SMALL NUMBERS-LARGE NUMBERS
(Only a Part of the Article Appears Here)
BIMAL
K. SRIVASTAVA
When we were young,
we were taught to count from 1 to 100. Beyond that we used to learn of our own the numbers up to 1000. However, it was considered
unnecessary to learn any number beyond 1000, as the same was considered as self explained. Perhaps a `four-figure salary'
was taken as a status symbol during earlier days, which of-course has now been replaced by a more realistic term of `five-figure
salary' or may be `six-figure salary'.
Then in English language we heard
about millionaires (representing a number as 106) and `Lakhpati'
or `Crorepati' (105 or 107) in Hindi. Certain luckier names as `Henry Ford' or `Tata' and `Birla' or
to that matter ‘Bill Gates’ or ‘Ajim Premjis’ in today’s
scenario are called `billionaires' which is a degree higher than a `millionaires'. However the term `billionaire' itself is
a confusing term, since whereas in American dictionaries a billion represented 1000 Million (or 109), the British
dictionaries defined it as `a million million' (or 1012). Even the population of India and China and the budgets
of countries were expressed in `billions'. The combined population of world itself is around 5.8 billion (5.8X108).
So far no practical use of numbers beyond billion was considered worth learning. However, a trillion (thousand billion or 1012) was used to cater
for something larger. In Hindi schools, somehow, they went further beyond to cover numbers like `arab' (109), `kharab'
(1011), `neel' (1013), `padma' (1015), `shankh' (1017), and even `mahashankh'
(1019), which can be awkwardly translated as `10 million trillion' in English.
FRACTIONAL REPRESENTATIONS
In terms of fractions these number were expressed as `deca (da)' (101),
`hecto (h)' (102),`kilo (k)' (103), `mega (M)' (106), `giga (G)' (109), `tera
(T)' (1012), `peta' (1015), `exa' (1018) etc. In the similar manner we used to have our limited
dictionary for small fractional numbers also starting from `deci (d)' (10-1), `centi (c)' (10-2), `milli
(m)' (10-3), `micro (m)' (10-6), `nano (n)' (10-9), `pico (p)' (10-12), `femto' (10-15),
`atto' (10-18) etc.
While we are thoroughly familiar with some of the fractional terms like milli,
centi, kilo etc. such as 2 millimetres, 5 centimetres, 15 kilometres, 10 kilograms (we have even abbreviated them as mm, cm,
km, kg etc.), we find it non-conventional and unacceptable to use the words as mega meter (no book will give the distance
between Delhi and Bangalore as 2 mega meters rather than 2000 kilometres). Also, the distance to Sun 150 billion metres or 150 giga metres.
On the other hand all the above terms have been accepted by International
Union of Weight & Measures, and small fractions like micro, nano, pico etc. are finding place in scientific literatures.
Even most of us are also familiar with megahertz, gigabyte, etc. but not with gigagram.
MORE NUMBERS
With the advancement of science and technology our requirement of
having still larger and smaller numbers has increased manifold. It was easily possible to express these numbers by using the
term 10x, where `x' could be any positive or negative integer. However, for general purpose some new terms were
also introduced for naming very big and very small numbers.
Some of the terms, which are in use in this field, are Zetta (1021),
Yotta (1024), Xenna (1027), Vendeka (1033), Googol (10100) etc. Similarly, for
smaller numbers we have Zepto (10-21), Yocto (10-24), Xenno (10-27), Vendeko (10-33)
etc. (However, some names like xenno have not yet been approved by the international
satndards committee for scientific nomenclature.)
For example instead of writing 1021 meters, it is more
figurative to write l yottameter. Similarly the distance travelled by light in one year is 9.45X1012 Kms. This
number can be expressed as 9.45 petameters, which although is difficult to apprehend, yet appears easier to read. In the similar
manner the weight of an atom is 1.66x10-27 kgs. or 1.66 yoctograms.
PRACTICAL UTILITY
Now the question arises
that what is the utility of having such huge numbers and such fractional minute numbers. Well, this is definitely needed in
today's vast and expanding world of scientific imagination and analysis.
..................
...................
.....................
If we try to calculate the mass of earth it comes as 6 xennagrams
which is nothing but 6X1024 kilograms or (6 x 1027 gms). The volume of the earth is 1.08X1021 cu
metres. Similarly, if we try to count all the letters of the books of all the libraries of the world it comes as one zetta
(1021). If we express this number in Hindi it comes as 10000 sankhs. The age of Sun is 5 billion years and its
expected life is 10 billion years. This can be expressed as 3X1017 seconds or 300 petaseconds. The age of this
Universe (big-bang) itself has been estimated between 12 to 15 billion years or say 0.4 exaseconds. In other words it can
be said that while the Universe was formed about 0.4 exasecs back, our Sun (and hence the whole earth with its inhabitants)
would perish after 0.3 exaseconds.
...............
................
..................
.................
of 109. Now it is reaching the age of Terabytes or in
terms of 1012. Surely our next generation computers will be in multiples of `petabytes' (1015) and `exabytes'
(1018) or higher and higher.
LARGER AND LARGER
However, the numbers do not end here. The number of words printed since in the
500 years after the Gutenberg Bible (so, say, 1456 to 1956) is around 1017. If we compare the smallest particle in this universe with
the size of our universe it comes as 1051. And the famous scientist Archimedes had estimated about 2000 years back
that the total number of particles in this universe is 1051. In the beginning of this century the famous scientist
Arthur Edington had calculated and revised the number as 1080. If the entire universe were filled with protons and electrons so that there was no vacant space, the total number
would be about 10110 .
We can further enlarge our scope of counting.
The first and foremost
large number in real sense, which we come across, is a googol, which is 10100.
It is believed that the term Googol
was invented by Milton S..........irotta, the 9-year nephew of mathematician Edward Kasner, who had asked
........
his nephew what he thought such a large number should
be called. Such a number,............ Milton apparently replied after a short thought, could only be called something as silly
as...a googol!
There is one more number, devised by another mathematician, which is even larger than a goggle, is called a googolplex, which is 10googol (or ten to the power ten
to the power hundred). A googolplex is much bigger than a googol, much bigger even
than a googol times a googol. A googol times a googol would be 1 with 200 zeros, whereas a googolplex is 1 with a googol of
zeros. You will get some idea of the size of this very large but finite number from the fact that there would not be enough
room to write it, if you went to the farthest star, touring all the nebulae and putting down zeros every inch of the way. Frank
Pilhofer has
...........
............
...........
Certain scientists
are of the opinion that the game of chess can be played with one googolplex methods.
The largest prime number that has been calculated by mathematicians
is as large as 1065000. Now let us see the possibility of something impossible, and then we can go even to larger
scales. For example, what is the probability that a chimpanzee will learn typing and will type all the novels of Shakespeare?
The scientists have found that the possibility of this incident is one in 101,00,000.
Another branch of mathematics is called `number theory'. Under this
branch there is a number called `Skewes number'. This mind boggling Number is so big that it is not possible to write it by
simple means. Somehow it can be written as 10 I 10 I (10 I 34), where 10 I 10 represents 1010.
But still larger number has already been defined which is known
as `Graham Number'. This Number is used under `Ramsay Theory'. `Graham number' is so large that it is just impossible for
us to write it on papers by simple means. Perhaps God Brahma may be able to write it or scientists may have to device a new
theory for writing this Number. This can, however, be understood in a manner that if complete universe is converted into paper
and if we start writing Graham Number, the number will not be complete even after exhausting all the papers spread all over
the universe.
SMALL NUMBERS
..........
.............
..............
Similarly Blink
of an Eye is 25 milli Seconds,
Camera Flash is 10 Micro Seconds; Optical Microscope resolution is 200 Nanometers,
One tick of a Pentium Processor Clock is
280 Pico Seconds, Mass of E.Coli Bacterium is 665 Femtograms ,
and
mass of virus
is 10-17 gms or 10 Attograms
Thus we find that it is really interesting to enter the strange
world of small and large numbers. However for understanding such numbers occasionally we may need the help of a mathematician.
(BIMAL K.
SRIVASTAVA)
ADDRESS
148, DIN APARTMENTS,
SECTOR-4,
(PLOT No. 7)
DWARKA
NEW DELHI-110075.
Phone (Res) : (011)-25075876
