ARITHMETICS FOR COMPUTERS (NUMBER SYSTEM & OPERATIONS)

Introduction 

Number System Base.

Table : Number System Conversion

Binary ( X2 )	Decimal ( X10 )	Hexadecimal ( X16 )
0000	0	0
0001	1	1
0010	2	2
0011	3	3
0100	4	4
0101	5	5
0110	6	6
0111	7	7
1000	8	8
1001	9	9
1010	10	A
1011	11	B
1100	12	C
1101	13	D
1110	14	E
1111	15	F
10000	16	10

REMEMBER THIS!!

210	29	28	27	26	25	24	23	22	21	20
1024	512	296	128	64	32	16	8	4	2	1

20	2-1	2-2	2-3	2-4	2-5
1	0.5	0.25	0.125	0.0625	0.03125

Conversion of Decimal to Binary.

Conversion of Binary to Decimal.

Conversion of Decimal point to Binary

Conversion of Hexadecimal to Binary

Conversion of Binary to Hexadecimal

Sign Integer Representation

~ represent negative (-ve) values, computer system allocate the high order bit to indicate the sign of value.

~ 0 (+ve sign) ; 1 (-ve sign)

high order bit :

- left most in a byte .   called the most significant bit.

- remaining bits contain the value of the number.

3 ways signed binary numbers been expressed:

-          signed magnitude

-          1’s complement

-          2’s complement

Negative Number Conversion

Second complement

Binary Addition

The binary number operation rules (addition)

Binary Rules	Sum	Carry
0 + 0 = 0	0	0
0 + 1 = 1	1	0
1 + 0  =  1	1	0
1 + 1 = 1	0	1

 Binary Subtraction

The binary number operation rules (subtraction)

Binary Rules	Sum	Borrow
0 – 0 = 0	0	0
0 – 1 = 1	1	10
1 – 0 = 1	1	0
1 – 1 = 1	0	1

 

Binary Multiplication

The binary number operation rules (multiplication)

Binary Rules	Multiply
0 x 0 = 0	0
0 x 1 = 1	0
1 x 0 = 1	0
1 x 1 = 1	1

Hexadecimal  Addition

If sum number  >  15₁₀, the amount of the sum that exceeds 16₁₀ will carry a 1 to 

the next column.

Hexadecimal  Subtraction

 1^st step :  convert the hexadecimal number to binary number

2^nd step : take the 2^nd complement of the binary number and change it back to hexadecimal number

3^rd step : add both numbers to get the result

Post by Nurul Syifa Binti Zainudin (B031210018)

DIGITAL LOGIC

Login design.

·           Digital electronics operate with only two voltage levels

-          high voltage and a low voltage

·           Computer used binary system (1,0)

·           Combinational logic block contains no memory

·           Logic block with n inputs, there are 2^nd entries (possibility) in the truth table.

Boolean algebra.

• P1: X = 0 or X = 1
• P2: 0 . 0 = 0 
• P3: 1 + 1 = 1 
• P4: 0 + 0 = 0 
• P5: 1 . 1 = 1 
• P6: 1 . 0 = 0 . 1 = 0 
• P7: 1 + 0 = 0 + 1 = 1

Laws of Boolean algebra.

Example.

Basic Logic Gates.

Boolean Equation Forms.

Sum-of-product (SOP)

-          Combination of input values that produce 1s is convert into equivalent variables, ANDed together then ORed with other combination variables with same output.

-          SOP is easier to derive from truth table.

Example:

F = ABC + (ABC)’

The truth table:

A	B	C	ABC	(ABC)’	F
0	0	0	0	1	1
0	0	1	0	1	1
0	1	0	0	1	1
0	1	1	0	1	1
1	0	0	0	1	1
1	0	1	0	1	1
1	1	0	0	1	1
1	1	1	1	0	1

SOP expression:

F = ABC + (ABC)’

Product-of-sums (POS)

-          Input combinations that produce 0 in sums terms (ORed variables) are ANDed together.

-          Convert input values that produce 0s into equivalent variables, ORed the variables, then ANDed with other ORed.

-          Usually are if more 1s produce in output function.

Example:

F = (A + B + C)(A + B + C’)(A + B’ + C)(A’ + B + C)

A	B	C	F
0	0	0	0
0	0	1	0
0	1	0	0
0	1	1	1
1	0	0	1
1	0	1	0
1	1	0	1
1	1	1	1

POS expression:

F = (A + B + C)(A + B + C’)(A + B’ + C)(A’ + B + C)

Post by Marissa Aefdiani Effendi (B031210235)