solution, namely \(i\), though it is not real  \(\left( {i \notin \mathbb{R}} \right)\). Now, the next question that arises is, is such a definition (which we seemed to have made out of our own free will) justified? If yes, what significance do we attach to \(i\)? Is it even useful?  Will we be able to do useful mathematics with \(i\) or is it just a useless mathematical construction on our part? We can associate real numbers with geometric lengths. What do we associate \(i\) with? The answers to these questions will soon become clear. Let us first try to give a graphical (or geometrical) significance to \( i.\)