Digital Representation

 Number System

• Computers use binary (base 2) number system, as they are made from binary digital components known as transistors.
• There are also hexadecimal (base 16) and octal (base 8) number systems.

Decimal (Base 10) Number System

• It has ten symbols: 0-9, called digits.
• It uses positional notation.
• We can denote it with an optional suffix D if ambiguity arises.

Binary (Base 2) Number System

• It also uses positional notation.
• It has only 2 notations: 0 and 1.
• We can denote it with the symbol B.
• Eight bit is equal to a byte.

Hexadecimal (Base 16) Number System

• It also uses positional notation.
• It uses 16 symbols: 0-9 and A-F.
• We can denote it with a suffix H.
• Each hexadecimal digit is also called a hex digit.
• Most programming language accept both lower case a-f and upper case A-F.

Number Representation-Unsigned

• If there are N bits in the binary number, the range of the number is:
• 0 .. 2 **N 1.  Assume 3 bits: 2**3- 1 = 8 -1=7
• 000, 001, 010, 011
• 100, 101, 110, 111
• Range for 8 bits? 0..2 8 1 = 0..256 1=0..255
• Why unsigned numbers?
• Memory address, cluster number (file system) and process identifier (PID).

• Each group of binary bits can be converted into a single hexadecimal digit.

Number Representation - Signed

• For signed integers , the leftmost bit is used to indicate the sign
• 0 for positive
• 1 for negative
• There are three ways to represent signed integers:
• 1. Sign and magnitude
• 2. 1’s complement
• 3. 2’s complement                                                                                                                                   Negative numbers have different representations
• Sign and magnitude: negative values are represented by changing the most significant bit (b 3)
• 1’s complement: negative values are obtained by complementing each bit of the corresponding positive number.
• 2’s complement: obtain by forming bit complement of that number, then add 1. 

Addition

There are four rules that need to be followed when adding two binary numbers. These

are:

0 + 0 = 0

1 + 0 = 1

1 + 1 = 10 (binary for 2)

1 + 1 + 1 = 11 (binary for 3)

Subtraction

Rules and tricks: Binary subtraction is much easier than the decimal subtraction when

you remember the following rules:

0-0 = 0

1-0 = 1

1-1 = 0

0-1 = 1 (Borrow 1)

For signed integers, the leftmost bit is used to indicate the sign:

• 0 for positive
• 1 for negative
• Range: -2**n**(- 1) to +2**n**(-1)- 1
• 4 bits: -8 to 7 (2**3 to 2**3**(-1)) , 8 bits: -128 to 127 (2**7 to 2**7**(-1))

OVERFLOW occurs when the answer does not fir in the number range.

Overflow (Addition)

●If 2 Two's Complement numbers are added, and they both have the same sign (both

positive or both negative), then overflow occurs if and only if the result has the

opposite sign.

●Overflow never occurs when adding operands with different signs.

• Adding two positive numbers must give a positive result
• Adding two negative numbers must give a negative result

Overflow occurs if:

(+A) + (+B) = −C

(−A) + (−B) = +C

Overflow (Subtraction)

●If 2 Two's Complement numbers are subtracted, and their signs are different, then

overflow occurs if and only if the result has the same sign as the subtrahend (what is

being subtracted).

●Overflow occurs if

(+A) − (−B) = −C

○

(−A) − (+B) = +C