Exploring Beyond ∞

Copyright © Rahul Shankar Bharadwaj 2025 

Chapter 2

Exploring Beyond ∞

Number line having negative & positive directions in one dimension is illustrated below. Let us call it the ‘original number line’. It has been constructed by plotting the values of f(x) = x in one dimension, where x is a real number.

Figure 2.1

Now let us reconstruct the number line to incorporate more directions or dimensions by plotting the values of f(x) = 1/√(x) , where x is a real number appearing on the original number line.

Starting from extreme right on original number line to input values of ‘x’: -

1/√(+∞) = ±0 gives coincidental point 0, since 0 has no direction.

Figure 2.2

Further as we move left on original number line to input values of ‘x’ and keep extrapolating: -

 1/√(+100) = ±0.1, giving point 0.1 on either side of 0.

Figure 2.3

1/√(+1) = ±1, giving point 1 on either side of 0.

Figure 2.4

1/√(+0.01) = ±10, giving point 10 on either side of 0.

Figure 2.5

1/√(0) = ±∞, giving point ∞ on either side of 0.

Figure 2.6

Moving further and inputting negative values: -

1/√(-0.01) = ±10i , giving point 10i on either side of 0.

Figure 2.7

It is pertinent to note that we were plotting the positive roots of f(x) on the right side and negative roots of f(x) on the left side of 0.

The positive root of 1/√(-0.01) = 1/(+0.1i) = 10/i = (10× i)/(i × i) = -10i, thus -10i has been plotted to the right side of 0.

Similarly, the negative root of 1/√(-0.01) = 1/(-0.1i) = +10i has been plotted to the left side of the 0.

Here one more interesting fact comes to light; 0 can have both positive and negative signs, therefore,

1/√(0) = 1/√(-0)

Taking positive square roots of both sides,

+∞ = +∞/i 

or +∞ = -∞i

Similarly, equating the negative roots of both sides

-∞ = -∞/i

or -∞ = +∞i

Adding this information to the number line,

Figure 2.8

Continuing with the extrapolation: -

1/√(-1) = ±1i , giving point 1i on either side of 0.

Figure 2.9

1/√(-100) = ±0.1i , giving point 0.1i on either side of 0.

Figure 2.10

1/√(-∞) = ±0i , but 0×i=0 thus giving point 0 on either side of 0.

Figure 2.11

Combining zeros at a common point we can redraw the number line as a loop.

Figure 2.12

It may be noted that after ∞ the numbers tend to reduce back to zero albeit with ‘i’. This return path constitutes a new dimension which may be called the ‘i’ dimension. These two dimensions cross over at three points viz. 0, +∞ (or -∞i) and -∞ (or +∞i).

Conventional Imaginary Number Line

The conventional imaginary number line lies perpendicular to the original number line. Here multiplication with ‘i’ is considered as a 90° rotation and two times ‘i’ becomes 180° rotation or reversal of sign.

The conventional imaginary number line system is quite like the number loop except for the fact that it does not show any relationship between the ∞ points on respective axes. However, if we try to extend this correlation on number line system then it begins to resemble the number loop.

Figure 2.13

Thus, it can be deduced that the multiplication with ‘i’ does not rotate a quantity by 90° in the two-dimensional plane but rotates it in an altogether different dimension. Multiplying second time brings back the quantity in original dimension but with inverted sign or direction.

However, it is not possible to plot complex numbers on the number loop because the real numbers component and imaginary numbers component do not lie in the same two-dimensional plane but in two different planes which exist in different realms, one the real dimension and other the ‘i’ dimension. Though, the existing conventional imaginary line system is working accurately to carry out calculations involving complex numbers because it exactly resembles the number loop stretched to ∞.

We will try to uncover the meaning and implications of the ‘i’ dimension in the subsequent chapters.

        Chapter 3: Connotation of 'i'

Copyright © Rahul Shankar Bharadwaj 2025