ࡱ> XZW}[@ 'bjbj44 "ViVi"888L7777Td7|Lzn89999#H#H#Hyyyyyyy$u{R}py8H{F#HHHy99 yLLLH989yLHyLLMno8r98 p(2a7I$p sy0zp7~JT7~@rLL7~8rT#H7HLEH QHS#H#H#HyyLL27LLL7Number Systems Purposes: To understand how does a digital computer work. Binary digital computers only work with 1s and 0s, or high and low voltage, or true and false. To convert among different number systems. We use decimal numbers everyday. Computers understand only binary numbers, which are lengthy and inconvenient to human beings. Octal and Hexadecimal numbers are introduced to make both happy: they are easier to be converted to binary numbers and also easier for us to handle. Classification:  SHAPE \* MERGEFORMAT  Unsigned Numbers Radices and Characters: Binary: 0, 1 Decimal: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 Octal: 0, 1, 2, 3, 4, 5, 6, 7 Hexadecimal: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F Structure of a number:  SHAPE \* MERGEFORMAT  Note: If no fractional part, the radix point can be omitted! Positional Notation:  EMBED Equation.3 , where  EMBED Equation.3 , and  EMBED Equation.3  is the radix. Practice Problem 1: Given the positional notations of the following numbers: (1101.1)2, (724)8, and (BCD)16. Converting Numbers Between Bases: Conversion to Decimal: write the number using the positional notation and then perform decimal arithmetic to compute the result, which is the decimal number. To speedup, using HORNERs RULE. Example: Convert (6734)8 into decimal. Example: Convert (A6B3)16 into decimal using Horners rule. Example: Convert (1101)2 into decimal. Conversion from Decimal: Integer part: Divide the decimal number by the base to which we want to convert and cast out the reminders. Fractional part: Multiply the decimal number by the base to which we want to convert and cast out the integer part. Rationale: based on the positional notation. Example: Convert (29)10 into Binary, to Octal, and to Hexadecimal. Example: Convert (217)10 into a hexadecimal, an octal and a binary number. Example: Convert (0.57)10 to binary, octal and hexadecimal Example: Convert (0.6328125)10 to binary, octal and hexadecimal. Note: When converting to binary, the process may never stop for some irrational decimal numbers. In this case, we need to (1) figure out the precision of the decimal number; (2) determine when to stop based on the precision. Eg. (0.57)10 =0.10010001 since the precision of (0.57)10 is 1/100 and we need at least 7 bits for this precision. Binary to/from Octal and Hexadecimal: Starting at the binary point, cast off three (four) bits at a time and convert each group to its octal (hexadecimal) equivalent. Padding 0s to the left for the integer part and to the right for the fractional part when necessary. Example: Convert (10110111)2 to an octal and a hexadecimal number. Example: Convert (0.10110111)2 to an octal and a hexadecimal number Question: How to convert octal to/from hexadecimal? Practice Problem 2: Fill in the missing entries DecimalBinaryOctalHexadecimal13610101100357BC23.1570.10101100110.532AF.234500000000000000 Signed Numbers Signed-Magnitude: 0 for positive, 1 for negative, followed by the magnitude in different radix. A signed-magnitude number can be represented as  EMBED Equation.3  where  EMBED Equation.3  is the sign bit and  EMBED Equation.3  is the ith character in the number. Note: Generally we use floating-point numbers to represent a mixed radix-point number. Thus we focus on signed integers now. Example: Given the signed-magnitude notations of (+9)10 and (-9)10 in binary, octal, decimal, and hexadecimal form. (+9)10 = ( ) 2sm and (-9)10 = ( )2sm (+9)10 = ( ) 8sm and (-9)10 = ( )8sm (+9)10 = ( ) 10sm and (-9)10 = ( )10sm (+9)10 = ( ) 15sm and (-9)10 = ( )15sm Diminished Radix-Complement (for integer): 0 for positive, followed by the magnitude in different radix; (r-1) for negative, followed by the complement of the magnitude of the number:  EMBED Equation.3  where r is the radix and N is the number. The complement is computed by taking each character in the number and subtracting the character from (r-1). Example (Nines Complement): (+9574) 10 = ( ) 9C and (-9574)10 = ( )9C Example (Ones Complement): (+9)10 = ( )1C and (-9)10 = ( )1C What is the result of the sum of (+9)10 and (-9)10 in ones complement? Negative zero Example (15s Complement): (+30) 10 = ( ) 15C and (-30)10 = ( )15C Radix-Complement (for integer): For a negative number, adding one to its Diminished-Complement; For positive numbers, radix-complement = diminished radix-complement. Example (Tens Complement): (+9574) 10 = ( ) 10C and (-9574)10 = ( )10C Example (Twos Complement): (+8)10 = ( )2C and (-8)10 = ( )2C What is the result of the sum of (+8)10 and (-8)10 in ones complement? unique zero Example (16s Complement): (+30) 10 = ( ) 16C and (-30)10 = ( )16C Fill in the entries of the following table: DecimalSigned-magnitude1s complement2s complementnotes01234567-7-6-5-4-3-2-1 Remarks: The sign bit is part of the number in ones and twos complement notation and is included in any computations. In twos complement , the number 0 is expressed as all zeros, the number -1 is expressed as all 1s. Digital computers use 2s complement for arithmetic operation. Question: What is the largest positive/negative signed 32-bit numbers in twos complement? Floating-Point Numbers Remark: We only consider binary numbers. Floating-point expands the range of numbers representable in a digital computer and improves precision. It is the binary scientific notation: Examples:  EMBED Equation.3  and  EMBED Equation.3  sign: + Mantissa one non-zero digit followed by the radix point: 5.42713 Base: 10 Exponent signed integers: -23 IEEE 754 Single-Precision Floating-Point Standard: IEEE 754 Double-Precision Floating-Point Standard: Remarks: The hidden 1 must be included in the significand to get the mantissa. Use biased exponent to facilitate sorting. For single-precision fp, the bias is (127)10, for double-precision, it is (1023)10. The value of a floating-point number is: Example: Show the IEEE 754 binary representation for the floating point number (-0.75)10 in single-precision for a 32-bit computer. Practice Problem 3: What decimal number does the following floating-point number represent? 1100 0001 1101 0000 0000 0000 0000 0000 Special representations in IEEE 754 Standard   ExponentSignificandExponentSignificandObject000000Non-zero0Non-zeroDenomalized numbers1-254Anything1-2046anythingNormalized FP numbers255020470infinity255Non-zero2047Non-zeroNaN Overflow: The exponent of the number is too large to be put in the exponent field. Underflow: The exponent is too small (or the absolute value is too large) to be put in the exponent field. Double-Precision FP increases both range and precision because there are more number of bits in the exponent and the significand fields. Question: What are the smallest normalized numbers (positive and negative) that a single-precision FP can represent? Practice Problem: Fill in the entries in the following table. 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