“You are travelling on a highway. You are afraid your car might run out of petrol. You only have a bank note of 500 Rupees in your pocket that you can use for a refill. At the same time, the fuel tank in your car was recently repaired but may start to leak again (although very slowly). Even though you don’t have a choice, do you know it would be better to lose the tank to leakage than to lose the banknote? Why?”
For the absolute naïve, let’s start off with the definition of a discrete system:
Discrete system
A discrete system is a system with a countable number of states.
What does this mean? It means that there is always a limit to the accuracy of a discrete system. Discrete is synonymous with digital. Digital devices also have a limit to their accuracy. They have a preset frequency. A digital device (like a microcontroller or a microprocessor) with a frequency of 16 MHz will watch out for input after every (1/16) microsecond. NOT in between this interval. A digital clock shows the time 10:08:51 at one point in time and the next second, it shows 10:08:52. It does not show the milliseconds, microseconds, nanoseconds, picoseconds and the femtoseconds that pass in between.
Continuous system
Now moving on to a continuous system; it simply is a system with an uncountable number of states. Continuous is synonymous with analog. You must have seen analog clocks with the seconds hand moving continuously (without “jerks”). If you think about it, the hand actually DOES pass through the points which are equivalents of femtoseconds and picoseconds. An analog system has NO limit to its accuracy. You cannot count the states of a continuous system, they are infinite.
Note: Don’t think that by adding extra accuracy to a digital clock you can make it a continuous system. Even if it shows femtoseconds, there is still a limit to its accuracy. And that limit is one femtosecond.
People often use the terminology discrete and continuous variables. They think that the progressive integers 1 2 3 4 5 and 6 are part of a discrete system since the limit to accuracy is 1 unit. On the other hand, irrational numbers like 3.1416 and 2.71828 are continuous since they are more accurate. In fact, both are discrete. You cannot really express the continuous form of ANY number, since it would be infinitely long.
The continuous count from 0 to 1 would probably be like
0
0.1
0.11
0.111
0.1111
0.11111
0.111111
0.1111111
0.11111111…….we don’t seem to be getting there yet….
And yes, as you might have guessed, getting from 0 to 0.1 itself requires an infinite number of states.
Now let’s get back to the original topic.
From the driver’s point of view, the note being lost indicates that he won’t be able to get a refill at all in case he loses the note. Let’s represent it mathematically:
Let T denote the point in time when the driver loses his note. Immediately before time T, he can get a refill. But immediately after time T, he cannot get a refill. It’s not like after time T, he can get half the refill, or 5 seconds after T, he can get one quarter of a refill. The note being lost means that he won’t get a refill at all. DISCRETE variation. If 1 means a full refill and 0 means no refill, then either there is a 0 or a 1, not anything in between.
Now suppose we have two possibilities at time T and T0
In situation A:
* The driver does not lose his money at time T (and hence can get his car refueled) but loses his tank to leakage at T0.
In situation B:
* The driver loses his money at time T (and hence cannot get his car refueled) but does not lose his tank to leakage at T0.
Let’s plot a graph of the fuel in the driver’s car against the time in situation A. (click here for graph)
The blue segment shows the normal fuel consumption (the negative gradient is less significant). The purple segment is the car being refueled. The mauve segment is the car moving with fuel leakage (the fuel loss is greater and has a more significant gradient.
The area under the graph at any point shows the total fuel in the car from time t=0. This is directly proportional to the distance the driver can travel.
Now let’s move on to situation B. The driver could not get a refill at time T but did not even start losing his fuel to the leakage after time T0. This is pretty much a straight graph. (click here for graph)
Let’s put both graphs on the same axis for your comparison. (click here for graph).
Can you compare the two graphs? The area shaded represents the difference in the potential fuel that the car can hold in each situation, which of course is in favor of situation A. This area is subject to two discrete moments in time (at T and T0). This area shows the discrete lead of situation A over B minus the continuous lag of situation A behind B (which in this case is zero).
The idea of discrete and continuous systems is found in other areas as well. Suppose you had money to buy a new cell phone. You were afraid that the cell phone might get damaged and you should buy it later. At the same time, you were afraid that the money might get stolen. In the latter, there is a tradeoff between a 0 and a 1. Either you have a cell phone at the time when you can keep it safe, or you don’t get a cell phone at all (since your money is stolen). DISCRETE system. In the former, you can get a cell phone but its condition can fall slightly below the 1 (which means the best). Here, you don’t have a tradeoff. You can get both states at one time. CONTINUOUS system. You can get a cell phone; but one that can get slightly damaged with time.
Discrete and continuous variation is also found in the human genes. Discrete variation is simply the survival of the better gene. Biologically, all the better genes being used will cause evolution, but on the molecular level, the genes are all the same from either of the parents. In continuous variation, you talk about the co dominance factor. The two genes will both be used and result in a completely new gene. If one is a 0 and the other is a 1, then you can get anything ranging from 0 < x < 1.
There is a multitude of discrete and continuous systems out there. All you need is a little bit of observation.
