CHAPTER V NUMBER SYSTEMS AND ARITHMETIC R.M. Dansereau; v.1.0 - PowerPoint PPT Presentation

INTRO. TO COMP. ENG. •CHAPTER V CHAPTER V-1 NUMBERS & ARITHMETIC CHAPTER V NUMBER SYSTEMS AND ARITHMETIC R.M. Dansereau; v.1.0

NUMBER SYSTEMS INTRO. TO COMP. ENG. •NUMBER SYSTEMS CHAPTER V-2 RADIX-R REPRESENTATION NUMBERS & ARITHMETIC • Decimal number expansion ( × ) ( × ) ( × ) ( × ) ( × ) 10 4 10 3 10 2 10 1 10 0 73625 10 7 3 6 2 5 = + + + + • Binary number representation ( × ) ( × ) ( × ) ( × ) ( × ) 2 4 2 3 2 2 2 1 2 0 10110 2 1 0 1 1 0 22 10 = + + + + = • Hexadecimal number representation ( × ) ( × ) ( × ) ( × ) ( × ) 16 4 16 3 16 2 16 1 16 0 3E4B8 16 3 14 4 11 8 = + + + + 255160 10 = R.M. Dansereau; v.1.0

NUMBER SYSTEMS INTRO. TO COMP. ENG. •NUMBER SYSTEMS -NUMBER REPRES. CHAPTER V-3 DECIMAL REPRESENTATION NUMBERS & ARITHMETIC Radix-10 Representation 73625.4385 10 10 3 10 2 10 1 10 0 – 10 2 – 10 3 – 10 4 10 5 10 4 10 1 10 5 – – ... ... 0 7 3 6 2 5 . 4 3 8 5 0 ( × ) ( × ) ( × ) ( × ) ( × ) 10 4 10 3 10 2 10 1 10 0 73625.4385 10 7 3 6 2 5 = + + + + ( × ) ( × ) ( × ) ( × ) 10 1 10 2 10 3 10 4 – – – – 4 3 8 5 + + + + R.M. Dansereau; v.1.0

NUMBER SYSTEMS INTRO. TO COMP. ENG. •NUMBER SYSTEMS -NUMBER REPRES. CHAPTER V-4 -DECIMAL REPRES. BINARY REPRESENTATION NUMBERS & ARITHMETIC Radix-2 Representation 10110.0011 2 2 5 2 4 2 3 2 2 2 1 2 0 2 1 2 2 2 3 2 4 2 5 – – – – – ... ... 0 1 0 1 1 0 . 0 0 1 1 0 MSB LSB ( × ) ( × ) ( × ) ( × ) ( × ) 2 4 2 3 2 2 2 1 2 0 10110.0011 2 1 0 1 1 0 = + + + + ( × ) ( × ) ( × ) ( × ) 2 1 2 2 2 3 2 4 – – – – 0 0 1 1 + + + + 22.1875 10 = R.M. Dansereau; v.1.0

NUMBER SYSTEMS INTRO. TO COMP. ENG. •NUMBER SYSTEMS -NUMBER REPRES. CHAPTER V-5 -DECIMAL REPRES. OCTAL REPRESENTATION NUMBERS & ARITHMETIC -BINARY REPRES. Radix-8 Representation 26516.1731 8 8 5 8 4 8 3 8 2 8 1 8 0 8 1 8 2 8 3 8 4 8 5 – – – – – ... ... 0 2 6 5 1 6 . 1 7 3 1 0 ( × ) ( × ) ( × ) ( × ) ( × ) 8 4 8 3 8 2 8 1 8 0 26516.1731 8 2 6 5 1 6 = + + + + ( × ) ( × ) ( × ) ( × ) 8 1 8 2 8 3 8 4 – – – – 1 7 3 1 + + + + 11598.24 10 = R.M. Dansereau; v.1.0

NUMBER SYSTEMS INTRO. TO COMP. ENG. •NUMBER SYSTEMS -DECIMAL REPRES. CHAPTER V-6 -BINARY REPRES. HEXADECIMAL REPRES. NUMBERS & ARITHMETIC -OCTAL REPRES. Radix-16 Representation 19AD6.F411 16 16 3 16 2 16 1 16 0 – 16 2 – 16 3 – 16 4 16 5 16 4 16 1 16 5 – – ... ... 0 1 9 A D 6 . F 4 1 1 0 ( × ) ( × ) ( × ) ( × ) ( × ) 16 4 16 3 16 2 16 1 16 0 19AD6.F411 16 1 9 A D 6 = + + + + ( × ) ( × ) ( × ) ( × ) 16 1 16 2 16 3 16 4 – – – – F 4 1 1 + + + + ≈ 105174.95 10 R.M. Dansereau; v.1.0

NUMBER SYSTEMS INTRO. TO COMP. ENG. •NUMBER SYSTEMS -BINARY REPRES. CHAPTER V-7 -OCTAL REPRES. BINARY <-> HEXADECIMAL NUMBERS & ARITHMETIC -HEXADECIMAL REPRES. BINARY -> HEXADECIMAL BINARY <-> HEXADECIMAL Group binary by 4 bits from radix point 0000 2 = 0 16 1000 2 = 8 16 Examples: 0001 2 = 1 16 1001 2 = 9 16 0010 2 = 2 16 1010 2 = 10 ( A 16 ) 0111 1011 2 = 7B 16 0011 2 = 3 16 1011 2 = 11 ( B 16 ) 7 B 0100 2 = 4 16 1100 2 = 12 ( C 16 ) 0101 2 = 5 16 1101 2 = 13 ( D 16 ) 0110 2 = 6 16 1110 2 = 14 ( E 16 ) 10 1010 0110.1100 01 2 = 2A6.C4 16 0111 2 = 7 16 1111 2 = 15 (F 16 ) 2 A 6 C 4 R.M. Dansereau; v.1.0

NUMBER SYSTEMS INTRO. TO COMP. ENG. •NUMBER SYSTEMS -BINARY REPRES. CHAPTER V-8 -OCTAL REPRES. BINARY <-> OCTAL NUMBERS & ARITHMETIC -BINARY<->HEXADECIMAL BINARY -> OCTAL BINARY <-> OCTAL Group binary bits by 3 from LSB 000 2 = 0 8 Examples: 001 2 = 1 8 010 2 = 2 8 10 100 110 2 = 246 8 011 2 = 3 8 100 2 = 4 8 2 4 6 101 2 = 5 8 110 2 = 6 8 10 101 111 011.011 11 2 = 2573.36 8 111 2 = 7 8 2 5 7 3 3 6 R.M. Dansereau; v.1.0

NUMBER SYSTEMS INTRO. TO COMP. ENG. •NUMBER SYSTEMS -OCTAL REPRES. CHAPTER V-9 -BINARY<->HEXADECIMAL BINARY -> DECIMAL NUMBERS & ARITHMETIC -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: ( × ) ( × ) ( × ) ( × ) ( × ) 2 4 2 3 2 2 2 1 2 0 10111 2 1 0 1 1 1 23 10 = + + + + = ( × ) ( × ) ( × ) ( × ) ( × ) 2 4 2 3 2 2 2 1 2 0 10110.0011 2 1 0 1 1 0 = + + + + ( × ) ( × ) ( × ) ( × ) 2 1 2 2 2 3 2 4 – – – – 0 0 1 1 + + + + 22.1875 10 = R.M. Dansereau; v.1.0

NUMBER SYSTEMS INTRO. TO COMP. ENG. •NUMBER SYSTEMS -BINARY<->HEXADECIMAL CHAPTER V-10 -BINARY<->OCTAL DECIMAL -> BINARY NUMBERS & ARITHMETIC -BINARY->DECIMAL • Integer part: Example: Convert 41.828125 10 • Modulo division of decimal 41 mod 2 1 LSB = 20 mod 2 0 = integer by 2 to get each bit, 10 mod 2 0 = starting with LSB. 5 mod 2 1 = • Fraction part: 2 mod 2 0 = 1 mod 2 1 • Multiplication decimal MSB = fraction by 2 and collect × 0.828125 2 1.65625 MSB = × resulting integers, starting 0.65625 2 1.3125 = × 0.3125 2 0.625 = with MSB. × 0.625 2 1.25 = × 0.25 2 0.5 = × LSB 0.5 2 1.0 = Therefore 41.828125 10 101001.110101 2 = R.M. Dansereau; v.1.0

NUMBER SYSTEMS INTRO. TO COMP. ENG. •NUMBER SYSTEMS -BINARY<->HEXADECIMAL CHAPTER V-11 -BINARY->DECIMAL FLOATING POINT NUMBERS NUMBERS & ARITHMETIC -DECIMAL->BINARY • Floating point numbers can be represented with a sign bit, a fraction (often referred to as the mantissa), and an exponent. × 10 3 • Example 1: 267.426 0.267426 , where the sign is negative, – = – the fraction is 0.267426 and the exponent is . 3 × 2 6 • Example 2: 0101011.1001 0.1010111 , where the sign is = positive, the fraction is 0.1010111 , and the exponent is 0110 . • Sample IEEE Floating-Point Formats s e f 32-bit 1 8 23 64-bit s e f 1 11 52 R.M. Dansereau; v.1.0

BINARY NUMBERS INTRO. TO COMP. ENG. •NUMBER SYSTEMS -DECIMAL->BINARY CHAPTER V-12 -POWERS OF 2 UNSIGNED INTEGER NUMBERS & ARITHMETIC -FLOATING POINT • The range for an -bit radix- unsigned integer is n r [ , ] n 0 r 10 1 – • Example: For a 16-bit binary unsigned integer, the range is [ , ] [ , ] 0 2 16 1 0 65535 – = which has a binary representation of 0000 0000 0000 0000 = 0 0000 0000 0000 0001 = 1 0000 0000 0000 0010 = 2 . . . 1111 1111 1111 1110 = 65534 1111 1111 1111 1111 = 65535 R.M. Dansereau; v.1.0

BINARY NUMBERS INTRO. TO COMP. ENG. •NUMBER SYSTEMS •BINARY NUMBERS CHAPTER V-13 -UNSIGNED INTEGERS SIGNED INTEGERS (1) NUMBERS & ARITHMETIC • The range for an -bit radix- signed integer is n r [ , ] n 1 n 1 – – r 10 r 10 1 – – • 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 [ , ] [ , ] 2 16 1 2 16 1 – – 1 32768 32767 – – = – R.M. Dansereau; v.1.0

BINARY NUMBERS INTRO. TO COMP. ENG. •NUMBER SYSTEMS •BINARY NUMBERS CHAPTER V-14 -UNSIGNED INTEGERS SIGNED INTEGERS (2) NUMBERS & ARITHMETIC -SIGNED INTEGERS • There are a number of different representations for signed integers, each which has its own advantage • Signed-magnitude representation: • 1010 0001 0110 1111 • Signed-1’s complement representation: • 1101 1110 1001 0000 • Signed-2’s complement representation: • 1101 1110 1001 0001 • The above examples are all the same number, 8559 10 . – R.M. Dansereau; v.1.0

BINARY NUMBERS INTRO. TO COMP. ENG. •NUMBER SYSTEMS •BINARY NUMBERS CHAPTER V-15 -UNSIGNED INTEGERS SIGNED-MAGNITUDE NUMBERS & ARITHMETIC -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 6 10 can be represented as the – 7-bit magnitude of using 6 10 000 0110 and then the sign bit appended to the MSB to form 1000 0110 R.M. Dansereau; v.1.0

BINARY NUMBERS INTRO. TO COMP. ENG. •BINARY NUMBERS -UNSIGNED INTEGERS CHAPTER V-16 -SIGNED INTEGERS RADIX COMPLEMENTS NUMBERS & ARITHMETIC -SIGNED-MAGNITUDE • The radix complement , or r’s complement , of an integer representation for an -digit integer is defined as n r n 10 number 10 – • The diminished radix complement , or (r - 1)’s complement , of an integer representation for an -digit integer is defined as n ( ) r n 10 1 10 number 10 – – • Example: Find the r’s and (r - 1)’s complement for 3764 10 r’s complement (r - 1)’s complement ( ) 10 5 10 5 3764 96236 1 3764 96235 – = – – = R.M. Dansereau; v.1.0