1. Computer Architecture CST 250
Number Systems in Brief
Prepared by:Omar Hirzallah

2. Contents The Number Systems Conversions ASCII Coding BCD Address Range
Binary Numbers
Binary Arithmetic (Add. & Sub.)
S & M, 1’s & 2’s Complement Methods

3. THE NUMBER SYSTEM (1) Binary Number System (0,1) Base 2
(2) Octal Number System
(0,1,2,3,4,5,6,7) Base 8
(3) Decimal Number System (Denary)
(0,1,2,3,4,5,6,7,8,9) Base 10
(4) Hexadecimal Number System
(0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F) Base 16

4. The ASCII Code
ASCII stands for “American Standard Codes for Information Interchange”. H= e=
l =
l = o=
It’s a bit method. 7 bits for code values and 1 bit for Parity check.

5. Binary Coded Decimal Decimal Symbol BCD Digit 0000 1 0001 2 0010 3
0000 1
0001 2
0010 3
0011 4
0100 5
0101 6
0110 7
0111 8
1000 9
1001

6. 8 7 1000 0111 The Binary Coded Decimals (BCD)
ASCII Codes use 1 byte for 1 character to store Whereas BCD can be used to save memory space by putting two characters in one byte.
Example:
87 can be written as 8 7
1000
0111

7. No. of Different Codes = 2n
THE ADDRESS RANGE:
The Formula to calculate the no. of different combinations /addresses range according to the no. of bits:
No. of Different Codes = 2n
(Where n is no. of bits.)
For Example: If there are two bits then:
No. of Different Codes = 22 = 2 x 2
= 4
For Example: If there are three bits then:
No. of Different Codes = 23 = 2 x 2 x 2
= 8

8. The Binary Numbers The Example of Unsigned Binary Numbers:
Signed Binary Numbers (+ or -)
The Example of Unsigned Binary Numbers:
Decimal Equivalent 25
M.S.B. L.S.B.
M.S.B. Most Significance Bit
L.S.B. Least Significance Bit

9. Signed (- or +) binary numbers
There are two very famous notations in dealing with
Sign & Magnitude Method (7 + 1 bit method):
Sign Bit
Magnitude
Complement Method (2’s complement):
2’s Complement = 1’s complement +1
1’s complement : Convert all 1s to 0s and a 0s to 1s

10. 1’s Complement : Convert all 0’s into 1’s and all 1’s into 0’s
1’s Complement : Convert all 0’s into 1’s and all 1’s into 0’s. For Example: ’s Complement:
2’s Complement : Convert all 0’s into 1’s and all 1’s into 0’s and then add 1. For Example: (37) 1’s Complement: __________1 2’s Complement: (-37) +

11. BINARY ARITHMETIC:
BINARY ADDITION:
There are four Basic Rules for Binary Addition:
FOR EXAMPLE: 1 1 1 1 1

12. BINARY ARITHMETIC: 0 1 1 0 - 1 1 BINARY SUBTRACTION:
1)There are four Basic Rules for Binary Subtraction:
Borrow
FOR EXAMPLE:
If any, otherwise impossible to solve 1 10 10 1 1

13. BINARY ARITHMETIC: =0101 + (-0011) =0101 + (1100+1) =0101 + 1101
BINARY SUBTRACTION:
2-Use the 2’s complement method:
FOR EXAMPLE: (a 4-bit number)
A – B can be expressed as A + (-B)
(-B) is the 2’s complement of B –
= (-0011)
= (1100+1) =
=1 0010
(5)
(3)
(2)

14. References
Mano, (2008). Logic and Computer Design Fundamentals, 4th ed., Prentice-Hall.
Mano, (2006),Digital Design, 4th ed, Prentice Hall.
Kifer, M., &Smolka, S. A. (2007).Introduction to Operating System Design and Implementation, Springer
memory.asp‏
en.wikipedia.org/wiki/Bus_(computing)
en.wikipedia.org/wiki/Addressing_mode‏