Computer Architecture
Topic 1:
Numeral systems with different
bases

2
Positional Numeral System
In Positional Numeral System the position of digit plays important role.
The number in this system has the following view
X = x
s
x
s-1
...x
1
x
0
,x
-1
...x
-m
.
Point divides an integer part and a fraction of number.
Quantitative equivalent of this equation:
X = k
s
x
s
+k
s-1
x
s-1
+...+k
1
x
1
+k
0
x
0
+k
-1
x
-1
+...+k
-m
x
-m
,
Where k – base of the numeral system;
s+1 – precision of the integer part of number;
m - precision of the fraction of number;
xi – digit i of the number (xi = 0, 1, ..., k-1);
ki – weight of digit i.

3
Decimal Numbers
In decimal system any number can be presented by digits from 0 to 9.
The position of digit plays important role. The rule of data
representation in decimal system has the following view:
Where:
N – a number (quantity) of digits in the integer part of number (from
left side of point);
M – quantity of digits in the fractio nal part of number (from right side
of point);
Di – volume of i-th digit in the integer part of number;
D'i – volume of i-th digit in the fractional part of number;
D – volume of the number;
Integer or fractional part can be absent in the number (N or M = 0).
M
M
N
N
N
N
D DDDD D DD
   



 10...10 101010...10 10
' 2'
2
1'
1
0
0
1
1
2
2
1
1

4
Binary Numbers
In computers all information is presente d by the binary digits. The reason is
that its base element has two states.
The rule of data representation in binary system has the following view:
Where:
N – quantity of binary digits in the integer part of number (from left side of
point);
M – quantity of binary digits in the fractional part of number (from right
side of point);
Bi – volume of i-th digit in the integer part of number;
B'i – volume of i-th digit in the fractional part of number;
B – volume of the number.
Integer or fractional part can be absent in the number (N or M = 0).
M
M
N
N
N
N
B BBBB B BB
   



 2...2222...2 2
' 2'
2
1'
1
0
0
1
1
2
2
1
1

5
Examples of Binary Numbers
10
2 1 0 1 2 3 4 5 6
2
25.9025.0281664
21202021202121202101,1011010


 
10 2
10
5 3 2 0 2
2
75.10211.1100110
0,40625.52121212121 01101,101


  

6
Transformation of numbers from numeral system
with base k to decimal
A number has the following view
X = x
s
x
s-1
...x
1
x
0
,x
-1
...x
-m
.
The point divides integer part and fraction of the number. Quantitative
equivalent of this equation is:
X = k
s
x
s
+k
s-1
x
s-1
+...+k
1
x
1
+k
0
x
0
+k
-1
x
-1
+...+k
-m
x
-m
,
Where k – base of the numeral system;
s+1 – precision of the integer part of number;
m - precision of the fraction of number;
xi – digit i of the number (xi = 0, 1, ..., k-1);
ki – weight of the digit i.
Example:
1011,1001
2
= 1·2
3
+ 0·2
2
+1·2
1
+1·2
0
+1·2
-1
+0·2
-2
+0·2
-3
+1·2
-4
=
= 8 + 0 + 2 + 1 + 0,5 + 0 + 0 + 0,0625 = 11,5625;
X
10
= 11,5625.

7
Transformation of numbers from decimal
to numeral system with base k
Example. Decimal 11,5625 to binary
Integer part:
11 : 2 = 5, rest 1 (low-order bit of result),
5 : 2 = 2, rest 1,
2 : 2 = 1, rest 0,
1 : 2 = 0, rest 1 (high-order bit of result).
Result X
i
= 1011.

8
Transformation of numbers from decimal
to numeral system with base k
Example. Decimal 11,5625 to binary
Fractional part:
Result X2f =0,10010.
Full result
X2= X2i+X2f = 1011 + 0,10010 = 1011,10010.

9
Hexadecimal numbers
binary
Hexadecimal
binary
Hexadecimal
0000
0
1000
8
0001
1
1001
9
0010
2
1010
A
0011
3
1011
B
0100
4
1100
C
0101
5
1101
D
0110
6
1110
E
0111
7
1111
F