Jesús de la Peña Hernández
9.18 DIVISION OF A PAPER STRIP BY MEANS OF BINOMIAL NUMERATION
As we know, in binary numeration the only numbers that exist are 0 and 1. Besides, to
convert an integer of the decimal system to the binary system, we must divide the integer and
successive quotients by 2 till the moment a quotient 0 is reached (and, therefore a remainder of
1).
The resultant binary number is made up of those remainders: the first one (always 1, at
left) is the last obtained and, successively towards the right, the others up to the first produced.
Example:On the other hand, to convert a decimal fraction to binary system it ́s better to follow
the procedure shown in this example:
To convert into binary the decimal 43.42.
The first thing is to have available the successive powers of 2 (positive as well as nega-
tive).
20 = 1 25 = 32 2 −^5 = 0. 03125
21 = 2 2 −^1 = 0. 5 26 = 64 2 −^6 = 0. 015625
22 = 4 2 −^2 = 0. 25 27 = 128 2 −^7 = 0. 0078125
23 = 8 2 −^3 = 0. 125 28 = 256 2 −^8 = 0. 00390625
24 = 16 2 −^4 = 0. 0625 29 = 512 2 −^9 = 0. 001953125Then subtract the greatest possible power of 2, to the given number.With the bases above, let ́s divide a paper strip in 7 equal parts. First of all we have toconvert to binary the fraction
71
.71
= 0,142857143,4211,42- 1
- 0,25
- 0,125
- 0,03125
3,421,420,420,170,01375- 2
- 32
- 8
0,045101011,01101101011,011101011,01101011,101011011( 2( 2( 2( 2( 2( 2( 2^5
310- 2
- 3
- 5
) ) ) ) ) ) )..................................................Put down 2 to the left of that result; this^5
is the power taken into account; besides, one 1 at
right (the first digit that always appears in any
binary expression). Continue likewise. At the right
of 3,42 repeat the 1 above and add one zero (one
power of 2 has been dropped – 2 -); also add the^4
1 corresponding to the line we are dealing with.
It can be seen that the process would con-
tinue till the difference zero would be found, what
not always happens.(^132)
(^162)
(^03)
1
2
1
1
2
0
(^131101) 2 )