Page 57 - Scientech 2014-15

BINARY SYSTEM INFINITE SERIES
There are several number systems in use: Decimal Sum of all the natural numbers 1+2+3+4+5+----
(base 10), hexadecimal (16 as base, 0-9, A-F), octal is a divergent series. When you sum up the series
(base 8), sexagesimal (base 60), binary and so on. 1+2+3+4+5+---- to no limit, the successive digits
We will make a simple note here on the binary add up to 1, 3, 6, 10, 15, and so on, ever increasing,
system. without limit: Intuitively, there is no sum to this
series. On the other hand the series 1-1+1-1+1-
Decimal counting uses the ten symbols 0 through 1+1---- (known as Grandi's series) will sum up to
9. Binary system on the other hand uses only 0 and either 0 or 1 depending on whether the last number
1 (alternately "on" and "off").
is -1 or +1. Logical, one would say.
Decimal: 00, 01, ---, 09, 10, 11, and so on.
Here is the surprise: There are summation methods
Binary: 0000, 0001, 0010, 0011, 0100, 0101, 0110, in mathematics that assign values to some of the
and so on. divergent series. 1-1+1-1+1- --- sums up to 1 /2
Binary 0 1 10 11 100 101 110 111 1000 1001 1010 and 1-2+3-4 --- sums up to 1 /4.
Decimal 0 1 2 3 4 5 6 7 8 9 10 And now get ready for this one: Sum of all the
These are equal to 0, 1, 2, 3, 4, 5, 6, and so on in natural numbers 1+2+3+4+5+ ---- is, hold your
the decimal system we are so used to. Each position breath, -1/12. And do you know who arrived at
from right to left represents the multiplier 2 , 2 , 2 , this result? None other than the great Indian
0
1
2
2 and so on. mathematician, Srinivasa Ramanujan!
3
KING CHALLENGES A WISE MAN IN CHESS: GRAPHICAL APPROACH TO EASY MULTIPLICATION.
This is an old Indian legend about a vain King Say we want to multiply 12 by 13.
challenging a visiting sage in a chess game. The
King looses and offers a "carte blanche" as was the Draw one line and two lines after
custom of the days. The humble sage says all he some gap (black).
wants is some grains of rice: one on the first square Draw lines in other direction,
of the chess board, 2 on the second, 4 on the third 1 followed by 3 lines again after
and so on, doubling on each square. The King some gap (blue).
happily consented – the sage could have asked for Now group together the lines and count the dots at
the kingdom!
intersection:
THIS IS WHAT THE KING DID NOT RECKON: 156 is the answer.
Square Grains Total
1 1 1 It is that simple.
2 2 3 You can get
3 4 7 additional
4 8 15 information from
10 512 1,027 a ~3 min video at
20 524,288 1,048,575 www.youtube.com/watch?v=AjIlpp9aQDk
30 536,870,912 1,073,741,823 Summing Up

In the above table, The Total column = Grains + The Mathematics has developed into a vast field today,
Previous Total, for example Grains = 4, Total = 3, whose roots go back to unknown epochs. Apart
hence the Grains total adds up to 7. By the 30th from making everyday life less chaotic, it has helped
square itself it is already a billion grains! That is in cracking genetic code, launch sattelites, predict
~ 25 tons of rice. So, to fill all 64 squares would weather, generate wealth in number of ways, The
need: 2 −1= 18,446,744,073,709,551,615 grains – numbers are there everywhere. Get charmed by the
64
A whopping 460 billion tons of rice. No king ever magic.
had that kind of wealth!

[In our story, the sage tells the humbled King that
he doesn't have to pay the debt immediately but can
pay him over time, just serve rice to pilgrims every
day until the debt is paid off.]

So, the power of binary doubling is nothing to be
taken lightly. The advantage of the binary codes
is in the computers and other electronic devices
where only two distinct states help build up a new
universe! Dr. Prakash M Dolas
Rajiv Gandhi Science & Technology Commission
Government of Maharashtra, India

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