3.1 3.2 Unit 3 Binary Representation ANALOG VS. DIGITAL 3.3 3.4 Analog vs. Digital Analog vs. Digital • The analog world is based on continuous • Q. Which is better? events. Observations can take on (real) any • A. Depends on what you are trying to do. value. • Some tasks are better handled with analog • The digital world is based on discrete events. data, others with digital data. Observations can only take on a finite number – Analog means continuous/real valued signals with of discrete values an infinite number of possible values – Digital signals are discrete [i.e. 1 of n values]
3.5 3.6 Analog vs. Digital Analog vs. Digital • How much money is in my checking account? • How much do you love me? – Analog: I love you with all my heart!!!! – Analog: Oh, some, but not too much. – Digital: 3.2 x 10 3 MegaHearts – Digital: $243.67 3.7 3.8 The Real (Analog) World Digital is About Numbers • The real world is inherently analog. • In a digital world, numbers are used to represent all the possible discrete events • To interface with it, our digital systems need – Numerical values to: – Computer instructions (ADD, SUB, BLE, …) – Convert analog signals to digital values (numbers) – Characters ('a', 'b', 'c', …) at the input. – Conditions (on, off, ready, paper jam, …) – Convert digital values to analog signals at the • Numbers allow for easy manipulation output. – Add, multiply, compare, store, … • Analog signals can come in many forms • Results are repeatable – Voltage, current, light, color, magnetic fields, – Each time we add the same two number we get pressure, temperature, acceleration, orientation the same result
3.9 3.10 Interpreting Binary Strings • Given a string of 1’s and 0’s, you need to know the representation system being used, before you can understand the value of those 1’s and 0’s. • ______________________________________ 01000001 = ? Unsigned DIGITAL REPRESENTATION Binary system ASCII BCD System system 65 10 ‘A’ ASCII 41 BCD 3.11 3.12 Binary Representation Systems • Codes • Integer Systems – Text – Unsigned • ASCII / Unicode • Unsigned (Normal) binary – Decimal Codes – Signed • Signed Magnitude • BCD (Binary Coded Decimal) / (8421 Code) • 2’s complement • Excess-N* • 1’s complement* • Floating Point OVERVIEW – For very large and small (fractional) numbers * = Not fully covered in this class
3.13 3.14 4 Skills • We will teach you 4 skills that you should know and be able to apply with confidence – Convert a number in any base (base r) to decimal (base 10) – Convert a decimal number (base 10) to binary Using positional weights/place values – Use the shortcut for conversion between binary BASE R TO BASE 10 (base 2) and hexadecimal (base 16) – Understand the finite number of combinations that can be made with n bits (binary digits) and its implication for codes including ASCII and Unicode 3.15 3.16 Number Systems Anatomy of a Decimal Number • Number systems consist of • A number consists of a string of explicit coefficients (digits). • Each coefficient has an implicit place value which is a _______ 1. ________________ of the base. 2. ___ coefficients [__________] • The value of a decimal number (a string of decimal • Human System: Decimal (Base 10): coefficients) is the sum of each coefficient times it place value 0,1,2,3,4,5,6,7,8,9 radix (base) • Computer System: Binary (Base 2): 0,1 (934) 10 = 9*___ + 3*___ + 4*____ = _____ • Human systems for working with computer systems (shorthand for human to read/write binary) Implicit place values Explicit coefficients – _____________________________________ (3.52) 10 = 3*____ + 5*____ + 2*____ = ____ – _____________________________________
3.17 3.18 Anatomy of a Binary Number General Conversion From Base r to Decimal • Same as decimal but now the coefficients • A number in base r has place values/weights are 1 and 0 and the place values are the that are the powers of the base powers of 2 • Denote the coefficients as: a i Most Significant Least Significant (a 3 a 2 a 1 a 0 .a -1 a -2 ) r = a 3 *r 3 + a 2 *r 2 + a 1 *r 1 + a 0 *r 0 + a -1 *r -1 + a -2 *r -2 Digit (MSB) Bit (LSB) (1011) 2 = 1*__ + 0*_ + 1*__ + 1*__ Right-most digit = Left-most digit = Least Significant Most Significant Digit (LSD) Digit (MSD) radix (base) place values coefficients = powers of 2 N r => _______=> D 10 Number in base r Decimal Equivalent 3.19 3.20 Examples Binary Examples (746) 8 = (1001.1) 2 = (10110001) 2 = (1A5) 16 = (AD2) 16 =
3.21 3.22 Powers of 2 Unique Combinations 2 0 = 1 • Given n digits of base r , how many unique numbers 2 1 = 2 can be formed? __ 2 2 = 4 2 3 = 8 – What is the range? [________] 2 4 = 16 2 5 = 32 2-digit, decimal numbers (r=10, n=2) 2 6 = 64 0-9 0-9 2 7 = 128 1024 512 256 128 64 32 16 8 4 2 1 3-digit, decimal numbers (r=10, n=3) 2 8 = 256 2 9 = 512 4-bit, binary numbers (r=2, n=4) 2 10 = 1024 0-1 0-1 0-1 0-1 6-bit, binary numbers (r=2, n=6) Main Point: Given n digits of base r, ___ unique numbers can be made with the range [________] 3.23 3.24 Approximating Large Powers of 2 • Often need to find decimal approximation of a large powers of 2 2 16 = 2 6 * 2 10 like 2 16 , 2 32 , etc. ≈ • Use following approximations: 2 24 = – 2 10 ≈ _________________ – 2 20 ≈ _________________ – 2 30 ≈ _________________ 2 28 = – 2 40 ≈ _________________ "Making change" • For other powers of 2, decompose BASE 10 TO BASE 2 OR BASE 16 into product of 2 10 or 2 20 or 2 30 and a 2 32 = power of 2 that is less than 2 10 – 16-bit half word: 64K numbers – 32-bit word: 4G numbers – 64-bit dword: 16 million trillion numbers
3.25 3.26 Decimal to Unsigned Binary Decimal to Unsigned Binary 73 10 = • To convert a decimal number, x, to binary: 128 64 32 16 8 4 2 1 – Only coefficients of 1 or 0. So simply find place values that add up to the desired values, starting with larger 87 10 = place values and proceeding to smaller values and place a 1 in those place values and 0 in all others 145 10 = 25 10 = 0.625 10 = 32 16 8 4 2 1 .5 .25 .125 .0625 .03125 3.27 3.28 Decimal to Another Base • To convert a decimal number, x, to base r: Use the place values of base r (powers of r). Starting – with largest place values, fill in coefficients that sum up to desired decimal value without going over. Shortcuts for Converting Binary (r=2), Hexadecimal (r=16) and Octal (r=8) SHORTHAND FOR BINARY 75 10 = hex 256 16 1
3.29 3.30 Binary, Octal, and Hexadecimal Binary to Octal or Hex • Make groups of 3 bits • Make groups of 4 bits • Octal (base 8 = 2 3 ) • Hex (base 16=2 4 ) starting from radix point starting from radix point • 1 Octal digit ( _ ) 8 can • 1 Hex digit ( _ ) 16 can and working outward and working outward represent: ________ represent: 0-F (_____) • Add 0’s where • Add 0’s where necessary necessary • 3 bits of binary (_ _ _) 2 • 4 bits of binary • Convert each group of 3 • Convert each group of 4 can represent: (_ _ _ _) 2 can represent: to an octal digit to an octal digit 000-111 = ________ 0000-1111= ______ • Conclusion… • Conclusion… 101001110.11 101001110.11 __Octal digit = __ bits __ Hex digit = ___ bits 3.31 3.32 Octal or Hex to Binary Hexadecimal Representation • Since values in modern computers are many bits, we • Expand each octal digit • Expand each hex digit use hexadecimal as a shorthand notation (4 bits = 1 to a group of 3 bits to a group of 4 bits hex digit) – 11010010 = D2 hex or 0xD2 if you write it in C/C++ 317.2 8 D93.8 16 – 0111011011001011 = 76CB hex or 0x76CB if you write it in C/C++
3.33 3.34 Binary Representation Systems • Integer Systems • Codes – Unsigned – Text • Unsigned (Normal) binary • ASCII / Unicode – Signed – Decimal Codes • Signed Magnitude • BCD (Binary Coded Decimal) • 2’s complement / (8421 Code) • 1’s complement* ASCII & Unicode • Excess-N* • Floating Point BINARY CODES – For very large and small (fractional) numbers * = Not covered in this class 3.35 3.36 Binary Codes BCD (If Time Permits) • Rather than convert a decimal number to binary which may lose some precision (i.e. 0.1 10 = infinite binary fraction), BCD • Using binary we can represent any kind of represents each decimal digit as a separate group of bits (exact information by coming up with a code decimal precision) • Using n bits we can represent 2 n distinct items – Each digits is represented as a ___________ number (using place values 8,4,2,1 for each dec. digit) – Often used in financial and other applications where decimal precision is needed (439) 10 Colors of the rainbow: Letters: This is the Binary Coded Decimal (BCD) •Red = 000 •‘A’ = 00000 representation of 439 •Orange = 001 •‘B’ = 00001 •Yellow = 010 •‘C’ = 00010 •Green = 100 . BCD Representation: •Blue = 101 . This is the binary •Purple = 111 . representation of 439 110110111 2 Unsigned Binary Rep.: •‘Z’ = 11001 (i.e. using power of 2 place values) Important: Some processors have specific instructions to operate on #’s represented in BCD
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