l15_6 - Number systems ECE 15A Fundamentals of Logic Design...

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1 ECE 15A Fundamentals of Logic Design Lecture 15 Malgorzata Marek-Sadowska Electrical and Computer Engineering Department UCSB 2 Number systems •Everyday arithmetic: decimal number system. •Number: a string of digits. •Each digit position determines the integer value of power that 10 is raised to. 1 0 1 2 10 3 10 6 10 5 10 8 3 . 856 : + + + = Example Convention: Write digits and deduce the corresponding powers of 10 from their positions. General form of a decimal number: L L 2 1 0 1 2 3 4 5 A A A A A A A A Each } 9 , 8 , 7 , 6 , 5 , 4 , 3 , 2 , 1 , 0 { i A Here, the base or radix is 10. 3 Binary system •Base: 2. Digits: 0,1 (called bits) •Each number is a string of 0s and 1s. •Binary - decimal conversions • Expand the string of bits into a power series with a base 2. Example: (?) 10101 2 = = + + + + ) 2 1 ( ) 2 0 ( ) 2 1 ( ) 2 0 ( ) 2 1 ( 0 1 2 3 4 16+0+4+0+1 = 21 4 Example: Convert to binary ) 13 ( 10 Check: 10 2 ) 13 ( 1 1 2 0 4 1 8 1 ) 1101 ( = + + + = 1 6 2 13 + = 0 3 2 6 + = 1 1 2 3 + = 1 0 2 1 + = Step 1 Step 2 Step 3 Step 4 1 1 0 1 5 Example: Convert to binary 10 ) 31 . 0 ( 62 . 0 62 . 0 2 31 . 0 + = = 24 . 1 24 . 1 2 62 . 0 + = = 48 . 0 48 . 0 2 24 . 0 + = = 96 . 0 96 . 0 2 48 . 0 + = = 92 . 1 92 . 1 2 96 . 0 + = = 84 . 1 84 . 1 2 92 . 0 + = = 68 . 1 68 . 1 2 84 . 0 + = = step1 step2 step3 step 4 step 5 step 6 step 7 2 10 0100111 . 0 ) 31 . 0 ( = 6 Octal number system Base: 8, digits: 0,1,2,3,4,5,6,7 Example 10 0 1 2 3 8 ) 533 ( 5 16 0 512 ) 8 5 ( ) 8 2 ( ) 8 0 ( ) 8 1 ( ) 1025 ( = + + + = + + + = Example Convert to octal 10 ) 189 ( 5 23 8 189 + = 7 2 8 23 + = 2 0 8 2 + = 2 7 5 8 10 ) 275 ( ) 189 ( = 189 5 56 128 1 5 8 7 64 2 8 5 8 7 8 2 0 1 2 = + + = + + = + +
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2 7 Hexadecimal system Base: 16, digits: 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F Example 10 0 1 2 3 ) 15042 ( 2 192 2560 12288 ) 16 2 ( ) 16 12 ( ) 16 10 ( ) 16 3 ( 2
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l15_6 - Number systems ECE 15A Fundamentals of Logic Design...

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