Binary Number System

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BINARY NUMBER SYSTEM
Just as the standard decimal system is based upon the powers of ten to express any
number, the binary system is based on the powers of two to express a number. The binary
system was first studied in detail by G. Leibnitz in 1678 and forms the basis for all
computer and digital manipulations. As an example of a binary representation, consider
the decimal base number
13.
It is equivalent to
2^3+2^2+2^0
and so has the binary
designation
1101
with the
1
designating an existing power of two and
0
if it does not exist.
Likewise twelve is expressed as
1100
since the number equals
2^3+2^2
and no other
powers of two. To double a binary number one needs to only add a zero at the right. Thus
decimal
26=2x13
is
11010
in binary and decimal
416=2x2x2x2x2x13
will be
110100000.
One
can also reverse things and ask what decimal number corresponds to 101011. Here we have
six digits so the number can be read off as
2^5+2^3+2^1+2^0=32+8+2+1=43.
To help in
these conversions it is a good idea to be familiar with the first twelve powers of two as
shown in the following table-
Number, n 0 1 2
3
4
5
6
7
8
9
10
11
12
n
2
1 2 4
8
16
32
64
128
256
512
1024
2048 4096
Also, when adding binary numbers, use the addition table-
0 + 0 = 0
1 + 0 = 1
0 + 1 = 1 1 + 1 = 10
and when multiplying, use the multiplication table-
1 x 1 = 1 1 x 0 = 0 0 x 1 = 0 0 x 0 = 0
Following these rules, one can construct a table for addition and subtraction of the
numbers 27 and 14 as follows-
2^6 2^5 2^4 2^3 2^2 2^1 2^0
N1=27
0
0
1
1
0
1
1
N2=14
0
0
0
1
1
1
0
N1+N2 0
1
0
1
0
0
1
N1-N2
0
0
0
1
1
0
1
The Japanese Soroban and Chinese Abacus are mechanical calculating devices whose
origin lies in a table of this sort when adjusted to a decimal base and using bead positions
to indicate 1 or 0. A binary abacus will require just one bead per column instead of the five
in the standard decimal Soroban. Here is a schematic of a Soroban-

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