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# COMPUTR HRDW FOUNDATNS CSCE 210

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This 45 page Class Notes was uploaded by Trace Mante MD on Monday October 26, 2015. The Class Notes belongs to CSCE 210 at University of South Carolina - Columbia taught by C. Huang in Fall. Since its upload, it has received 38 views. For similar materials see /class/229583/csce-210-university-of-south-carolina-columbia in Computer Science and Engineering at University of South Carolina - Columbia.

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Date Created: 10/26/15

CSCE 210 iComputer Hardware Foundations ChinTser Huang huanclctcsescedu University of South Carolina Chapter 3 Number Systems Counting and Arithmetic Decimal or base 10 number system Origin counting on the fingers Digit from the Latin word a igtus meaning finger Base the number of different digits including zero in the number system Example Base 10 has 10 digits 0 through 9 912009 3 Counting and Arithmetic Binay or base 2 Btbinary digit 2 digits 0 and 1 Octal or base 88 digits 0 through 7 Hexadecimal or base 16 16 digits 0 through 9 followed by A through F 912009 Why Binary Early computer design was decimal Mark I and ENIAC John von Neumann proposed binary data processing 1945 Simplified computer design Used for both instructions and data Natural relationship between 0 0nOff switches and True YES calculation using Boolean logic 912009 Keeping Track of the Bits Bits commonly stored and manipulated in groups 8 bits 1 byte 4 bytes 1 word in many systems Number of bits used in calculations Affects accuracy of results Limits size of numbers manipulated by the computer 912009 Numbers Physical Representation Same number of oranges different numerals Caveman IIIII Roman V I Arabic 5 different bases 510 1012 123 912009 7 Number System Roman position independent Modern based on positional notation place value Decimal system system of positional notation based on powers of 10 Binary system system of positional notation based powers of 2 Octal system system of positional notation based on powers of 8 Hexadecimal system system of positional notation based powers of 16 912009 Positional Notation 434x1013x10 912009 Base 10 10 s place 1 s place Place 101 10 Value 10 1 Evaluate 4 x 10 3 x1 Sum 40 3 f Positional Notation Base 10 5275x1022x1017x10 100 s place 10 s place 1395 Place Place 102 101 100 Value 100 10 1 Evaluate 5 x 100 2 x 10 7 x1 Sum 500 20 7 912009 10 g Positional Notation Octal 6248 40410 64 s place 8 s place 1 s place Place 82 81 8 Value 64 8 1 Evaluate 6 x 64 2 x 8 4 x 1 Sum 1 384 16 4 Base 10 912009 11 Positional Notation HexadeCI mal 670416 2637 210 4096 s place 256 s place 16 s place 1 s place Place 163 162 161 16 Value 4096 256 16 1 Evaluate 6 x 4096 7 x 256 0 x 16 4 x 1 Sum 039 24576 1792 o 4 Base 10 912009 Counting in Base 2 110 111 1000 1 x 23 1001 1 x 23 150140009 1 x 23 O 1 2 3 4 5 6 7 8 9 H O 13 Positional Notation Binary 1101 01102 21410 Place 27 26 25 24 23 22 21 2 Value 128 64 32 16 8 4 2 1 Evaluate 1x 1x 0x32 1x16 0x8 1x4 1x2 0x1 128 64 Sum W 128 64 o 16 o 4 2 0 Base 10 912009 14 i Estimating Magnitude Binary 1 01 01102 21410 1101 01102 gt 19210 128 64 additional bits to the righ Place gt 26 25 24 23 22 21 2 Value 128 64 32 16 s 4 2 1 Evaluate 1x 1x 0x32 1x16 0x8 1x4 1x2 0x1 128 64 Sum W 128 64 o 16 o 4 2 0 Base 10 0117an 1 I Range of Possible Numbers R BK where R range B base K number of digits Example 1 Base 10 2 decimal digits R 102 100 different numbers O99 Example 2 Base 2 16 binary digits R 215 65536 or 64K 16bit PC can store 65536 different number values 912009 16 Decimal Range for Bit Widths 8 10 16 20 32 64 128 912009 2 O and 1 16 0 to 15 256 1024 1K 65536 64K 1048576 1M 4294967296 4G Approx 16 x 1019 Approx 26 x 1038 17 Base The number of different symbols required to represent any given number The largerthe base the more numerals are required Base 10 O123456789 Base 2 01 Base 8 O1234567 Base 16 O123456789ABCDEF 912009 18 Number of Symbols vs Number of Digits In general the largerthe base the more symbols required but the fewerdigits needed to represent a number Example 1 6516 10110 1458 110 01012 Example 2 11C16 28410 4348 1000111002 912009 19 Base Problem Largest Single Digit 6 DeCImaI 3 9 O t 6 7 C a 1 6 Hexadecnmal F 9 B39 1 1 mary O 912009 g Addition with a Carry Base Problem Carry Answer L 6 DeCImaI 4 Carry the 10 10 O 6 C th 8 10 cta 2 arry e 6 Hexadecnmal Carry the 16 10 A B39 1 C th 2 10 Inary 1 arry e 912009 Base 10 Addition Table 310 quot39 610 910 22 Base 8 Addition Table 3868118 0 1 2 3 4 5 0 0 1 2 3 4 5 1 1 2 3 4 5 6 2 2 3 4 5 6 7 912009 10 11 12 13 12 13 14 11 12 13 14 15 16 no 8 or 9 of course 23 Binary Addition Table It s very simple 912009 24 1 1 1 1 1 1 0 1 912009 25 Base 10 Multiplication Table 310 x 610 1810 912009 21 etc 42 49 18 27 36 45 54 63 26 Base 8 Multiplication Table 38x68228 x012345u7 16 25 34 43 6 6 14 22 3O 36 44 52 7 7 16 25 34 43 52 61 912009 Binary Multiplication Table Again it s very simple 912009 28 g Binary Multiplication 1 1 1 1 1 1 5 place 0 2 5 place 110 H 4 s place bits shifted to line up with 4 s place of multiplier 1 0 0 0 Result 0 O H 912009 29 i Binary Arithmetic using Boolean Logic Addition Boolean using XOR ancl AND Multiplication AND Shift Division Shift 912009 30 g Binary Addition Boolean logic without performing arithmetic EXCLUSIVE0R Output is 1 only if either input but not both inputs is a 1 AND carry bit Output is 1 if and only both inputs are a 1 912009 31 Binary Multiplication Boolean logic without performing arithmetic AND cary bit Output is 1 if and only both inputs are a 1 Shift Shifting a number in any base left one digit multiplies its value by the base Shifting a number in any base right one digit divides its value by the base Examples a 1010 shift left 10010 n 1010 shift right 110 n 102 shift left 1002 n 102 shift right 12 912009 32 Converting from Base 10 Powers Table 2 256 128 64 32 16 8 4 2 8 32768 4096 512 64 8 16 65536 4096 256 16 912009 33 From Base 10 to Base 2 4210 1010102 Base 39 l 2 64 32 16 8 4 2 1 1 0 1 0 1 0 101 6 1 08 2g 99 01 Integer 4232 39 O v 1 0 1 O 1 1O 1O 2 2 O O Remainder 912009 From Base 10 to Base 2 Base 10 42 Remainder Quotient 2 42 0 Least significant bit 2 Z 1 2 10 0 2 E 1 2 2 0 2 1 Most significant bit Base 2 101010 912009 35 V From Base 10 to Base 16 573510 166716 Base 16 65536 4096 256 16 1 6 6 Integer 5735 4096 1639 256 103 16 Remainder 5735 4096 1639 1536 103 96 1639 103 7 912009 From Base 10 to Base 16 Remainder Base 10 5735 QUOtlent rsq 5735 7 Least significant bit 16 358 6 16 22 6 16 1 1 Most significant bit 16 0 Base 16 1667 912009 37 From Base 8 to Base 10 376310 Power I 83 I 82 I 81 I 8 512 64 8 1 X 7 X 2 X 6 X 3 Sum for Base 10 3584 128 48 3 912009 5 From Base 8to Base 10 376310 7 Q 56 2 58 g 464 6 470 Q 3760 3 3763 912009 39 From Base 16 to Base 2 The nibble approach Hex easier to read and write than binary Base 16 1 F 6 7 Base2 0001 1111 0110 0111 Why hexadecimal Modern computer operating systems and networks present variety of troubleshooting data in hex format 912009 40 Fractions Number point or radix point Decimal point in base 10 Binary point in base 2 No exact relationship between fractional numbers in different number bases Exact conversion may be impossible 912009 41 Decimal Fractions Move the number point one place to the right Effect multiplies the number by the base number Example 139310 139010 Move the number point one place to the left Effect divides the number by the base number Example 133010 13910 912009 42 Fractions Base 10 and Base 2 258910 Place 10391 10392 10393 10394 Value 110 1100 11000 110000 Evaluate 2 x 110 5 x 1100 8 x 11000 9 x11000 Sum 2 05 008 0009 1010112 0671875 Place 2391 2392 2393 2394 2395 2396 Value 12 14 18 116 132 164 Evaluate 1x12 0x14 1x18 0x116 1x132 1x164 Sum 5 0125 003125 0015625 912009 43 Fractions Base 10 and Base 2 No general relationship between fractions of types 110k and 12k Therefore a number representable in base 10 may not be representable in base 2 But the converse is true all fractions of the form 12k can be represented in base 10 Fractional conversions from one base to another are stopped If there is a rational solution or When the desired accuracy is attained 912009 44 Mixed Number Conversion Integer and fraction parts must be converted separately Radix point fixed reference for the conversion The digit to the left of radix point is a unit digit in every base B0 is always 1 regardless of the base 912009 45

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