# Decimal Numbers: Base 10 Integer Numbers & Arithmetic

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1 Decimal Numbers: Base 10 Integer Numbers & Arithmetic Digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 Example: 3271 = (3x10 3 ) + (2x10 2 ) + (7x10 1 )+(1x10 0 ) Ward 1 Ward 2 Numbers: positional notation Number Base B => B digits: Base 10 (Decimal): 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 Base 2 (Binary): 0, 1 Number representation: d 31 d 30 d 2 d 1 d 0 is a 32 digit number value = d 31 x B 31 + d 30 x B d 2 x B 2 + d 1 x B 1 + d 0 x B 0 Binary: 0, = 1x x x x x x2 + 0x1 = = 90 Notice that 7 digit binary number turns into a 2 digit decimal number A base that converts to binary easily? Hexadecimal Numbers: Base 16 Hexadecimal: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F Normal decimal digits + 6 more: picked alphabet Conversion: Binary <-> Hex 1 hex digit represents 16 decimal values 4 binary digits represent 16 decimal values => 1 hex digit replaces 4 binary digits Examples: (binary) =? (hex) (binary) = (binary) =? (hex) 3F9(hex) =? (binary) Ward 3 Ward 4

2 Decimal vs Hexadecimal vs Binary Converting Decimal to Binary [1] Examples: (binary) =? (hex) (binary) = (binary) =? (hex) 3F9(hex) =? (binary) A B C D E F 1111 Converting from base 10 to base 2: Take largest binary place in decimal number, put a binary digit there and subtract it from decimal number and continue until decimal number is (64) 2 5 (32) 2 4 (16) 2 3 (8) 2 2 (4) 2 1 (2) 2 0 (1) = = = = Ward 5 Ward 6 Converting Decimal to Binary [2] Converting from base 10 to base 2: Or continually divide decimal number by 2 and keep remainder Number Remainder 85 / / / / / / / What to do with representations of numbers? Just what we do with numbers! Add them Subtract them Multiply them Divide them Compare them Example: = 17 so simple to add in binary that we can build circuits to do it subtraction also just as you would in decimal Ward 7 Ward 8

3 Which base do we use? Decimal: great for humans, especially when doing arithmetic Hex: if human looking at long strings of binary numbers, its much easier to convert to hex and look 4 bits/symbol Terrible for arithmetic Binary: what computers use; you learn how computers do +,-,*,/ To a computer, numbers always binary Doesn t matter base in C, just the value: == 0x20 == Use subscripts ten, hex, two in book, slides when might be confusing Limits of Computer Numbers Bits can represent anything! Characters? 26 letter => 5 bits upper/lower case + punctuation => 7 bits (in 8) rest of the world s languages => 16 bits (unicode) Logical values? 0 -> False, 1 => True colors? locations / addresses? commands? but N bits => only 2 N things Ward 9 Ward 10 Comparison How to Represent Negative Numbers? How do you tell if X > Y? See if X - Y > 0 So far, unsigned numbers 3 potential systems for handling negative numbers Sign and Magnitude 1 s Complement 2 s Complement Ward 11 Ward 12

4 Sign and Magnitude Obvious solution: define leftmost bit to be sign! 0 => +, 1 => - Rest of bits can be numerical value of number Representation called sign and magnitude MIPS uses 32-bit integers +1 ten would be: And - 1 ten in sign and magnitude would be: Shortcomings of sign and magnitude Arithmetic circuit more complicated Special steps depending whether signs are the same or not Also, two zeros 0x = +0 ten 0x = -0 ten What would it mean for programming? Sign and magnitude abandoned Ward 13 Ward 14 Another try: complement the bits Shortcomings of 1 s complement Example: 7 10 = = Called 1 s Complement Note: positive numbers have leading 0s, negative numbers have leadings 1s What is ? How many positive numbers in N bits? How many negative ones? Ward 15 Arithmetic not too hard Still two zeros 0x = +0 ten 0xFFFFFFFF = -0 ten What would it mean for programming? 1 s complement eventually abandoned because another solution was better Ward 16

5 Search for Negative Number Representation Obvious solutions didn t work, find another What is result for unsigned numbers if tried to subtract large number from a small one? Would try to borrow from string of leading 0s, so result would have a string of leading 1s With no obvious better alternative, pick representation that made the hardware simple: leading 0s positive, leading 1s negative xxx is 0, xxx is < 0 This representation called 2 s complement 2 s Complement Number line N-1 nonnegatives 2 N-1 negatives one zero how many positives? comparison? overflow? Ward 17 Ward 18 2 s Complement two = 0 ten two = 1 ten two = 2 ten two = 2,147,483,645 ten two = 2,147,483,646 ten two = 2,147,483,647 ten two = 2,147,483,648 ten two = 2,147,483,647 ten two = 2,147,483,646 ten two = 3 ten two = 2 ten two = 1 ten One zero 1st bit => 0 or <0, called sign bit 2 s Complement Formula Can represent positive and negative numbers in terms of the bit value times a power of 2: d 31 x d 30 x d 2 x d 1 x d 0 x 2 0 Example two = 1x x x x2 2 +0x2 1 +0x2 0 = = -2,147,483,648 ten + 2,147,483,644 ten = -4 ten Note: need to specify width: we use 32 bits but one negative with no positive 2,147,483,648 ten Ward 19 Ward 20

6 2 s complement shortcut: Negation Invert every 0 to 1 and every 1 to 0, then add 1 to the result Sum of number and its one s complement must be two two = -1 ten Let x be the inverted representation of x Then x + x = -1 x + x + 1 = 0 x + 1 = -x Example: -4 to +4 to -4 x : two x : two +1: two () : two +1: two Signed vs Unsigned Numbers C declaration int Declares a signed number Uses two s complement C declaration unsigned int Declares a unsigned number Treats 32-bit number as unsigned integer, so most significant bit is part of the number, not a sign bit Ward 21 Ward 22 Example of Values in Unsigned & 2 s Complement Representations Unsigned & 2 s Complement Implementation A computer can use a single piece of hardware to provide unsigned or 2 s complement integer arithmetic; software running on the computer can choose an interpretation for each integer Example: Adding 1 to binary 1001 produces 1010 Unsigned interpretation goes from 9 to 10 2 s complement interpretation goes from 7 to 6 Ward 23 Ward 24

7 Signed vs Unsigned Comparisons Numbers are stored at addresses X = two Y = two Memory is a place to store bits Is X > Y (or X Y > 0)? unsigned: YES signed: NO Converting to decimal to check Unsigned comparison: 4,294,967,292 ten > 1,000,000,000 ten? Signed comparison: -4 ten > 1,000,000,000 ten? = 2 k -1 A word is a fixed number of bits (eg, 32) or bytes (eg, 4) at an address Address is also fixed no of bits Addresses are naturally represented as unsigned numbers Ward 25 Ward 26 Numbering Bits and Bytes Need to choose order for Storage in physical memory system Transmission over serial medium (eg, data network) Bit order Handled by hardware Usually hidden from programmer Byte order Affects multi-byte data items such as integers Visible and important to programmers Possible Byte Orders Least significant byte of integer in lowest memory location Known as little endian Most significant byte of integer in lowest memory location Known as big endian Ward 27 Ward 28

8 Illustration of Byte Orders Big Endian & Little Endian Examples Storage of 32-bit integer 1 Note: difference is especially important when transferring data between computers for which the byte ordering differs Ward 29 Ward 30 What if integer too big? Binary bit patterns above are simply representatives of numbers Numbers really have an infinite number of digits with almost all being zero except for a few of the rightmost digits Just don t normally show leading zeros If result of add (or -,*/) cannot be represented by these rightmost HW bits, overflow is said to have occurred Sign extension Convert 2 s complement number using n bits to more than n bits (eg, int to long int) Simply replicate the most significant bit (sign bit) of smaller to fill new bits 2 s comp positive number has infinite 0s 2 s comp negative number has infinite 1s Bit representation hides leading bits; sign extension restores some of them 16-bit -4 ten to 32-bit: two two Ward 31 Ward 32

9 Consequence for Programmers Because 2 s complement hardware performs sign extension, copying an unsigned integer to a larger unsigned integer changes the value; to prevent such errors from occurring, a programmer or a compiler must add code to mask off the extended sign bits And in Conclusion We represent things in computers as particular bit patterns: N bits =>2 N numbers, characters, (data) Decimal for human calculations, binary for computers, hex for convenient way to write binary 2 s complement universal in computing: cannot avoid, so learn Computer operations on the representation correspond to real operations on the real thing Numbers infinite but computers finite, so errors occur (eg, overflow, underflow) Knowing the powers of 2 (eg, 2 0, 2 1, 2 2, 2 3, ) and values of K (2 10 ) and M (2 20 ) will prove invaluable, so learn them Ward 33 Ward 34 Additional Basics: Metric Prefixes Clocks/Seconds T tera G giga 10 9 M mega 10 6 K kilo 10 3 m milli 10-3 μ micro 10-6 n nano 10-9 p pico Bytes/bits Note the difference T tera 2 40 (= 1099 * ) G giga 2 30 (= 1073 * 10 9 ) M mega 2 20 (= 1048 * 10 6 ) K kilo 2 10 (= 1024 * 10 3 ) Ward 35

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