Copyright © Rahul Shankar Bharadwaj 2025
Chapter 1
Understanding
∞
Infinity (∞) is the largest and the smallest
numerical quantity depending on the direction or sign attached. On number line it
appears at the extremes in each direction.
Negative and positive signs or
directions are only conventions, so what is negative for one observer may be
positive for another depending on their perspective. For example, when two cars
are moving in opposite direction at speed ‘s’ each, both drivers feel
that they are moving in positive direction with speed ‘+s’ and the other
one is moving in negative direction with speed ‘-s’.
Thus, there is no higher side on a
number line. The absolute numerical value of ∞ on both sides is equal, only their
directions are different as per our convention.
Here, a query may occur that, can ∞ have
an absolute numerical value? When we see the number line it appears that this
value can never be ascertained. However, the numerical value of ∞ was
discovered inadvertently by Aryabhata and Brahmagupta when they discovered and
analysed the use and properties of 0 (zero). The reciprocal of 0 (zero) is the absolute value of ∞, thus, ∞ = 1/0
Hence, the numerical value of ∞ is not
something alien and ∞ very much exists around us. To understand the numerical
value of ∞ further, let us perform a thought experiment.
Assume a car at point ‘O’ moving
towards another point ‘P’, 100 kms away as shown in the figure 1.2.
In table 1.1 we will tabulate the time
take by car to reach ‘P’, when it travels at various constant speeds from ‘O’
to ‘P’. Henceforth, also whenever the term ‘speed’ is used this book, it may
please be understood as constant speed.
|
Speed (kms/hr) |
Time (hrs) to reach P |
|
1000 |
0.1 |
|
100 |
1 |
|
10 |
10 |
|
1 |
100 |
|
0.1 |
1000 |
|
0 |
|
Table 1.1
We can see that at 1000 kms/hr, the car
will take 0.1 hr to reach ‘P’. For slower speeds, the time taken by car to reach
‘P’ increases and as the speed tends to become 0 (zero), the time taken by car
to reach ‘P’ tends to become ∞. When the speed of the car is absolute 0 (zero),
the time taken by car to reach ‘P’ = 100/0
Now let us try to understand the
quantity, ‘speed of an object’, increasing to infinity on a time vs space
diagram shown in figure 1.3.
Here
the ‘x’ axis represents distances and ‘y’ axis represents time. ‘0’ represents
the origin point and zero time. ‘A’, ‘B’ and ‘C’ are three points in space at
increasing distances in the same order from the origin point.
At zero speed an object will only move
along time axis and will stay at its original location in space. However, as
its speed increases, it tends to displace both on time axis as well as space
axis. As the speed increases further, the object covers larger distance on
space axis in shorter time. At infinite speed, no time or zero time should be
spent travelling between two locations. Thus, at a given time ‘t₁’, if an object could travel from the origin to point ‘C’ at
infinite speed, it will reach ‘C’ at time ‘t₁’ itself without spending any time in
travelling. However, while doing so it will also exist at point ‘A’ and ‘B’ at
the same time. Hence, this object will simultaneously exist at ‘0’, ‘A’, ‘B’
and ‘C’ at time ‘t₁’. Therefore, we can deduce that an object
travelling at infinite speed will exist at all locations at a given time.
Another interesting conclusion can be
that travel at speed greater than infinity may lead the object back in time i.e.
to the possibility of ‘time travel’. In the forthcoming chapter we will try to
find what can be greater than infinity or in other words what may lie beyond
infinity?
Chapter 2: Exploring Beyond Infinity
Copyright © Rahul Shankar Bharadwaj 2025
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