Index  Chapter1  Chapter2  Chapter3  Chapter4 
Digital Background  Semiconductor Background  CMOS Processing 
1.1  1.2  1.3a  1.3b  1.4  1.5  1.6 
Number System  Digital Arithmetic  Logic Gates  Logic Gates  Combinational Circuits  Multiplex (MUX) 
Binary Arithmetic
Below are the basics which most of us know very well. But still for reference point of view I have mentioned everything into a table followed by examples.
Binary addition

Binary subtraction

Binary multiplication

Binary Division

0+ 0 =0

0 – 0 = 0

0 x 1 = 0

0 ÷ 1 = 0

0 + 1 =1

1 – 0 = 1

1 x 0 = 0

1 ÷ 1 = 1

1 + 0 = 1

1 – 1 = 0

0 x 0 = 0
 
1 + 1 = 10
1 in (10) is Carry bit
Carry it to the next
higher order column

0 – 1 = 10
1 in (10) is Borrow bit
Carries from to the next higher order column

1 x 1 = 1

Hexadecimal Arithmetic
Hex addition rule

Subtraction

F + 1 = 10

10 – 1 = F

F + F = 1E

A – 1 = 9

F + F + 1 = 1F
 
1 + 1 = 2
 
9 + 1 = A

Hexadecimal Arithmetic (Best Way)
Best way to do hexadecimal arithmetic (Subtraction  Addition  Multiplication Division) is –
 First convert the number into decimal equivalents
 Perform the operation
 Convert back from decimal to hexadecimal
BCD Addition
There is a difference in binary addition and BCD addition. In binary maximum possible number is 1111 but in BCD, it is 1001. When the binary sum is equal to or less than 1001 (without a carry), corresponding BCD digit is correct. However, when binary sum is greater than or equal to 1010, the result is an invalid BCD digit. The addition of 6 = (0110)_{2} to the binary sum converts it to the correct digit & also produces a carry as required. This is because the difference between a carry in the most significant bit position of the binary sum & a decimal carry differ by 16  10 = 6
Example 6: Add 184 & 576 in BCD
BCD carry

1

1
 
0001

1000

0100

184
 
+0101

0111

0110

+576
 
Binary sum

0111

10000

1010
 
Add 6

0110

0110
 
BCD sum

0111

0110

0000

760

Boolean properties
AND function

X. 0 = 0
0. X = 0
X. 1 = X
1. X = X

OR function

X + 0 = X
0 + X = X
X + 1 = 1
1 + X = 1

Commutative laws

x. y = y. x
x + y = y + x

Distributive laws

x(y +z) = x.y + x.z
x + y. z = ( x+y) (x + z)

Associative laws

x(y.z) = (x. y) z
x + ( y + z) = (x + y) +z

Absorption laws

x + xy = x
x(x + y) = x
x + x'y = x+ y
x(x' + y) = xy

Demorgan’s laws

(x + y)' = x'. y'
(x. y)' = x' + y'

Duality Principle

x + x = x => x .x = x by duality
x + 1 = 1 => x. 0 = 0 by duality
x + xy = x => x(x + y) = x by duality
x + y = y + x => xy = yx by duality
x + (y+ z) = (x + y) + z => x(yz) = (xy)z by duality

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