The Touch of Infinity - Govt. Victoria College

The Touch of Infinity
T K Rajan, Associate Professor
Department of Mathematics, GVC Palakkad
Here we pursue the concept of infinity and the impact that this alien made in the real world.
The question of infinity arose as soon as people began to think about the world they lived in. The
questions like whether the space would go on or there was a finite end, what would happen if a
piece of wood cut into halves continuously or is it possible to do so etc. puzzled them. All these
questions led them to think of the concept of infinity. The early Greeks had come across the problem
of infinity at an early stage in their development of mathematics.
We all know that Mathematics is the area where the concept of infinity is handled and
scrutinised. What is infinity? Some people consider infinity as the largest number. In ancient period
itself People were very much crazy about finding and naming large numbers. The Indians too were
not behind in this venture. In Vedic literature we can find Sanskrit names for powers of 10 such as
laksha (105), koti (107), ayuta(109), niyuta(1011), pakoti(1013), vivara(1015), kshobya(1017),
vivaha(1019) etc. Even the names of dasavathara of Lord Mahavishnu in Hindu mythology like
Matsya, Kurma, Varaha etc. were used to represent very large numbers like 10600 ,102000 ,103600
respectively. Western people started naming large numbers in terms of illion In 17th century. These
names such as Million(106), Billion(109), Trillion(1012), Quadrillion(1015), Quintillion(1018),
Sextillion(1021), Septillion(1024) etc. are used now a days world widely. The largest number named
in this pattern is Millinillion (103003).
Really is there any need for such large numbers? We know the number of particles in the
whole universe is only somewhere between 1072 and 1087, so we don’t have to use a number larger
than this. At present 10100 known as Googol is universally accepted as the largest number and the
credit of naming it goes to Edward Kasner, a professor in Columbia University. You may know that
the internet search engine Google got its name from Googol, as it is expected to display large
number of links in a single hit. We know Googol is not the largest, for if we add 1 to it or multiply it
by 10 we get a larger number. People are still finding and naming large numbers. Googolplex (10
Googol
i.e. the number 1 followed by Googol number of zeros), Googolplexian (10 Googolplex) etc. are
some of them. But remember that there is no enough material in this universe to write number like
Googolplex. Where is the position of infinity? It is not big, it is not large and it is not enormously big,
but it is endless, it is boundless, it is eternal, it is poornatha. In Hindu mythology there is a mention
of infinity in terms of Anantha. Lord Vishnu is said to rest in the coils of Ananta, the great serpent of
Infinity, while he waits for the universe to recreate itself. Here infinity refers to God, who is the
personification of fullness or poornatha. The Isha Upanishad of Yajurveda states that “If you remove
a part from infinity or add a part to infinity, still what remains is infinity”. This is chanted as the
following sloka
OM POORNAMADAH POORNAMIDAM
POORNAAT POORNAMUDACHYATE
POORNASYA POORNAMAADAAYA
POORNAMEVAAVASHISHYATE
OM SHANTI SHANTI SHANTIH
Infinity in the real world
All the large numbers mentioned above are finite and one can eventually get there. But
none of these are close to infinity. The word infinity is derived from the Latin word infinitas which
means endless. In Mathematics we can see the examples of endlessness in various situations. The
natural numbers {1,2,3, …… } never ends and is infinite. When we write the fraction 1/3 in decimal
form (0.33333 ….. ), the digit 3 repeats endlessly. The points on a straight line segment, the real
numbers between 0 and 1, Oscillation of Topologist’s sine curve, sample space of tossing of a coin
until head turns up etc. are some cases of infinity. Though infinity seems complicated, it is simpler
than finite (which has ends and they must be specified). For example in Geometry to draw a straight
line segment of finite length, the additional information of its ends are to be mentioned where as it
is easier to draw a line, which has infinite length and goes in both directions without an end.
Though finiteness V/s infinity, in finiteness we can see infinity. You may be knowing the
method of finding the area of a finite circle by dividing it into many sectors and arranging them in a
rectangular form. But the arranged pieces will take a rectangle shape only if the size of sector is
negligibly small. This is possible only if the circle is cut into infinitely many sectors. Similarly in the
decimal expansion of the fraction 1/3 a finite number, the digit 3 repeats infinitely.
History of Infinity
Though ancient Indians had the notion of infinity, its attestable account goes to Greek
Mathematicians. It was Zeno, a member of Eleatic school of Greek Philosophers, who brought the
concept of infinity through his Paradoxes – Achilles and Tortoise, The dichotomy Paradox, etc.
The paradox of Achilles and Tortoise
In a running race Achilles allowed his opponent
a tortoise to start its run 100m ahead of him,
thinking that as tortoise is very slow in running
he could anyhow win the race. But according to
Zeno’s argument Achilles would not win. Since
Achilles starts running 100m behind the
tortoise, as he reaches the starting point of the
tortoise, his opponent will cover a some
distance say 10m (Assuming that Achilles is 10
times faster than the tortoise). Again when
Achilles covers this 10m, the tortoise runs
ahead 1m. It continues like this.
The net result is that Achilles could never overtake the tortoise. With this paradox Zeno claimed that
in a race, the quickest runner can never overtake the slowest, since the pursuer must first reach the
point whence the pursued started, so that the slower must always hold a lead.
The dichotomy Paradox
The kernel of the paradox is that which is in locomotion must arrive at the half-way stage before it
arrives at the goal. Imagine a person tries to catch a bus which stopped 100m past him. To get there
first he has to run the half of the distance (50m). Before he can get halfway there, he has to cover
quarter of the way (25m) there. Before running quarter way he has to cover one eighth of the
distance and so on. The resulting sequence is {. . . . . , 1/8, 1/4, ½, 1} which describes an infinite tasks.
According to Zeno’s argument even the motion is impossible as it is not able to predict the first
distance he has to travel.
Both these paradoxes reveal the involvement of infinity. If we rephrase Zeno’s dichotomy Paradox
we will get the infinite series ½ + ¼ + 1/8 + 1/16 + …….. and its sum is 1, a finite number . But Zeno
and his friends had never thought that the sum of infinite terms could be finite and hence they
concluded that all motion was impossible.
The Greek Philosopher Aristotle had also had vision of infinity. According to him “If everything has a
place, place too will have a place and so on ad infinitum. Infinity is something that is already fully
formed and can never compete even with the faraway galaxies.
Visualise infinity
How can we visualize infinity? For this let us see Grand Hotel Paradox of Hilbert, a German
Mathematician who is famous for axiomatic geometry and Hilbert Space.
His grand hotel is a hypothetical one with countably infinitely many rooms all of which are occupied.
Everybody is tempted to think that the hotel could not accommodate any newly arrived guest as in
the case of a hotel with finite number of rooms. But the fact is that the Grand hotel could
accommodate any number of new guests. This can be done by moving the occupant in room 1 to
room 2, in room 2 to room 3 and so on. Now it is possible to give room number 1 to the newly
arrived guest. It can even accommodate a countably infinite number of new guests: just move the
person occupying room 1 to room 2, the guest occupying room 2 to room 4, and in general room n
to room 2n, and all the odd-numbered rooms will be free for the new guests. Through this Paradox
Hilbert showed infinity’s behaviour is different from regular numbers.
The symbol of infinity
The symbol sideways 8 or lemniscate for infinity
(∞) was introduced in 1655 by the English
Mathematician John Wallis. One belief is that he
chose the symbol from Etruscan numeral for
1000 which looks like CIƆ and used to mean
many. Another conjecture is that it is the
reformation of the last Greek Alphabet omega
(ω). It is also said that the symbol of infinity is a
variation of the ancient Ouroboros in Greek
mythology. It is a serpent which eats its own tail
taking the shape of infinity, a unique symbol of
endlessness. It represents the cyclic renewal of
life – life, death and rebirth – leading to
immortality.
Is Infinity a number?
To many people infinity is considered as the largest number. Some people take infinity as a number
which is the reciprocal of zero. What is interesting about infinity is that when you add or subtract a
number from it, infinity remains. Due to this if we accept ∞ as a number several contradictory
statements will follow. For example
∞ + 1 = ∞
∞ + 1 + 1 = ∞ + 1 (adding 1 on both sides)
∞ + 2 = ∞ + 1
2 = 1 (Cancelling ∞ from both sides)
We know every real number has an additive inverse (y is said to be the additive inverse of x
if x + y = 0. The additive inverse of x is usually denoted by -x), but infinity doesn’t have this
property as ∞ + -∞ = ∞ - ∞ ≠ 0. Also
Thus ∞ breaks the
fundamental axioms that define numbers. So it is not a real number (A real number is either
rational or irrational, which can be represented on a real line. Corresponding to any point on
a real line there exists a real number and vice versa) and it has no representation on a real
line.
Reciprocal of Zero
Some people take
as the reciprocal of zero. Is this right? A multiplicative inverse or
reciprocal of a real number x is denoted by
or x-1 is a number which when multiplied by x
yields the multiplicative identity 1. We can see that Zero does not have a reciprocal because no real
number multiplied by 0 produces 1. The following graph depicts the reciprocal function
.
Here we can see that when x comes closer to 0 from right its reciprocal becomes larger
and larger. If we think it in terms of limit concept in Calculus this can be interpreted as
=
Sill we cannot say the reciprocal 0 is , as division by 0 is not permissible in
the field of real numbers. But in the extended complex plane division by zero is considered
as defined and it is known as the complex infinity. Here the formal statement
is
permitted, but it doesn’t mean that 1 = 0
. This idea was mentioned in Brahmasphuta
Siddantha of the great Indian Mathematician Brahma-guptha by the term khachheda. It
means ‘divided by kha; kha being ‘space’ or ‘void’, one of the names for ‘zero’. ‘Division by
zero’ gives an intuitive definition of infinity.
Types of Infinity
In Indian Mathematics infinity was termed in two different contexts. Asankhyatha which
represents countless or innumerable and Anantha which is endless or unlimited. These are
nothing but the nowadays concepts countable and uncountable introduced by the great
German Mathematician Georg Cantor (1845 -1918), the founder of Set Theory and one who
tamed infinity.
Using the idea of one-one correspondence between
sets, he proved that real numbers are more numerous
than natural numbers. More precisely saying the set of
real numbers is uncountable infinite whereas the set of
natural numbers is countably infinite. It was Cantor who
finally put infinity on a firm logical foundation and
described a way to do arithmetic on infinite quantities.
With the concept of Cardinal numbers, he demonstrated
that there are different sizes of infinity; say infinity of
natural numbers, infinity of real numbers, infinity of
power set of real numbers and so on. Their cardinalities
are
which are called transfinite numbers that go beyond infinity. Any set
that has the same size of the set of natural numbers is termed a countable set. The set of
even numbers, the set of squares of natural numbers, the set of rational numbers etc. are all
countable sets with cardinality ℵ0 (aleph-not). The set of irrational numbers, the set of real
numbers, Cantor set etc. are uncountable sets with cardinality ℵ1 (aleph-one).
Limit of a sequence.
You may be familiar with the concept function in Calculus. A function from a set A to a set B
is a rule that assigns to every element of A, a unique element of B. It is sometimes
expressed by f(x), where x represents elements of A.
A sequence on a set S is a function from the set of natural numbers N to S. Generally a
sequence is denoted by < xn > or < x(n )>. The terms of the sequence are countably infinite
and written as x1, x2, x3 . . . . .
Examples:
1
1,2, 3, 4, ………
2
1, ,
…….
3
,
…….
4
0.1, 0.01, 0.001, ………
5
3, 3.1, 3.14, 3.14, 3.141, ………
If we examine the terms of each of the above sequences, do we have any idea where it
leads to. In the case of example 1, we can see that it is going to endless, where as in
example 2, the terms become smaller and smaller as position of the term increases and
there is a tendency to approach 0. This concept is known as limit of a sequence. Here we
write
= 0 or the sequence < > converges to zero as n tends to . Here is the nth
term of the sequence. Similarly in example 3
= 1, i.e. the sequence <
> is
converging to 1 as n tends to infinity. The sequence in example 5 is one which converges to
π. As the sequence in example 1 goes to endless, we say it is divergent.
π is a signature of infinity
The irrational number π is the limit of a sequence. To understand more about it let us take
the problem of evaluating π using perimeter of a circle. For this we approximate the circle
by a regular polygon inscribed in it. Then each side of the polygon is a chord of the circle.
If θ is the central angle made by the chord AB and CD (of length d) is the diameter, then
so that AB = CD
then Pn = n
sin
AB = n
= d
d
sin
sin
= sin ,
. If Pn is the perimeter (the sum of all chords) of the polygon,
, which can be calculated for various values of n.
perimeter of the circle πd is approximately equal to Pn , we have π
Since The
. Note that the more
approximated value of π is got by increasing the value of n (number of sides of the polygon). i.e.
= π.
The other side of infinity
As the people were concerned of infinitely large quantities, some were also keen on infinitely small
quantities, called infinitesimals. It is a positive real number which is smaller than the smallest
positive real number one can imagine. In number system incorporating infinitesimals, the reciprocal
of an infinitesimal is an infinite. George Cantor who introduced infinitely large quantities, claimed
that infinitesimals are logically impossible. But Archimedes defined infinitesimal as real number x
such that | |>n. These numbers were originally developed to create the differential and integral
calculus for solving problems of finding slopes, areas under curves, minima and maxima and other
geometrical concepts. But later these were replaced by systems using limits, as they were shown to
lack theoretical rigor. The infinitesimal Calculus was the independent contributions of Gottfried
Leibniz and Isaac Newton in the 1660s. The paradox of Achilles and tortoise, dichotomy paradox etc.
could be resolved with the idea of infinitesimal numbers and limit concept.
Infinity in Indian Mathematics
As early as the 8th century BC in Sulva Sutras there is a remarkably accurate figure for the square
root of 2 obtained by adding 1 + 1⁄3 + 1⁄(3 x 4) - 1⁄(3 x 4 x 34), which yields a value of 1.4142156, correct to 5
decimal places.
Bhaskara II, who lived in the 12th Century, was one of the most accomplished of all India’s great
mathematicians. He is credited with explaining the previously misunderstood operation of division
by zero. He noticed that dividing one into two pieces yields a half, so 1 ÷ 1⁄2 = 2. Similarly, 1 ÷ 1⁄3 = 3.
So, dividing 1 by smaller and smaller factions yields a larger and larger number of pieces. Ultimately,
therefore, dividing one into pieces of zero size would yield infinitely many pieces, indicating that 1 ÷
0 = ∞. This is nothing but the concept
=
in Calculus.
The great mathematician Madhava (1340-1425) from the town of Sangamagrama in Kerala,
contributed many ideas of infinity by linking it with geometry and trigonometry. By finding infinite
series expansion of trigonometric functions like sine, cosine etc. he made the first step from the
traditional finite processes of algebra to the considerations of infinity, that led to the development
of Calculus and Mathematical Analysis. The famous Leibnitz’s expansion 1-1/3 + 1/5 - 1/7 +............ of
(1646-1716) was known to Madhava.
We shouldn’t forget the incredibly brilliant Indian Mathematician Srinivasa
Ramanujan (1887-1920), the man who knew infinity and his contributions to
Number Theory and Analysis. He posed and then supplied solutions to many
problems concerning infinite series like
to the world and
put forward the idea of partial sums for finding the sum of infinite series.
Conclusion
The framework of number system allows one to express numbers having finite, infinite, and
infinitesimal parts, numbers having only infinite and infinitesimal parts, etc. But remember that
human beings or even fast computers are able to execute only a finite number of operations even if
they have to handle infinite mathematical objects and processes.
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