Whenever we come across the word “infinity”, the first thing that comes to our mind is something extraordinarily huge, that always seems to catch up and exceed any measurable number no matter how large it is. Ever since its discovery by mathematician John Wallis in 1657 , it has been studied and explored by generations of mathematician and still continues to baffle us due to it inherent incomprehensible nature.
1. THE PARADOX OF QUANTIFYING INFINITY
Since, infinity seems to exceed any measurable number, we might call it extra-ordinarily large, humongous or assign a myriad of similar adjectives and yet fail to define it sufficiently, because it is simply the concept of “fully-formed endlessness” and that is not something very simple to grasp. This is because to call something large (irrespective of the adjective used) would ultimately, at some point beg the question “how much?” and that in itself would be a paradox.
The symbol “∞” very accurately suggests the said characteristic. i.e. a feeling of looping constantly (in a horizontal eight shape and meeting no end)
Some properties of infinity:
- It is fully formed, and definite (not in the sense of being” finite” but something that is fully formed). It is not something that is growing, but has grown. This is because something that is “growing” is larger than what it was the previous moment. It will then be arguable whether it is infinity in the first place because it will then,logically, hold a greater value in the future.
- It has a direction, indicate by positive or negative sign. Infinity can be positive or negative. We can say that -∞ will always be smaller than any finitely defined number.
- ∞ + ∞ = ∞
- ∞ x ∞ = ∞
- ∞ x k = ∞
- ∞ x (-k) = ∞
- ∞ x (-∞) = -∞
Some invalid operations:
- ∞ + (-∞)
- ∞ x k = ∞
- ∞ x (-k) = ∞
- It is a mathematical tool that can defined according to scope. i.e. the cardinal number 5 holds a definite value, yet when it comes to application it requires a dimension. ‘5 grams of rice’ makes better sense. The number in itself, despite being complete, fails to convey a certain idea, until its scope is defined. Such is the case with infinity. When we use “infinity” to describe a line, we have defined its scope and can say that it is a geometrical shape that extends in two directions opposite to each other “infinitely” or “indefinitely”.
2. CONVERGING INFINITE SERIES
Infinity doesn’t have to be vast and complex and out of the reaches of the human brain.In certain cases, such as the converging series, infinity can actually be resolved into something finite . The sentence “the sky has infinite number of stars” is sure to baffle us, but such is not the take when we define a converging infinite series. In this kind of series, with each step (increase in the term value n) the difference in the values of consecutive summation (Sn and Sn+1) decreases in a predictable fashion defined in the function (where S k is defined as the sum of all terms up to k). Before we get too caught up with its definition, let us understand its nature with an example. Consider two variables A and B, where
A = 0.9999999……. whose infinite series can be written as
A = 9/ (101) + 9/ (102) + 9/ (103) + 9/ (104) + 9/ (105) ……………………………….
and B =1
We can sufficiently say that A=B because we are able to draw the following set of conclusion.
- With successive addition of 9, the number gets closer and closer to 1 but also remains lesser for any finitely defined term value k no matter how large it is.
- There is no possible number of 9s adding which (after the decimal point) will let it exceed the value of 1.
Another way of thinking why the answer must be 1 is by the following method of elimination:
- Any real numbers lesser than 1 is sure to be exceeded by the converging series, at some point.
- Any real number greater than 1 is sure to remain greater irrespective of how large the term value of the converging series is chosen.
- The only number that falls in-between the two is 1 and therefore, it must be correct.
But doesn’t the above series give us a hunch that no matter how small, there will always remain a slight difference in the two values (0.9999…. and 1)?
The mathematical exactitude of the above series is counter intuitive, but true. This is because the fact that, “successive addition of 9s only brings it closer to 1, but never meets” holds true only when the series is stopped before infinity and calculated up to a finite number of term no matter how large it is (violating the very nature of the series that is supposed to extend upto infinity).
It does eventually after and endless number of terms, because as discussed earlier – infinity is fully formed. The infinite series of 9s after the decimal point already exists and is not growing (as discussed earlier)!
Thus, the value of a converging infinite series is finite because the difference between consecutive terms decreases with increasing term value k that finally collapses to zero at k=∞. Two converging series can therefore be compared with each other due to the finite value.
Another example of converging series is:
A = 1/2 + 1/4 + 1/8 + 1/16 +………….
=> A = 1/(21) + 1/(22) +1/(23) +1/(24) +1/(25) +……………
These kind of series exist as geometric progression and can be calculated using the formula
where a1 = first term of the series
r = geometric progression
Here the value of rn = 0
Let us understand the above problem through a diagram.
Let us consider a square cake to be consumed by an infinite number of individuals. With each step, someone comes in slices the cake consuming the half of what is left.Theoretically, there will always be a piece of cake remaining no matter how small for someone. Thinking conversely in terms of infinity, it would take an infinite number of individuals to consume the entire cake!
A practical example where converging series are used – The law of radioactive decay where with each half-life, the number of nuclei reduces to half of the previous states in the following fashion. Say, if it starts with 100 nuclei, it becomes 50 followed by 25,12.5,6.25 and so on ,theoretically requiring an infinite number of half-lives before it becomes zero.
3. DIVERGING INFINITE SERIES
Let us consider the series:
A = 101 + 102 + 103 + 104………
In this case, the summation value increases , but so does the difference between consecutive terms . This value, most definitely approaches infinity and becomes so, after an infinite number of terms.
Thus, a series can be called diverging series if, and only if, for any “threshold” M of our choosing, after a finite number of term values all the elements of the sequence are above the threshold. That is, no matter how high a threshold you fix, the sequence will eventually reach it, and stay above it afterwards.
4. TYPES OF INFINITY
Now, that we are able to fairly wrap our heads around the concept of infinity, we shall now analyze the different types of infinity and an unique method of comparing them. Two infinite quantities (just like two finite quantities) can be compared to each other, given that their scopes are already defined. Unlike finite sets however, the method used to do so will be stricter and will use bijection (mathematical correspondence).
Let us understand the necessity of having to use this method over the standard counting technique with the help of an example. Let us consider two bags each containing an arbitrary number of marbles. Each bag can either be in – “counted” state or “not counted” state.
As the name suggests, the bag is in “not counted” states by default and reaches the “counted” states, only after all the number of marbles have been accounted for (There is no such marble that has been left out).
We are able to compare their numbers simply because it would take us a finite time to reach the “counted” states in both bags, for a finite number of marbles. This would not be the case if both of the bags had infinite number of marbles. We cannot simply say that the two bags are equal, because despite having “infinite” values, we are never able to reach a “counted” state in each.
This is why the value ∞ – ∞ cannot be thought of something finite (e.g. 0) and remains undefined.
Thus, we are forced to use a much more formal method discussed in the next sub-topic.
METHOD OF BIJECTION/MATHEMATICAL CORRESPONDENCE
Two sets, by this method, can be proved to equal by establishing a correspondence with one another with having to individually count them. This eliminates the necessity for each of the sets to be finite. For e.g. say in a bus with a given number of seats, there are three possible cases:
Case A: Some people are standing and all seats are occupied
Case B: All the people are seated and some seats are empty
Case C: All the people are seated and all seats are occupied
In Case A, we are able to conclude under the assumption that no person will remain standing unless all seats are occupied that the number of people in the bus are greater than the number of seats without having to count either of the seats or the number of people. Similarly, in B, the number of seats are greater than number of people and in C, both are equal to one another.
Bijection is, therefore, defined as a mapping that is both one-to-one (an injection) and onto (a surjection), i.e. a function which relates each member of a set S (the domain) to a separate and distinct member of another set T (the range), where each member in T also has a corresponding member in S.
This type of infinity is smallest type and is the size of integer number line. This is countable because of two reasons.
- Finite elements exist in a finite interval. i.e.
Although the total number of distinct integers is infinite, the number of integers between any two integers is finite. E.g. between 5 and 8(not including) there exists two integers.
2. Minimum steps and distinct
The next integer with respect to a current integer lies in +1/-1 distinct interval that makes its counting possible. Also, we are able to say with certainty that no integer lie between two consecutive integers say, 16 and 17.
- Is the set of integers Z larger than set if even integers E(including zero)?
Ans: – No. If we were to rely, purely, on our intuition, it might seem that even integers are twice as sparse as integers because it leaves out odd integer integers to form its own set and therefore, must be exactly half. This is not true and both the sets are equal, which we can be understood using the previously discussed bijection rule.
For every element z in Z, there exists an even integer e twice as large that exists in set E, and for every element e in E, there exists a corresponding element z in Z that is exactly half.
Thus, the condition is all z in Z evaluates to some e (in E) = F1(z)
And z = F2(e) is also true.
Also, F1 = F2-1 and no two e corresponds to one z and vice versa.
In the above example, F1(z) = 2z, F2(e)= e/2.
This is the larger infinity which is infinitely dense and is uncountable irrespective of the interval chosen. E.g. the real number line
How many distinct numbers are there in the real number line? The answer is infinity. This is also true for the distinct digits in the interval say, 2 and 4, or even extremely small ones (e.g. 1 and 1.0000000000000000001)
WHY REAL NUMBER LINE IS GREATER THAN THE INTEGER NUMBER LINE
Through our above discussion, we have understood how old integers, even integers, and integers are equal to one another. But this is not the case with real numbers R. They are larger than any countably infinite sets such as integer Z.
This is because, for any given rule any integer z in Z will always correspond to an element r in R, but the converse is not true. There will always be infinitely many(uncountable) number of elements r’ left out.
A much more formal proof is given in 1891 by Georg Cantor known as Cantor’s Diagonal proof is a formal proof which explains that there are infinitely many sets that cannot be put be into a one-to-one correspondence, shall be given in the next article, where we shall discuss uncountable infinite and Cantor’s diagonal argument.
The concept of infinity is not only limited to mathematics, but is also is also applicable in the metaphysical and purely scientific domain. For e.g. it is debated in metaphysical discussion of God that there exists an ultimate entity in a infinite state of being. In physics, temporal infinity is used in discussion if the universe will last forever. Ultimately infinity for now, has remained only as a concept, yet it is widely applied in the field of mathematics and physics.It never ceases to amaze us and as mathematician G.H. Hardy said in the popular film The Man Who Knew Infinity(2015) , “We are merely explorers of infinity in the pursuit of absolute perfection.”