DIGITAL BASIC - 1.2 : DIGITAL ARITHMETIC

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

Note: Above method can be used for any number system. 

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