CHAPTER V NUMBER SYSTEMS AND ARITHMETIC


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1 CHAPTER V1 CHAPTER V CHAPTER V NUMBER SYSTEMS AND ARITHMETIC
2 CHAPTER V2 NUMBER SYSTEMS RADIXR REPRESENTATION Decimal number expansion = ( ) + ( ) + ( ) + ( ) +( ) Binary number representation = ( ) + ( ) + ( ) + ( ) + ( ) = Hexadecimal number representation 3E4B8 16 = ( ) + ( ) + ( ) + ( ) + ( ) =
3 CHAPTER V3 NUMBER SYSTEMS DECIMAL REPRESENTATION NUMBER REPRES. Radix10 Representation = ( ) + ( ) + ( ) + ( ) + ( ) + ( ) + ( ) + ( ) + ( )
4 CHAPTER V4 NUMBER SYSTEMS BINARY REPRESENTATION NUMBER REPRES. DECIMAL REPRES. Radix2 Representation MSB LSB = ( ) + ( ) + ( ) + ( ) +( ) = + ( ) + ( ) + ( ) +( )
5 CHAPTER V5 NUMBER SYSTEMS OCTAL REPRESENTATION NUMBER REPRES. DECIMAL REPRES. BINARY REPRES. Radix8 Representation = ( ) + ( ) + ( ) + ( ) +( ) + ( ) + ( ) + ( ) +( ) =
6 CHAPTER V6 NUMBER SYSTEMS HEXADECIMAL REPRES. DECIMAL REPRES. BINARY REPRES. OCTAL REPRES. Radix16 Representation 19AD6.F A D 6. F AD6.F = ( ) + ( ) + ( A 16 2 ) + ( D 16 1 ) + ( ) + ( F 16 1 ) + ( ) + ( ) +( )
7 CHAPTER V7 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 V8 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 V9 NUMBER SYSTEMS BINARY > DECIMAL OCTAL REPRES. BINARY<>HEXADECIMAL BINARY<>OCTAL Perform radix2 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 V10 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 V11 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 FloatingPoint Formats 32bit s e f bit s e f
12 CHAPTER V12 BINARY NUMBERS UNSIGNED INTEGER DECIMAL>BINARY POWERS OF 2 FLOATING POINT The range for an nbit radix r unsigned integer is n 1 [ 0, r 10 ] Example: For a 16bit binary unsigned integer, the range is [ 0, ] = [ 0, 65535] which has a binary representation of = = = = = 65535
13 CHAPTER V13 BINARY NUMBERS SIGNED INTEGERS (1) BINARY NUMBERS UNSIGNED INTEGERS The range for an nbit radix r signed integer is rn 1 10 n 1 1 [, r 10 ] The mostsignificant bit is used as a sign bit, where 0 indicates a positive integer and 1 indicates a negative integer. Example: For a 16bit binary signed integer, the range is [ , ] = [ 32768, 32767]
14 CHAPTER V14 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 Signedmagnitude representation: Signed1 s complement representation: Signed2 s complement representation: The above examples are all the same number,
15 CHAPTER V15 BINARY NUMBERS SIGNEDMAGNITUDE BINARY NUMBERS UNSIGNED INTEGERS SIGNED INTEGERS The signedmagnitude binary integer representation is just like the unsigned representation with the addition of a sign bit. For instance, using 8bits, the number can be represented as the 7bit magnitude of using and then the sign bit appended to the MSB to form
16 CHAPTER V16 BINARY NUMBERS RADIX COMPLEMENTS BINARY NUMBERS UNSIGNED INTEGERS SIGNED INTEGERS SIGNEDMAGNITUDE The radix complement, or r s complement, of an integer representation for an ndigit integer is defined as r n 10 number 10 The diminished radix complement, or (r  1) s complement, of an integer representation for an ndigit 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 V17 BINARY NUMBERS 1 S COMPLEMENT BINARY NUMBERS SIGNED INTEGERS SIGNEDMAGNITUDE RADIX COMPLEMENTS The 1 s complement (diminished radix complement) binary integer representation for an nbit 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 8bits, the number can be represented as the 8bit positive number using and then each of the bits flipped to form the 1 s complement
18 CHAPTER V18 BINARY NUMBERS 2 S COMPLEMENT BINARY NUMBERS SIGNEDMAGNITUDE RADIX COMPLEMENTS 1 S COMPLEMENT The 2 s complement (radix complement) binary integer representation for an nbit integer is defined as 2 n 10 number 10 In essence, this takes the 1 s complement and adds one. For instance, using 8bits, the number can be represented as the 8bit 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 V19 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 Signedmagnitude 1 s complement 2 s complement Notice that all representations are the same for positive numbers!!!!
20 CHAPTER V20 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 V21 BINARY ARITHMETIC UNSIGNED SUBTRACTION BINARY NUMBERS BINARY ARITHMETIC UNSIGNED ADDITION Unsigned binary subtraction follows the standard rules. Examples Borrows Borrows Borrows Borrows
22 CHAPTER V22 BINARY ARITHMETIC SIGNED ADDITION BINARY NUMBERS BINARY ARITHMETIC UNSIGNED ADDITION UNSIGNED SUBTRACT. Signedmagnitude 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 V23 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 rereprepresented as follows. SUM = A + ( B) Or in general any two numbers can be added as follows. SUM = ( ± A) + ( ± B)
24 CHAPTER V24 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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