Representing Numbers: Integers Humans use Decimal Number System Computers use Binary Number System Important to understand Decimal system before looking at binary system Decimal Numbers - Base 10 10 digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 Positional number system: the position of a digit in a number determines its value Take the number 1649 The 1 is worth 1000 The 9 is worth 9 units Formally, the digits in a decimal number are weighted by increasing powers of 10 i.e. they use the base 10. We can write 1649 in the following form: 1*103 + 6*102 + 4*101 + 9*100 Comp 1001: IT & Architecture - Joe Carthy

Representing Numbers: Integers weighting: 103 102 101 100 Digits 1 6 4 9 1649 = 1*103 + 6*102 + 4*101 + 9*100 Least Significant Digit: rightmost one - 9 above Lowest power of 10 weighting Digits on the right hand side are called the low-order digits (lower powers of 10). Most Significant Digit: leftmost one - 1 above Highest power of 10 weighting The digits on the left hand side are called the high-order digits (higher powers of 10) Comp 1001: IT & Architecture - Joe Carthy

Representing Numbers: Decimal Numbers Largest n-digit number ? Made up of n consecutive 9’s (= 10n -1 ) Largest 4-digit number if 9999 9999 is 104 -1 Distinguishing Decimal from other number systems such as Binary, Hexadecimal (base 16) and Octal (base 8) How do we know whether the number 111 is decimal or binary One convention is to use subscripts Decimal: 11110 Binary:1112 Hex: 11116 Octal: 1118 Difficult to write use keyboard Another convention is to append a letter (D, B, H, O) Decimal: 111D Binary:111B Hex: 111H Octal: 111O Comp 1001: IT & Architecture - Joe Carthy

Representing Numbers: Binary Numbers Binary numbers are Base 2 numbers Only 2 digits: 0 and 1 Formally, the digits in a binary number are weighted by increasing powers of 2 They operate as decimal numbers do in all other respects Consider the binary number 0101 1100 Weight 27 26 25 24 23 22 21 20 bits 0 1 0 1 1 1 0 0 01011100 = 0*27 + 1*26 + 0*25 + 1*24 + 1*23 + 1*22 + 0*21 + 0*20 = 0 + 6410 + 0 + 1610 + 810 + 410 + 0 + 0 = 9210 Comp 1001: IT & Architecture - Joe Carthy

Representing Numbers: Binary Numbers Leftmost bit is the most significant bit (MSB). The leftmost bits in a binary number are referred to as the high-order bits. Rightmost bit is the least significant bit (LSB). The rightmost bits in a binary number are referred to as the low-order bits. Largest n-bit binary number ? Made up of n consecutive 1’s (= 2n -1) e.g. largest 4-bit number: 1111 = 24 -1 = 15 Comp 1001: IT & Architecture - Joe Carthy

Representing Numbers: Binary Numbers Exercises Convert the following binary numbers to decimal: (i) 1000 1000 (ii) 1000 1001 (iii) 1000 0111 (iv) 0100 0001 (v) 0111 1111 (vi) 0110 0001 Joe Carthy Formatting Convention In these notes we insert a space after every 4 bits to make the numbers easier to read Comp 1001: IT & Architecture - Joe Carthy

Representing Numbers: Converting Decimal to Binary To convert from one number base to another: you repeatedly divide the number to be converted by the new base the remainder of the division at each stage becomes a digit in the new base until the result of the division is 0. Example: To convert decimal 35 to binary we do the following: Remainder 35 / 2 1 17 / 2 1 8 / 2 0 4 / 2 0 2 / 2 0 1 / 2 1 The result is read upwards giving 3510 = 1000112. Comp 1001: IT & Architecture - Joe Carthy

Representing Numbers: Converting Decimal to Binary Exercise: Convert the following decimal numbers to binary (1) 64 (2) 65 (3) 32 (4) 16 (5) 48 Shortcuts To convert any decimal number which is a power of 2, to binary, simply write 1 followed by the number of zeros given by the power of 2. For example, 32 is 25, so we write it as 1 followed by 5 zeros, i.e. 10000; 128 is 27 so we write it as 1 followed by 7 zeros, i.e. 100 0000. Remember that the largest binary number that can be stored in a given number of bits is made up of n 1’s. An easy way to convert this to decimal, is to note that this is the same as 2n - 1. For example, if we are using 4-bit numbers, the largest value we can represent is 1111 which is 24-1, i.e. 15 Comp 1001: IT & Architecture - Joe Carthy

Representing Numbers: Converting Decimal to Binary Binary Numbers that you should remember because they occur so frequently Binary Decimal 111 7 1111 15 0111 1111 127 1111 1111 255 Comp 1001: IT & Architecture - Joe Carthy

Comp 1001: IT & Architecture - Joe Carthy Review Decimal Number System - Base 10 Significant Digits Binary Number System - Base 2 Notation (B,D,H,O) Binary to Decimal Decimal to Binary Shortcuts and Common Binary Numbers Review Questions What is a positional number system ? What is the MSB and the LSB. Give an example of each one. Show how the weights of the bits in an 8-bit binary number 16-bit binary numbe What is the weight of the MSB in (a) 8-bit number (b) 16-bit number (c) 32-bit number Convert 48D, 65D, 31D, 15D to binary Convert 1111 0111B, 1010 1010B and 1110 0111B to decimal Comp 1001: IT & Architecture - Joe Carthy

Hexadecimal Number System Base-16 number System 16 digits: 0, 1, 2, .., 9, A, B, C, D, E, F. We use the letters A to F to represent numbers 10 to 15 using a single symbol A = 10; B = 11; C = 12; D = 13; E = 14; F = 15; Use H at right hand side to indicate Hexadecimal Used because binary numbers are very long and so are error prone Easy to convert between hexadecimal and binary than between decimal and binary Example: Convert 2FAH to decimal weighting: 162 161 160 digits 2 F A 2FA = 2 * 162 + F * 161 + A * 160 = 2 * 162 + 15 * 161 + 10 * 160 = 256 + 240 + 10 = 50610 Comp 1001: IT & Architecture - Joe Carthy

Hexadecimal Number System Comp 1001: IT & Architecture - Joe Carthy

Hexadecimal Number System Example 2: Convert FFFH to decimal weighting: 162 161 160 digits F F F 2FA = F * 162 + F * 161 + F * 160 = 15 * 162 + 15 * 161 + 15 * 160 = 3840 + 240 + 15 = 409510 Example 3: Convert 65D to Hexadecimal 65 / 16 = 4 Remainder 1 4 / 16 = 0 Remainder 4 65D = 41H Exercise: Convert 97D, 48D, 255D to Hexadecimal Comp 1001: IT & Architecture - Joe Carthy

Hexadecimal Number System Hexadecimal to Binary Convert each Hex digit to a 4-bit binary number Example: 7FAH 7 F A 0111 1111 1010 7FA = 0111 1111 1010B Exercise: Convert FFH, FEH, BBH to binary Comp 1001: IT & Architecture - Joe Carthy

Hexadecimal Number System Binary to Hexadecimal Break binary numbers into groups of 4-bits from right hand side Pad with 0’s on left if necessary Convert each group of 4-bits to its equivalent Hex digit Example 1: 111100111011B 1111 0011 1011 Decimal 15 3 11 Hex F 3 B 111100111011B = F3BH Exercise: Convert 11 1111 1010B; 111 1101 1111B and 1111 1111 1111 1111B to Hexadecimal Comp 1001: IT & Architecture - Joe Carthy

Comp 1001: IT & Architecture - Joe Carthy Signed Numbers How do we represent negative numbers ? Humans use a symbol to indicate number sign: “-” or “+” In computer we only have binary: 1’s and 0’s . Two common methods for representing signed numbers Signed Magnitude Two’s Complement (2’s Complement) In the following assume we are working with 8-bit numbers We designate the leftmost bit i.e the MSB as a sign bit The sign bit indicates whether a number is positive or negative 0 sign bit => positive number 1 sign bit => negative number The remaining bits give the magnitude of the number Comp 1001: IT & Architecture - Joe Carthy

Comp 1001: IT & Architecture - Joe Carthy Signed Numbers Example +15 and -15 as 8-bit numbers +15 => 0 000 1111B MSB = 0 => + -15 => 1 000 1111B MSB = 1 => - Note the magnitude is comprised of 7 bits Largest positive number is 0111 1111 => +127 Largest negative number is 1 111 111 => -127 Two representations of zero ! 0 000 0000 and 1 000 0000 Comp 1001: IT & Architecture - Joe Carthy

Signed Numbers: 2’s Complement In a complementary number system each number has a unique representation. Two’s complement is a complementary number system used in computers It is the most commonly used method for representing signed numbers Uses only one representation of zero: 0000 0000 Uses a sign bit as for signed magnitude 0 sign bit => positive number 1 sign bit => negative number In the case of positive numbers, the representation is identical to that of signed magnitude, the sign bit is 0 and the remaining bits represent the positive number. In the case of negative numbers, the sign bit is 1 but the bits to the right of the sign bit do not directly indicate the magnitude of the number. Comp 1001: IT & Architecture - Joe Carthy

Signed Numbers: 2’s Complement In negative numbers the sign bit carries a negative weight while all other bits carry a positive weight e.g. the 2’s complement number 1000 0011B is weighted as follows Bits 1 0 0 0 0 0 1 1 Weights -128 +64 +32 +16 +8 +4 +2 +1 Value = -128 + 2*1 + 1*1 = -128 +3 = -125D So 1000 0011B = -125D Exercise: Convert the 2’s complement numbers 1000 0111B and 1000 1111B to decimal. Comp 1001: IT & Architecture - Joe Carthy

Signed Numbers: 2’s Complement Example 2: The 2’s complement number 1111 1111B Bits 1 1 1 1 1 1 1 1 Weights -128 +64 +32 +16 +8 +4 +2 +1 Value = -128 + 64 + 32 +16 + 8 +4 + 2 _+1 = -128 +127 = -1D In 2’s complement each number has a unique representation i.e. the negative representation of a number uses a completely different bit pattern than its positive counterpart. In any number system: +x - x = 0 Example: +1 - 1 = 0 In 2’s complement: 0000 0001B 1111 1111B ------------------ 0000 0000B Comp 1001: IT & Architecture - Joe Carthy

Signed Numbers: 2’s Complement 2’s Complement Arithmetic In 2’s Complement, we do not need subtraction to compute x - y We simply add -y to x to get the result. This makes it easier to design the hardware to implement 2’s arithmetic It much more complicated with signed magnitude e.g to compute +2 - 6 we must always subtract the smaller number from the larger one and then take the sign of the larger number. Quick Conversion to/from 2’s Complement Use the rule: Flip the bits and Add 1 Comp 1001: IT & Architecture - Joe Carthy

Signed Numbers: 2’s Complement Quick Conversion to/from 2’s Complement Use the rule: Flip the bits and Add 1 Example 1 Convert 2’s complement number 1111 1111B to decimal Step 1: Flip the bits: Change 1’s to 0’s and change all 0’s to 1’s (complement of 1 is 0; complement of 0 is 1) 1111 1111B => 0000 0000B Step 2: Add 1 0000 0000B + 1B => 0000 0001B => 1D Remember the sign bit was 1 => negative So 1111 1111B => -1D Comp 1001: IT & Architecture - Joe Carthy

Signed Numbers: 2’s Complement Example 2: Convert -1D to 2’s complement First convert 1D to binary => 0000 0001B Step 1: Flip the bits 0000 0001B => 1111 1110B Step 2: Add 1 1111 1110B + 1B => 1111 1111B => -1D => 1111 1111B Exercise: Convert the following 2’s complement numbers to decimal: 1111 1110; 1000 0000; 100 0001; 1111 0000 Decimal to 2’s complement: -128 ; -65; -2 Comp 1001: IT & Architecture - Joe Carthy

Comp 1001: IT & Architecture - Joe Carthy 1000 0000B (i.e. -128) is the largest negative 8-bit 2’s complement number 0111 1111 (127) is the largest positive 8-bit 2’s complement number 256 numbers can be represented using 8-bit two’s complement numbers from -128 to 127. There is only one representation for zero. The table below lists the decimal equivalents of some 8-bit 2’s complement and unsigned binary numbers. Comp 1001: IT & Architecture - Joe Carthy

Number Range and Overflow The range of numbers (called the number range) that can be stored in a given number of bits is important. Given an 8-bit number, we can represent unsigned numbers in the range 0 to 255 (0 to 28-1) and two’s complement numbers in the range -128 to +127 (-27 to 27). Given a 16-bit number, we can represent unsigned numbers in the range 0 to 65,535 (0 to 216 -1) and two’s complement numbers in the range -32768 to 32767 (-215 to 215-1). In general given an n-bit number, we can represent unsigned numbers in the range 0 to 2n -1 and two’s complement numbers in the range -2n-1 to 2n-1 -1 Exercise: What is the number range of 4-bit, 10-bit, 20-bit, 30-bit and 32-bit numbers ? Comp 1001: IT & Architecture - Joe Carthy

Number Range and Overflow The magnitude of an unsigned number doubles for every bit added 10 bits can represent 1024 numbers (1K) 11 bits => 2048 numbers (2K) 12 bits => 4096 numbers (4K) …. 16 bits => 64K numbers .. 20 bits => 220 => 1 Mb 21 bits => 2 Mb 24 bit => 16Mb 30 bits => 230 => 1 Gb 31 bits => 2Gb 32 bits => 4 Gb Comp 1001: IT & Architecture - Joe Carthy

Number of bits in Memory Address The maximum amount of memory that a processor can access is determined by the number of bits that the processor uses to represent a memory address. This determines the maximum memory address that can be accessed i.e. is a limit on the maximum amount of RAM a computer can use For example, a processor that uses 16-bit addresses will only be able to access up to 65,536 memory locations (64Kb), with addresses from 0 to 65,535. A 20-bit address allows up to 220 (1Mb) memory locations to be accessed A 24-bit address allows up to 16Mb (224 bytes) of RAM to be accessed A 30-bit address allows up to 1Gb (230 bytes) of RAM to be accessed A 32-bit address allows up to 1Gb (230 bytes) of RAM to be accessed Most PCs now use 32-bit addresses. Original PC (1981) used 20-bit addresses. Early Macintoshs used 24-bit addresses. Comp 1001: IT & Architecture - Joe Carthy

Comp 1001: IT & Architecture - Joe Carthy Review Hexadecimal: 16 digits: 0 to 9 and A to F. Easy to convert to binary Signed Numbers: Signed Magnitude and 2’s Complement Sign bit: MSB 1 => negative 2 Complement: Flip the bits and add 1. Number range is important Given an n-bit number, we can represent unsigned numbers in the range 0 to 2n -1 and two’s complement numbers in the range -2n-1 to 2n-1 -1. A 20-bit address allows up to 220 (1Mb) memory locations to be accessed A 30-bit address allows up to 1Gb (230 bytes) of RAM to be accessed A 32-bit address allows up to 1Gb (230 bytes) of RAM to be accessed Comp 1001: IT & Architecture - Joe Carthy