The ENGLISH ELECTRIC Co. Ltd.
	

NELSON RESEARCH LABORATORIES                     Technical Memorandum: NS u 242
STAFFORD                                                         Date: 9/9/57
MATHEMATICS DEPARTMENT                                    
Tel. Stafford 700   

             Notes on Binary Decimal and Decimal Binary Conversion
                               (using Brunsviga)
						   
                             Report by: C. Robinson.

1.   SUMMARY.

     This report gives instructions for decimal to binary and binary to decimal 
conversions, using a hand operated Brunsviga calculating machine.

2.   DECIMAL-BINARY.

     (a) Decimal Integers.
     Place integer at extreme right hand side of register. Multiply by 0.03125. 
If last five figures of product are less than 0.75 divide the first two of these
figures by three and record (without round off).  If the last five figures are 
greater than or equal to 0.75 divide first two of these figures by 3, subtract 
1 and record.  Now transfer, losing last five figures.  Multiply again by 
0.03125 and carry on.  When integral portion of number in accumulator is less 
than 32, record and stop.

Example.   123456789

    123456789 x 0.03125 = 3858024.65625    Record 21

      3858024 x 0.03125 =  120563.25000    Record 8

       120563 x 0.03125 =    3767.59375    Record 19

         3707 x 0.03125 =     117.71875    Record 23

          117 x 0.03125 =       3.65625    Record 21

Therefore 123456789 = 21, 8, 19, 23, 21, 3 (Chinese binary. Groups
                                            of five binary digits.)

     (b) Decimal Fractions.

     Set number on extreme left hand side of register.  Multiply by 32 so that 
when the result is transferred the integral portion of the answer is not 
transferred.  Record the integral portion and carry on to the requisite number 
of binary places.

Example.  Calculate Binary equivalent of 0.853012 to 25 b.p.

       0.853012 x 32 = 27.296384    Record 27

       0.296384 x 32 =  9.484288    Record 9

       0.484288 x 32 = 15.497216    Record 15
       
       0.497216 x 32 = 15.910912    Record 15

       0.910912 x 32 = 29.149184    Record 29

Therefore    0.853023 = 29, 15, 15, 9, 27, 0    to 25 binary places.

3.   BINARY-DECIMAL.

     (a) Binary integers.

     Write down binary integer in usual form i.e. the decimal equivalent of each
group of five binary digits, most significant on the right. Take the most 
significant group of five binary digits and multiply the decimal equivalent by 
32, add the next significant group, transfer and multiply again by 32. Repeat 
until least significant group has been added in (Note the last operation is an 
addition not a multiplication)

Example.  Find decimal equivalent of the binary integer, 21, 8, 19, 23, 21, 3.

            3 x 32 = 96           Add 21

          117 x 32 = 3744         Add 23

         3767 x 32 = 120544       Add 19

       120563 x 32 = 3858016      Add 8

      3858024 x 32 = 123456768    Add 21

Therefore,  answer = 123456789

     (b) Binary Fractions.
     
     Set least significant group on left hand side of register. Multiply by 
0.03125. Transfer and add the next least significant group to the result as a 
decimal integer.  Multiply by 0.03125 and continue.

Example.

Find decimal equivalent of the binary fraction 29, 15, 15, 9, 27.0.

                  29 x .03125 = 0.90625

            15.90625 x .03125 = 0.4970703125

       15.4970703125 x .03125 = 0.4842834472

         .4842834472 x .03125 = 0.2963838577

       27.2963838577 x .03125 = 0.8530119956 = 0.853012 rounded off.




Signed: C Robinson
MATHEMATICS DEPARTMENT