Why Do We Call i^1sqrt{-1} an Imaginary Number?

Mathematics often requires expanding the number system to accommodate new types of numbers, such as the familiar real numbers. When this expansion was necessary to include the square roots of negative numbers, a new category of imaginary numbers was introduced. The symbol #39;i#39; was chosen to represent the unit iota, derived from the initial letter of the word imaginary.

Historical Context

The term imaginary was introduced by the famous mathematician Leonhard Euler, though the symbol #39;i#39; was adopted for this purpose. Before this, mathematicians such as Carl Friedrich Gauss had used the term but not the symbol #39;i#39;. Gauss himself discussed the issue of terminology, suggesting that it would be more appropriate to use terms like direct, inverse, and lateral to replace the existing terms.

Gauss once stated:

If we hadn't named (1, -1, sqrt{-1}) as the positive, negative, imaginary or even impossible unity, and had used instead say direct, inverse or lateral unity, then we wouldn't have to speak about such an obscurity.

However, the term imaginary has stuck and remains in widespread use today.

Pioneering Imaginary Numbers

My friend, all those numbers inside the square root with a negative sign are known as complex numbers. A complex number is a combination of a real part and an imaginary part. For simplicity, let's consider the formula:

(-1^1/2 x)

Squaring both sides, we get:

(x^2 -1)

Now, think about a number whose multiplication with itself gives a negative number. This is impossible in the realm of real numbers, so we define the solution as an imaginary number, which we denote as fst#39; #8730; -1, representing iota, where

Iota is simply a concept: it is akin to a phasor or a component of a vector. Geometrically, we can visualize it as follows:

Geometric Interpretation

Consider the real line, a straight line with numbers on it. To represent a complex number, it must be orthogonal to the real line, thus not having a real component.

Therefore, a complex number (x yi)

Where (y) is the imaginary part and (x) is the real part.

Geometric Representation

In a coordinate system, the real axis represents the horizontal axis or X axis. Real numbers correspond to points on this axis. Specifically, numbers towards the right of zero are positive, and those to the left are negative.

The numbers like (i), (i sqrt{-1}), (5i), (-2i), etc., are all imaginary numbers. These do not correspond to any point on the real axis. Each imaginary number corresponds to a point on the imaginary axis, which is the vertical line, conventionally denoted as the Y axis, perpendicular to the X axis and passing...

These concepts are fundamental in complex analysis and applied mathematics, providing a richer framework for understanding and solving a wide range of mathematical and engineering problems.