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Digital Design: From Gates to Intelligent Machines. Bruce F Katz Da Vinci Engineering Press. Number Systems. Numbers and Numerals A number is a quantity A numeral is a representation of a number Example (all representations of the quantity 5)

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## Digital Design: From Gates to Intelligent Machines

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**Digital Design:From Gates to Intelligent Machines**Bruce F Katz Da Vinci Engineering Press**Number Systems**Numbers and Numerals A number is a quantity A numeral is a representation of a number Example (all representations of the quantity 5) 5, V (Roman), 101 (binary), (Babylonian) Not equivalent in ease of computation, however**Number Systems**Positional Number Systems A quantity is a weighted sum of powers of a base b Compactness of representation and ease of computation Additional characteristics digits to the left of radix point are integral digits to the right of the radix point are fractional**Number Systems**Examples of Positional Number Systems base 10 102410 = 1*103 + 0*102 + 2*101 + 4*100 base 8 417.238= 4*82 + 1*81 + 7*80 + 2*8-1 + 3*8-2 = 256 + 8 + 7 + 2/8 + 3/64 base 2 1010.1 = 1*23 + 1*21 + 1*2-1 = 10.510**Number Systems**Commonly used bases**Number Systems**Which base is best for human use? Base 12 has the most divisors among the small numbers but we usually use base 10. Why?**Number Systems**Conversion between bases (special case) Principle If one base is an integer power of another base, can group by this integer to perform conversion. Examples 1011112 = ?8 solution: group by 3 bits {101}{111}2 = 578 F416 = ?2 solution: each hex digit represents 4 bits F416 = {1111}{0100}2 = 111101002**Number Systems**Conversion to and from base 10 To base 10 Use definition of a positional number Example: 1101.1012 to base 10 1101.1012 = 1*23 + 1*22 + 0*21 + 1*21 + 1*2-1 + 0*2-2 + 1*2-3 = 13.625 From base 10 Use reformulation of definition of a positional number Successive divisions by the base will yield then digits as remainders Example: 125 to base 3 125/3 = 41 remainder 2 41/3 = 13 remainder 2 13/3 = 4 remainder 1 4/3 = 1 remainder 1 1/3 = 0 remainder 1 therefore answer is 111223**Number Systems**Binary Number Systems Motivation Correspondence between 0 and 1 and logical values (true and false) Ease of constructing binary circuits Powers of 2**Number Systems**Binary Addition and Subtraction Addition Same as decimal addition with binary carries 0 0 1 1 0 0 1 0 carry 1 0 0 1 1 1 0 1 addend1 0 1 0 1 1 0 0 1 addend2 ----------------------- 1 1 1 1 0 1 1 0 sum Subtraction Same as decimal subtraction with binary borrows 0 1 0 0 0 1 1 0 borrow 1 1 0 1 1 1 0 0 minuend 0 1 1 0 1 0 0 1 subtrahend ----------------------- 0 1 1 1 0 0 1 1 difference**Number Systems**Binary Addition and Subtraction Tables Important:These tables are the foundation for computer arithmetic!**Number Systems**Binary Multiplication Easier than decimal multiplication because always multiplying by 0 or 1 Example 0 1 1 0 * 1 0 1 1 ---------- 0 1 1 0 0 1 1 0 0 0 0 0 0 1 1 0 ------------------- 1 0 0 0 0 1 0**Number Systems**Representing Negative Numbers System 1: Signed Magnitude Leftmost bit represents negative sign Examples 01010111 = 87 11010111 = - 87 Advantage Simplicity Disadvantage Mathematical operations clumsy, e.g. addition: if (signs same)then { add magnitudes give result this sign } else /* signs different */ { compare magnitudes subtract smaller from larger give result sign of the larger }**Number Systems**Representing Negative Numbers System 2: 2’s complement Positive numbers identical, negative numbers 2n - positive version, where n is the number of bits in the representation Examples 00010001 = 17 100000000 - 00010001 = 11101111 = - 17 Trick for computing negative number Flip all the bits and add 1 00010001 11101110 after flip 11101111 after adding 1 Note: Negative numbers will always begin with 1, positive with 0**Number Systems**Representing Negative Numbers System 2: 2’s complement Addition method Just add! (and ignore any bits > 2n-1) 0 0 1 0 1 0 1 1 (43) +1 1 1 0 1 1 1 1(-17) ------------------- 1 0 0 0 1 1 0 1 0 Overflow condition: If add 2 positive and get a negative or vice versa Example 1 0 0 0 1 1 1 1 (-113) +1 1 1 0 1 1 1 1(-17) ------------------- 1 0 1 1 1 1 1 1 0 overflow!**Number Systems**Codes A way of representing a set of quantities or a set of symbols within a given base Example in base 2**Number Systems**Codes BCD Each decimal digit is encoded by four binary digits Motivation ease of conversion Examples 0001 0100 14 1001 0111 97 Gray Coding Each successive number differs by only 1 bit from previous Motivation counting with CMOS Karnaugh maps**Number Systems**Codes Parity An extra bit is added to make the string always even or odd Motivation error checking ASCII**Number Systems**Codes Unicode 4 hex digits encode 216 characters Example**Number Systems**Summary of topics Numbers and numerals Positional number systems Conversion between bases Binary number systems Binary addition, subtraction, and multiplication Representation of negative numbers Codes

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