Scales, Units, and Conventions. ESD I : lesson 1. Positional Notations. Positional Notations. Numbers. Weighted-positional notation

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1 Scales, Units, and Conventions ESD I : lesson 1 Numbers Term K (kilo-) M (mega -) G (giga-) T (tera-) P (peta-) Term m (milli -)? (micro-) n (nano-) p (pico- ) Normal Usage As a power of = = 1,048, = 1,073,741, = 1,099,511,627, = 1,125,899,906,842,624 Usage Note the differences between usages. You should commit the powers of 2 to memory. Units: Bit (b), Byte (B), Nibble, Word (w), Double Word, Long Word, Second (s), Hertz ( Hz) Positional Notations Weighted-positional notation Decimal number system, symbols = { 0, 1, 2, 3,, 9 } Position is important Example:(7594) 10 = (7x10 3 ) + (5x10 2 ) + (9x10 1 ) + (4x10 0 ) The value of each symbol is dependent on its type and its position in the number In general, (a n a n-1 a 0 ) 10 = (a n x 10 n ) + (a n-1 x 10 n-1 ) + + (a 0 x 10 0 ) Positional Notations Fractions are written in decimal numbers after the decimal point. (2.75) 10 = (2 x 10 0 ) + (7 x 10-1 ) + (5 x 10-2 ) In general, (a n a n-1 a 0. f 1 f 2 f m ) 10 = (a n x 10 n ) + (a n-1 x10 n-1 ) + + (a 0 x 10 0 ) + (f 1 x 10-1 ) + (f 2 x 10-2 ) + + (f m x 10 -m ) 1

2 Base-R to Decimal Conversion Binary (base 2): weights in powers-of-2. Binary digits (bits): 0,1. Octal (base 8): weights in powers -of-8. Octal digits: 0,1,2,3,4,5,6,7. Hexadecimal (base 16): weights in powers-of-16. Hexadecimal digits: 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F. Base R: weights in powers-of-r. Base-R to Decimal Conversion ( ) 2 = 1? ? ? ? ?2-3 = = (13.625) 10 (572.6) 8 = 5? ? ? ?8-1 = = (392.75) 10 (2A.8) 16 = 2? ? ? 16-1 = = (42.5) 10 (341.24) 5 = 3? ? ? ? ? 5-2 = = (96.56) 10 Decimal-to-Binary Conversion Repeated Division-by-2 Method (for whole numbers) Repeated Multiplication-by-2 Method (for fractions) Sum-of-Weights Method Repeated Division-by-2 Method To convert a whole number to binary, use successive division by 2 until the quotient is 0. The remainders form the answer, with the first remainder as the least significant bit (LSB) and the last as the most significant bit (MSB). (43) 10 = (101011) rem 1 LSB 2 10 rem rem rem rem 0 0 rem 1 MSB 2

3 Repeated Multiplication-by-2 Method To convert decimal fractions to binary, repeated multiplication by 2 is used, until the fractional product is 0 (or until the desired number of decimal places). The carried digits, or carries, produce the answer, with the first carry as the MSB, and the last as the LSB. E.g. (0.3125) 10 = (.0101) 2 Carry = MSB = = = LSB Sum-of-Weights Method Determine the set of binary weights whose sum is equal to the decimal number. (9) 10 = = = (1001) 2 (18) 10 = = = (10010) 2 (58) 10 = = = (111010) 2 (0.625) 10 = = = (0.101) 2 Conversion between Bases In general, conversion between bases can be done via decimal: Base-2 Base-2 Base-3 Base-3 Base-4 Decimal Base-4. Base-R Base-R Shortcuts are available for conversion between bases 2, 4, 8, 16. Binary-Octal/Hexadecimal Conversion Binary? Octal: Partition in groups of 3 Octal? ( ) 2 = ( ) 8 Binary : reverse ( ) 8 = ( ) 2 Binary? Hexadecimal: Partition in groups of 4 ( ) 2 = (5D9.B8) 16 = 0x5D9.B8 Hexadecimal? Binary : reverse (5D9.B8) 16 = ( ) 2 3

4 Table of Binary, Decimal and Hexadecimal Numbers Decimal Binary Octal Hexadecimal A B C D E F Data Types Data Types in digital systems may be classified as: Digits (Numbers) used in arithmetic computations, Letters of the alphabet used in data processing, Discrete symbols ( Special Characters) used for specific purposes. All data types (except binary numbers) are represented in computer registers in binary-coded form. Basic Number Systems Integer Representation Binary Base or radix 2 number system Binary digit is called a bit. Numbers are 0 and 1 only. Numbers are expressed as powers of = 1, 2 1 = 2, 2 2 = 4, 2 3 = 8, 2 4 = 16, 2 5 = 32, 2 6 = 64, 2 7 = 128, 2 8 = 256, 2 9 = 512, 2 10 = 1024, 2 11 = 2048, 2 12 = 4096, 2 12 = 8192, Only have 0 & 1 to represent everything Positive numbers stored in binary e.g. 41= No minus sign No period Sign-Magnitude 1 s complement 2 s complement 4

5 Negative Numbers: Sign-and-Magnitude Example: an 8-bit number can have 1-bit sign and 7-bits magnitude. sign magnitude Sign-Magnitude Left most bit is sign bit 0 means positive 1 means negative +18 = = Problems Need to consider both sign and magnitude in arithmetic Two representations of zero (+0 and -0) 1s Complement (I) Two other ways of representing signed numbers for binary numbers are: 1s-complement 2s-complement 1s Complement (II) Shortcut: invert all the bits. Examples: -( ) 2 = ( ) 1s -( ) 2 = ( ) 1s Largest Positive Number: (127) 10 Largest Negative Number: (127) 10 Zeroes: Range: -(127) 10 to +(127) 10 The most significant bit still represents the sign: 0 = +value; 1 = -value. 5

6 1s Complement (III) Given a number x which can be expressed as an n-bit binary number, its negative value can be obtained in 1scomplement representation using: - x = 2 n - x - 1 Example: With an 8-bit number , its negative value, expressed in 1s complement, is obtained as follows: -( ) 2 = - (12) 10 = ( ) 10 = (243) 10 = ( ) 1s 2s Complement (I) Shortcut: invert all the bits and add 1. Examples: -( ) 2 = ( ) 1s (invert) = ( ) 2s (add 1) -( ) 2 = ( ) 1s (invert) = ( ) 2s (add 1) Two s Complement +3 = = = = = = = s Complement (II) Given a number x which can be expressed as an n-bit binary number, its negative number can be obtained in 2s-complement representation using: - x = 2 n - x Example: With an 8-bit number , its negative value in 2s complement is thus: -( ) 2 = - (12) 10 = (2 8-12) 10 = (244) 10 = ( ) 2s 6

7 Benefits Geometric Depiction of Twos Complement Integers One representation of zero Arithmetic works easily (see later) Negating is fairly easy 3 = Boolean complement gives Add 1 to LSB Negation Special Case 1 0 = Bitwise not Add 1 to LSB +1 Result Overflow is ignored, so: - 0 = 0 Negation Special Case = bitwise not Add 1 to LSB +1 Result So: -(-128) = -128 Monitor MSB (sign bit) It should change during negation 7

8 Range of Numbers 8 bit 2s compliment +127 = = = = bit 2s compliment = = = = Conversion Between Lengths Positive number pack with leading zeros +18 = = Negative numbers pack with leading ones -18 = = i.e. pack with MSB (sign bit) Addition and Subtraction Normal binary addition Monitor sign bit for overflow Take twos compliment of substraend and add to minuend i.e. a - b = a + (-b) So we only need addition and complement circuits Multiplication Example 1011 Multiplicand (11 dec) x 1101 Multiplier (13 dec) 1011 Partial products 0000 Note: if multiplier bit is 1 copy 1011 multiplicand (place value) 1011 otherwise zero Product (143 dec) Note: need double length result 8

9 Multiplying Negative Numbers This does not work! Solution 1 Convert to positive if required Multiply as above If signs were different, negate answer Solution 2 Booth s algorithm Division More complex than multiplication Negative numbers are really bad! Based on long division Division of Unsigned Binary Integers Real Numbers Divisor Partial Remainders Quotient Dividend Remainder Numbers with fractions Could be done in pure binary = =9.625 Where is the binary point? Fixed? Very limited Moving? How do you show where it is? 9

10 Floating Point Floating Point Examples Sign bit Biased Exponent Significand or Mantissa +/-. significand x 2 exponent Point is actually fixed between sign bit and body of mantissa Exponent indicates place value (point position) Signs for Floating Point Mantissa is stored in 2s compliment Exponent is in excess or biased notation e.g. Excess (bias) 128 means 8 bit exponent field Pure value range Subtract 128 to get correct value Range -128 to +127 Normalization FP numbers are usually normalized i.e. exponent is adjusted so that leading bit (MSB) of mantissa is 1 Since it is always 1 there is no need to store it (c.f. Scientific notation where numbers are normalized to give a single digit before the decimal point e.g x 10 3 ) 10

11 FP Ranges Expressible Numbers For a 32 bit number 8 bit exponent +/ ? 1.5 x Accuracy The effect of changing lsb of mantissa 23 bit mantissa 2-23? 1.2 x 10-7 About 6 decimal places IEEE 754 Standard for floating point storage 32 and 64 bit standards 8 and 11 bit exponent respectively Extended formats (both mantissa and exponent) for intermediate results FP Arithmetic +/- Check for zeros Align significands (adjusting exponents) Add or subtract significands Normalize result 11

12 FP Arithmetic x/? Check for zero Add/subtract exponents Multiply/divide significands (watch sign) Normalize Round All intermediate results should be in double length storage Binary Coded Decimal (BCD) (I) Decimal numbers are more natural to humans. Binary numbers are natural to computers. Quite expensive to convert between the two. If little calculation is involved, we can use some coding schemes for decimal numbers. One such scheme is BCD, also known as the 8421 code. Represent each decimal digit as a 4-bit binary code. Binary Coded Decimal (BCD) (II) Binary Coded Decimal (BCD) (III) Decimal digit BCD Decimal digit BCD Some codes are unused, eg: (1010) BCD, (1011) BCD,, (1111) BCD. These codes are considered as errors. Easy to convert, but arithmetic operations are more complicated. Suitable for interfaces such as keypad inputs and digital readouts. Decimal digit BCD Decimal digit BCD Examples: (234) 10 = ( ) BCD (7093) 10 = ( ) BCD ( ) BCD = (86) 10 ( ) BCD = (9472) 10 Notes: BCD is not equivalent to binary. Example: (234) 10 = ( ) 2 12

13 Alphanumeric Codes (I) Alphanumeric Codes (II) Apart from numbers, computers also handle textual data. Character set frequently used includes: alphabets: A.. Z, and a.. z digits: special symbols: *, non-printable: SOH, NULL, BELL, Usually, these characters can be represented using 7 or 8 bits. Two widely used standards: ASCII (American Standard Code for Information Interchange) EBCDIC (Extended BCD Interchange Code) ASCII: 7-bit, plus a parity bit for error detection (odd/even parity). EBCDIC: 8-bit code. Character ASCII Code : A B Z [ \ Alphanumeric Codes (III) ASCII table: (65 in decimal) MSBs LSBs NUL DLE SP P ` p 0001 SOH DC 1! 1 A Q a q 0010 STX DC2 2 B R b r 0011 ETX DC3 # 3 C S c s 0100 EOT DC 4 $ 4 D T d t 0101 ENQ NAK % 5 E U e u 0110 ACK SYN & 6 F V f v 0111 BEL ETB 7 G W g w 1000 BS CAN ( 8 H X h x 1001 HT EM ) 9 I Y i y 1010 LF SUB * : J Z j z 1011 VT ESC + ; K [ k { 1100 FF FS, < L \ l 1101 CR GS - = M ] m } 1110 O RS. > N ^ n ~ 1111 SI US /? O _ o DEL Required Reading Stallings Chapter 8 IEEE 754 on IEEE Web site 13