Connotation of ‘i’

Copyright © Rahul Shankar Bharadwaj 2025 

Chapter 3

Connotation of ‘i’

We have seen in last chapter that the number line does not end at ∞ but moves ahead into a new dimension denoted by i.

Let us try to understand the connotation of ‘i’ through a thought experiment of a car moving left to right as shown in the figure below.

Figure 3.1

The car is moving from left to right and is presently at point ‘O’ at a given point in time. Distance NO and OP are 100 kms each. For convention, distances and direction of motion from left to right are considered ‘positive’ and therefore right to left becomes ‘negative’. Therefore, NO is -100 kms and OP is +100 kms from the car. We will now make a table of speed vs time for the car’s motion at positive and negative speeds.

Speed (kms/hr)	Time (hrs) to reach N	Time (hrs) to reach P
+∞	?	0
+1000	?	0.1
+100	?	1
+10	?	10
0	∞	∞
-10	10	?
-100	1	?
-1000	0.1	?
-∞	0	?

Table 3.1

The time values in direction of motions are easily discernible. It is also evident that for 0(zero) speed, ∞ time will be taken by car to reach points in either direction i.e. the car will never reach there.

However, what needs to be seen is at negative speeds, the car moves towards the left or backwards away from ‘P’. Therefore, the car will never cross point ‘P’ but is this motion same as the car’s motion at 0(zero) speed wherein it will take ∞ time to reach ‘P’? Ideally by extrapolating the fact, that the time to travel is increasing, it should take time greater than ∞ to reach ‘P’. Vice versa, for positive speeds, should we conclude that the car will take more than ∞ time to reach point ‘N’?

The positive and negative motion for moving towards ‘N’ and ‘P’ respectively are certainly not same as 0(zero) speed motion. Mathematics must have a way to quantify these motion variables.

The time taken to reach ‘P’ at negative speeds can be obtained by extrapolating this motion back in time. If the car was travelling at speed +100 kms/hr in the past as well, then it would have crossed ‘N’, 1 hr back in time. Similarly, while moving at -100 kms/hr, it would have crossed ‘P’, 1 hr back in time.

The same result can simply be obtained by using the formula time= distance/speed. Time taken by car to cross ‘N’ at +100 kms/hr = -100/+100 = -1hr and vice versa to cross ‘P’ at -100 kms/hr= +100/-100 = -1 hr

So, shall we conclude that these time values are simply -1 hr or is there something we are missing? What happens at +∞ kms/hr?

At +∞ kms/hr, the car is moving towards ‘P’. Using the formula, time taken to reach ‘P’= +100/+∞ = 0 hr. At the same instance, as per the formula, time taken by car to reach ‘N’ is −100/+∞ = 0 hr. Hence, the car reaches both points ‘P’ and ‘N’ in 0(zero) time while moving at +∞ kms/hr.

This means that, while the car is moving in one direction at ∞ speed (left or right of O), somehow it also reaches the points in opposite direction (right or left of O) simultaneously. This cannot be explained by the simple diagram in two-dimensional plane that we are using to describe the cars motion.

Getting back to the table, lets fill the missing time values by extrapolation, using corresponding values beyond +∞ on number loop and see if that helps: -

Speed (kms/hr)	Time (hrs) to reach N	Time (hrs) to reach P
+∞	0	0
+1000	-0.1i	0.1
+100	-1i	1
+10	-10i	10
0	-∞i/∞	∞/-∞i
-10	10	-10i
-100	1	-1i
-1000	0.1	-0.1i
-∞	0	0

Table 3.2

The newly filled values in the table tell us that time taken to reach ‘N’ when the car is travelling at +100 kms/hr is −1i hr. Vice versa time taken to reach ‘P’ at −100 kms/hr is also −1i hr. Here the numerical values confirm to the results obtained using the formula; time = distance/speed, albeit the ‘i’ is additionally attached.

However, the table confirms the observation from the formula-based calculations that at ∞ speed, the car exists at all points in all directions.

To delve deeper into the issue and to understand what the mathematics is trying to tell us with ‘i’, let’s plot the car’s movement on a time vs space diagram.

 

Figure 3.2

  Here we can see that when the car moves to ‘P’ in time ‘t₁’, and assuming it was moving with same speed in same direction in the past as well, it’s movement can be traced to ‘N’ back in time and it happens to be there at time ‘−t₁’. So, does it confirm our calculation from the formula, time = distance/speed?

Before reaching a verdict let us now assume that with a futuristic technology, the car can travel at a speed faster than ∞ which is in ‘i’ dimension on number loop. Now let’s plot its movement in time and space.

Figure 3.3

Assuming the car is travelling faster than ∞, it will reach point ‘P’ at time ‘−t₁’. Now, how does this ‘−t₁’ differ from ‘−t₁’ taken by car to be at ‘N’ in its traced back motion shown in figure 3.2?

The difference is that in the previous case the motion was not actual but traced back in time assuming that the car was travelling at the same speed and in the same direction all the time in the past. Whereas, in the present case the car is physically moving in ‘negative’ time zone.

Now mathematics tells us this difference in the quantities by distinguishing the actual motion to be in real dimension and the traced back motion in ‘i’ dimension, i.e. when it is an actual motion whether positive or negative then we get real numbers as solution, indicating real dimension and when the motion is presumably traced back in time and is not actual then we get imaginary numbers as solution, indicating the inversion or reversal of orientation of time denoted by ‘i’ dimension. Before proceeding ahead let’s extend the table 3.2 by extrapolating the speed of car and the corresponding time and distances into ‘i’ dimension.

Speed (kms/hr)	Time (hrs) to reach N	Time (hrs) to reach P
0	+∞/-∞i	∞i/-∞
-10i	+10i	-10
-100i	+1i	-1
-1000i	+0.1i	-0.1
+∞/-∞i	0	0
+1000	-0.1i	+0.1
+100	-1i	+1
+10	-10i	+10
0	-∞i/+∞	+∞/-∞i
-10	+10	-10i
-100	+1	-1i
-1000	+0.1	-0.1i
-∞/+∞i	0	0
+1000i	-0.1	+0.1i
+100i	-1	+1i
+10i	-10	+10i
0	-∞/+∞i	+∞i/-∞

Table 3.3

We will try to understand the inverted or reversed quantities with the example of the mirror image of the car.

Let us assume that in our thought experiment depicted in figure 3.1, we place a large mirror exactly at ‘O’ as shown in figure 3.4.

Figure 3.4

The mirror image of the car is shown in grey colour which will appear inside the mirror. When the car moves at +100 kms/hr towards ‘P’, its mirror image will seem to move towards ‘N’ at −100 kms/hr. The mathematics depicts that it is the speed of mirror image and not real car by attaching ‘i’ to −100 i.e. −100i kms/hr as shown in table 3.3 Further, the car physically reaches ‘P’ after +1 hr and its mirror image appears to reach ‘N’ at the same time as shown in figure 3.5.

Figure 3.5

Here, the time for car’s mirror image to reach ‘N’ is depicted as +1i hr in table 3.3 which again signifies that it pertains to the mirror image of the car. Another point to be observed is that on retracing the movement of mirror image of car at −100i, we find that it should be at ‘P’ at −1 hr without the ‘i’, which means that it is again inverted like a mirror image. Thus, it may be concluded that when the mathematics yields result for a quantity with ‘i’, it means the inverted or reversed quantity like a mirror image.

The values in the table 3.3 confirm to our interpretation of the car’s movement as discussed above. It may be noted that if speed beyond infinity i.e. speed in ‘i’ dimension can be achieved, the car may reach ‘P’ physically back in time. Similarly at speed lesser than −∞, the car will reach ‘N’ physically back in time.

We have studied the above examples for speed and time, so now let us generalize the connotation of ‘i’ for all the numerical quantities. If the numerical quantity appears with ‘i’ attached to it, then it means that the quantity is inverted or reversed in orientation.

We will try to further decipher the conundrum of orientation of quantities through the example of faster than light speed travel in the next chapter.

Chapter 4: Faster than 'c'

Copyright © Rahul Shankar Bharadwaj 2025