Formula Sheets: Number System & Representation

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Digital Electronics: Num b er System &
Represen tation F orm ula Sheet f or Electrical GA TE
Num b er System Basics
• Decimal to Base-r Con v ersion (In teger P art) :
Divide b y r, collect remainders in rev erse order
where r is the base (e.g., 2 for binary , 8 for o ctal, 16 for hexadecimal).
• Decimal to Base-r Con v ersion (F ractional P art) :
Multiply b y r, collect in teger parts
• Base-r to De cimal Con v ersion :
N =
n
?
i=-m
d
i
·r
i
where d
i
is the digit in p osition i , r is the base, n is the n um b er of in teger digits, m is
the n um b er of fractional digits.
Binary Arithmetic
• Binary A ddition :
Sum =A+B, Carry = 1 if A+B= 2
• Binary Subtraction :
Difference =A-B, Borro w = 1 if A<B
• 2’s Complemen t Represen tation :
2’s Complemen t = 1’s Complemen t+1
where 1’s Complemen t = In v ert all bits.
• Subtraction using 2’s Complemen t :
A-B =A+( 2’s Complemen t of B)
Num b er Represen tations
• Signed Magnitude : MSB indicates sign (0 for p ositiv e, 1 for negativ e).
• 1’s Complemen t : Negativ e n um b er is bit wise complemen t of p ositiv e.
• 2’s Complemen t Range (n-bit) :
Range = [-2
n-1
,2
n-1
-1]
• Unsigned Binary Range (n-bit) :
Range = [0,2
n
-1]
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Digital Electronics: Num b er System &
Represen tation F orm ula Sheet f or Electrical GA TE
Num b er System Basics
• Decimal to Base-r Con v ersion (In teger P art) :
Divide b y r, collect remainders in rev erse order
where r is the base (e.g., 2 for binary , 8 for o ctal, 16 for hexadecimal).
• Decimal to Base-r Con v ersion (F ractional P art) :
Multiply b y r, collect in teger parts
• Base-r to De cimal Con v ersion :
N =
n
?
i=-m
d
i
·r
i
where d
i
is the digit in p osition i , r is the base, n is the n um b er of in teger digits, m is
the n um b er of fractional digits.
Binary Arithmetic
• Binary A ddition :
Sum =A+B, Carry = 1 if A+B= 2
• Binary Subtraction :
Difference =A-B, Borro w = 1 if A<B
• 2’s Complemen t Represen tation :
2’s Complemen t = 1’s Complemen t+1
where 1’s Complemen t = In v ert all bits.
• Subtraction using 2’s Complemen t :
A-B =A+( 2’s Complemen t of B)
Num b er Represen tations
• Signed Magnitude : MSB indicates sign (0 for p ositiv e, 1 for negativ e).
• 1’s Complemen t : Negativ e n um b er is bit wise complemen t of p ositiv e.
• 2’s Complemen t Range (n-bit) :
Range = [-2
n-1
,2
n-1
-1]
• Unsigned Binary Range (n-bit) :
Range = [0,2
n
-1]
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Co des
• Binary Co ded Decimal (BCD) :
Eac h decimal digit represen ted b y 4-bit binary(0000to1001)
• Gra y Co de : A djacen t n um b ers differ b y one bit.
Gra y Co de = Binary?( Binary» 1)
where? is X OR,» is righ t shift.
• Excess-3 Co de : BCD + 0011 (binary 3).
Excess-3 = BCD+0011
Arithmetic Op erations
• BCD A ddition :
If sum> 9 or carry = 1, add 0110 (6) to correct
• Hexadecimal A ddition/Subtraction :
Use base-16 arithmetic, carry/b orro w on = 16
• Signed Num b er Ov erflo w (2’s Complemen t) :
Ov erflo w if sign (A) = sign(B)?= sign( Result)
Key Notes
• Base V alues : Binary (r = 2 ), Octal (r = 8 ), Hexadecimal (r = 16 ).
• Con v ersion Shortcuts : Binary to Octal (group 3 bits), Binary to Hex (group 4
bits).
• BCD Range : 0–9 p er digit, 4 bits eac h.
• Gra y Co de Prop ert y : Minimizes errors in digital systems due to single-bit c hanges.
• GA TE F o cus : Practice 2’s complemen t arithmetic, BCD, and co de con v ersions.
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