CHAPTER V NUMBER SYSTEMS AND ARITHMETIC

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1 CHAPTER V-1 CHAPTER V CHAPTER V NUMBER SYSTEMS AND ARITHMETIC

2 CHAPTER V-2 NUMBER SYSTEMS RADIX-R REPRESENTATION Decimal number expansion = ( ) + ( ) + ( ) + ( ) +( ) Binary number representation = ( ) + ( ) + ( ) + ( ) + ( ) = Hexadecimal number representation 3E4B8 16 = ( ) + ( ) + ( ) + ( ) + ( ) =

3 CHAPTER V-3 NUMBER SYSTEMS DECIMAL REPRESENTATION -NUMBER REPRES. Radix-10 Representation = ( ) + ( ) + ( ) + ( ) + ( ) + ( ) + ( ) + ( ) + ( )

4 CHAPTER V-4 NUMBER SYSTEMS BINARY REPRESENTATION -NUMBER REPRES. -DECIMAL REPRES. Radix-2 Representation MSB LSB = ( ) + ( ) + ( ) + ( ) +( ) = + ( ) + ( ) + ( ) +( )

5 CHAPTER V-5 NUMBER SYSTEMS OCTAL REPRESENTATION -NUMBER REPRES. -DECIMAL REPRES. -BINARY REPRES. Radix-8 Representation = ( ) + ( ) + ( ) + ( ) +( ) + ( ) + ( ) + ( ) +( ) =

6 CHAPTER V-6 NUMBER SYSTEMS HEXADECIMAL REPRES. -DECIMAL REPRES. -BINARY REPRES. -OCTAL REPRES. Radix-16 Representation 19AD6.F A D 6. F AD6.F = ( ) + ( ) + ( A 16 2 ) + ( D 16 1 ) + ( ) + ( F 16 1 ) + ( ) + ( ) +( )

7 CHAPTER V-7 NUMBER SYSTEMS BINARY <-> HEXADECIMAL -BINARY REPRES. -OCTAL REPRES. -HEXADECIMAL REPRES. BINARY <-> HEXADECIMAL = = = = = = = = = = = 10 (A 16 ) = 11 (B 16 ) = 12 (C 16 ) = 13 (D 16 ) = 14 (E 16 ) = 15 (F 16 ) BINARY -> HEXADECIMAL Group binary by 4 bits from radix point Examples: = 7B 16 7 B = 2A6.C A 6 C 4

8 CHAPTER V-8 NUMBER SYSTEMS BINARY <-> OCTAL -BINARY REPRES. -OCTAL REPRES. -BINARY<->HEXADECIMAL BINARY <-> OCTAL = = = = = = = = 7 8 Examples: BINARY -> OCTAL Group binary bits by 3 from LSB = =

9 CHAPTER V-9 NUMBER SYSTEMS BINARY -> DECIMAL -OCTAL REPRES. -BINARY<->HEXADECIMAL -BINARY<->OCTAL Perform radix-2 expansion Multiply each bit in the binary number by 2 to the power of its place. Then sum all of the values to get the decimal value. Examples: = ( ) + ( ) + ( ) + ( ) + ( ) = = ( ) + ( ) + ( ) + ( ) +( ) = + ( ) + ( ) + ( ) +( )

10 CHAPTER V-10 NUMBER SYSTEMS DECIMAL -> BINARY -BINARY<->HEXADECIMAL -BINARY<->OCTAL -BINARY->DECIMAL Integer part: Modulo division of decimal integer by 2 to get each bit, starting with LSB. Fraction part: Multiplication decimal fraction by 2 and collect resulting integers, starting with MSB. Example: Convert mod 2 = 1 LSB 20 mod 2 = 0 10 mod 2 = 0 5 mod 2 = 1 2 mod 2 = 0 1 mod 2 = 1 MSB = = = = = = 1.0 MSB LSB Therefore =

11 CHAPTER V-11 NUMBER SYSTEMS FLOATING POINT NUMBERS -BINARY<->HEXADECIMAL -BINARY->DECIMAL -DECIMAL->BINARY Floating point numbers can be represented with a sign bit, a fraction (often referred to as the mantissa), and an exponent. Example 1: = , where the sign is negative, the fraction is and the exponent is 3. Example 2: = , where the sign is positive, the fraction is , and the exponent is Sample IEEE Floating-Point Formats 32-bit s e f bit s e f

12 CHAPTER V-12 BINARY NUMBERS UNSIGNED INTEGER -DECIMAL->BINARY -POWERS OF 2 -FLOATING POINT The range for an n-bit radix- r unsigned integer is n 1 [ 0, r 10 ] Example: For a 16-bit binary unsigned integer, the range is [ 0, ] = [ 0, 65535] which has a binary representation of = = = = = 65535

13 CHAPTER V-13 BINARY NUMBERS SIGNED INTEGERS (1) BINARY NUMBERS -UNSIGNED INTEGERS The range for an n-bit radix- r signed integer is rn 1 10 n 1 1 [, r 10 ] The most-significant bit is used as a sign bit, where 0 indicates a positive integer and 1 indicates a negative integer. Example: For a 16-bit binary signed integer, the range is [ , ] = [ 32768, 32767]

14 CHAPTER V-14 BINARY NUMBERS SIGNED INTEGERS (2) BINARY NUMBERS -UNSIGNED INTEGERS -SIGNED INTEGERS There are a number of different representations for signed integers, each which has its own advantage Signed-magnitude representation: Signed-1 s complement representation: Signed-2 s complement representation: The above examples are all the same number,

15 CHAPTER V-15 BINARY NUMBERS SIGNED-MAGNITUDE BINARY NUMBERS -UNSIGNED INTEGERS -SIGNED INTEGERS The signed-magnitude binary integer representation is just like the unsigned representation with the addition of a sign bit. For instance, using 8-bits, the number can be represented as the 7-bit magnitude of using and then the sign bit appended to the MSB to form

16 CHAPTER V-16 BINARY NUMBERS RADIX COMPLEMENTS BINARY NUMBERS -UNSIGNED INTEGERS -SIGNED INTEGERS -SIGNED-MAGNITUDE The radix complement, or r s complement, of an integer representation for an n-digit integer is defined as r n 10 number 10 The diminished radix complement, or (r - 1) s complement, of an integer representation for an n-digit integer is defined as ( r n ) number 10 Example: Find the r s and (r - 1) s complement for r s complement (r - 1) s complement = ( ) 3764 = 96235

17 CHAPTER V-17 BINARY NUMBERS 1 S COMPLEMENT BINARY NUMBERS -SIGNED INTEGERS -SIGNED-MAGNITUDE -RADIX COMPLEMENTS The 1 s complement (diminished radix complement) binary integer representation for an n-bit integer is defined as ( 2 n ) number 10 In essence, this takes the positive version of the number and flips all of the bits. For instance, using 8-bits, the number can be represented as the 8-bit positive number using and then each of the bits flipped to form the 1 s complement

18 CHAPTER V-18 BINARY NUMBERS 2 S COMPLEMENT BINARY NUMBERS -SIGNED-MAGNITUDE -RADIX COMPLEMENTS -1 S COMPLEMENT The 2 s complement (radix complement) binary integer representation for an n-bit integer is defined as 2 n 10 number 10 In essence, this takes the 1 s complement and adds one. For instance, using 8-bits, the number can be represented as the 8-bit positive number using and then each of the bits flipped to form the 1 s complement and then add 1 to form the 2 s complement

19 CHAPTER V-19 BINARY NUMBERS SIGNED EXAMPLES BINARY NUMBERS -RADIX COMPLEMENTS -1 S COMPLEMENT -2 S COMPLEMENT Below are some examples for the signed binary numbers using 6 bits. Decimal Signed-magnitude 1 s complement 2 s complement Notice that all representations are the same for positive numbers!!!!

20 CHAPTER V-20 BINARY ARITHMETIC UNSIGNED ADDITION BINARY NUMBERS -1 S COMPLEMENT -2 S COMPLEMENT -SIGNED EXAMPLES Unsigned binary addition follows the standard rules of addition. Examples Carries Carries Carries Carries

21 CHAPTER V-21 BINARY ARITHMETIC UNSIGNED SUBTRACTION BINARY NUMBERS BINARY ARITHMETIC -UNSIGNED ADDITION Unsigned binary subtraction follows the standard rules. Examples Borrows Borrows Borrows Borrows

22 CHAPTER V-22 BINARY ARITHMETIC SIGNED ADDITION BINARY NUMBERS BINARY ARITHMETIC -UNSIGNED ADDITION -UNSIGNED SUBTRACT. Signed-magnitude Add magnitudes if signs are the same, give result the sign Subtract magnitudes if signs are different. Absence or presence of an end borrow determines the resulting sign compared to the augend. If negative, then a 2 s complement correction must be taken. 2 s complement Add the numbers using normal addition rules. Carry out bit is discarded. 1 s complement Easiest to convert to 2 s complement, perform the addition, and then convert back to 1 s complement. This is done as follows: Add 1 to each integer, add the integers, subtract 1 from the result

23 CHAPTER V-23 BINARY ARITHMETIC SIGNED SUBTRACTION BINARY ARITHMETIC -UNSIGNED ADDITION -UNSIGNED SUBTRACT. -SIGNED ADDITION Typically want to do addition or subtraction of A and B as follows. SUM = A + B DIFFERENCE = A B If we use 2 s complement, we can make life easy on us since addition and subtraction are done in the same manner: with addition only!!! A subtraction can be re-reprepresented as follows. SUM = A + ( B) Or in general any two numbers can be added as follows. SUM = ( ± A) + ( ± B)

24 CHAPTER V-24 BINARY ARITHMETIC SIGNED MATH EXAMPLE BINARY ARITHMETIC -UNSIGNED ADDITION -UNSIGNED SUBTRACT. -SIGNED ADDITION Subtraction of signed numbers can best be done with 2 s complement. Performed by taking the 2 s complement of the subtrahend and then performing addition (including the sign bit). Example: = = s complement discard carry out Carries standard addition = -( ) = s complement

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