Chapter 2 - Binary Numbers

Representation of numbers in binary

(Ford & Topp call this the expanded form representation of a number)

How can you convert numbers from decimal to binary?

Subtraction of powers decimal-to-binary method

Flowchart (representing the subtraction of powers method)

In pseudo code the same algorithm can be documented as:

x = number_to_be_converted;
k = number_digits - 1;
/* number_digits = 8 so that while-loop starts at 7 */

while (k >= 0) do
   begin
      if (x-2^k) < 0 then d(k) = 0
         else
         begin
            d(k) = 1;
            x = x-2^k;
         end
      k = k-1;
   end;


trace loop:   
setup:        
x=217         
k=7           
              
looping:      
k=7           k=3
217-2^7=89    9-2^3=1
d(7)=1        d(3)=1
x=89          x=1
              
k=6           k=2
89-2^6=25     1-2^2=-3
d(6)=1        d(2)=0
x=25          
              
k=5           k=1
25-2^5=-7     1-2^1=-2
d(5)=0        d(1)=0
              
k=4           k=0
25-2^4=9      1-2^0=0
d(4)=1        d(0)=1
x=9           x=0

Even-odd decimal-to-binary conversion method

Flowchart (representing the even-odd method)

In pseudo code the same algorithm can be documented as:

N = number_to_be_converted;
x = quotient;
M = current_digit;

if (N >= 0) then
   begin
      x = N;
      k = 0;
      while (k < 8) do
         begin
            d(k) = mod(x,2)
            x = int(x/2)
            k = k+1
         end
   end;

trace loop:
setup:
x=217
k=0

looping:
k=1
d(0) = mod(217,2) = 1
x=int(217,2) = 108    odd
k=2
d(1) = mod(108,2) = 0
x=int(108,2) = 54     even
k=3
d(1) = mod(54,2)  = 0
x=int(54,2) = 27      even
k=4
d(1) = mod(27,2)  = 1
x=int(27,2) = 13      odd
k=5
d(1) = mod(13,2)  = 1
x=int(13,2) = 6       odd
k=6
d(1) = mod(6,2)   = 0
x=int(6,2) = 3        even
k=7
d(1) = mod(3,2)   = 1
x=int(3,2) = 1        odd
k=8
d(1) = mod(1,2)   = 1
x=int(1,2) = 0        odd

Binary fractions

How about fractions?
No one said we couldn't have negative powers of two!

Binary to decimal - conversion is always exact
Decimal to binary - conversions are not always exact

Next Section: Chapter 3 - Number Systems
Previous Section: Chapter 1 - Course Introduction
EEAP 282 Class Notes Table of Contents