Assoc. Prof. Dr. Ahmet Turan ÖZCERİT

 The base of numbers  Conversion between number bases  Arithmetic operations on different bases You will learn: 2

He/She can define the term of number bases 3  Computers use a number base other than base-10, namely binary  Each data used and stored in computer represented in binary numbers  Binary numbers are not easy to do arithmetic operations, so we use hex and octal numbers for the sake of simplicity  All characters, images, audio and video samples are also presented in binary numbers  Computers use a number base other than base-10, namely binary  Each data used and stored in computer represented in binary numbers  Binary numbers are not easy to do arithmetic operations, so we use hex and octal numbers for the sake of simplicity  All characters, images, audio and video samples are also presented in binary numbers

He/She can define the term of number bases 4  N: Digit value  d: Number digit  R: Number radix(base)  N=d n R n + d n-1 R n-1 + …+ d 2 R 2 + d 1 R 1 +d 0 R 0 (for integers)  N=d n R n +d n-1 R n-1 +…+d 1 R 1 +d 0 R 0, d 1 R -1 + d 2 R -2 +…+ d n R -n (for real numbers)  The number of digits in the R-based number system is R, the largest digit is R-1, and the least digit is 0.  The largest number for n-digit is R n -1 and the number of different value for n-digit is R n  N: Digit value  d: Number digit  R: Number radix(base)  N=d n R n + d n-1 R n-1 + …+ d 2 R 2 + d 1 R 1 +d 0 R 0 (for integers)  N=d n R n +d n-1 R n-1 +…+d 1 R 1 +d 0 R 0, d 1 R -1 + d 2 R -2 +…+ d n R -n (for real numbers)  The number of digits in the R-based number system is R, the largest digit is R-1, and the least digit is 0.  The largest number for n-digit is R n -1 and the number of different value for n-digit is R n

He/She can make operation on binary numbers 5  The largest digit in binary system R-1 => 2-1 => 1  The least digit in binary system is 0  Each radix in binary number systems is called BIT (BInary DigiT).  The most significant bit (MSB)  The binary number general form: B= d n 2 n + d n-1 2 n-1 + ……. + d d d 0 2 0, d d d n 2 -n  The largest digit in binary system R-1 => 2-1 => 1  The least digit in binary system is 0  Each radix in binary number systems is called BIT (BInary DigiT).  The most significant bit (MSB)  The binary number general form: B= d n 2 n + d n-1 2 n-1 + ……. + d d d 0 2 0, d d d n 2 -n Binary SystemDecimal System

He/She can make operation on binary numbers 6  What is binary value of following binary number: B= 1*2 8 +0*2 7 +1* * * * * * *2 0 B= B=  What is binary value of following binary number: , 1101 B= 1*2 8 +0*2 7 +1*2 6 +1*2 5 +0*2 4 +1*2 3 +1*2 2 +0*2 1 +1*2 0, 1* * * *2 -4 B= B= 365,  What is binary value of following binary number: B= 1*2 8 +0*2 7 +1* * * * * * *2 0 B= B=  What is binary value of following binary number: , 1101 B= 1*2 8 +0*2 7 +1*2 6 +1*2 5 +0*2 4 +1*2 3 +1*2 2 +0*2 1 +1*2 0, 1* * * *2 -4 B= B= 365,

He/She can make operation on octal numbers 7 Octal numbers are used to present binary numbers with 3-digit format The digits used in Octal Systems: 0, 1, 2, 3, 4, 5, 6, 7 O= d n 8 n + d n-1 8 n-1 + ……. + d d d 0 8 0, d d d n 8 -n Octal numbers are used to present binary numbers with 3-digit format The digits used in Octal Systems: 0, 1, 2, 3, 4, 5, 6, 7 O= d n 8 n + d n-1 8 n-1 + ……. + d d d 0 8 0, d d d n 8 -n BinaryOctal

He/She can make operation on hex numbers 8 Hex numbers are used to present binary numbers with 4-digit format The digits used in Octal Systems: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F O= d n 16 n + d n-1 16 n-1 + ……. + d d d , d d d n 16 -n Hex numbers are used to present binary numbers with 4-digit format The digits used in Octal Systems: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F O= d n 16 n + d n-1 16 n-1 + ……. + d d d , d d d n 16 -n BinaryHex BinaryHex A 1011B 1100C 1101D 1110E 1111F

He/She can convert number bases each other 9  The steps of converting of 117,86 10 decimal number into a binary number  First, the integer part of the number is resolved then real part  The steps of converting of 117,86 10 decimal number into a binary number  First, the integer part of the number is resolved then real part DivisionRemainderResult 117/2 =581B 0 =1 58/2 = 29 0B 1 =0 29/2 =141B 2 =1 14/2 =70B 3 =0 7/2 =31B 4 =1 3/2 =1 1B 5 =1 1/2 =0 1B 6 =1 MultiplyIntegerResult 0.86*2=1.72 1b 1 =1 0.72*2=1.44 1b 2 =1 0.44*2=0.88 0b 3 =0 0.88*2=1.76 1b 4 =1 0.76*2=1.52 1b 5 =1 0.52*2=1.04 1b 6 =1 0.04*2=0.08 0b 7 =0 (117,86) 10 = ( , ….) 2

He/She can convert number bases each other 10  Convert a real decimal number (0,513) 10 into an octal number OperationMultiplyIntegerResult 0.513* o 0 = * o 1 = * o 2 = * o 3 = * o 4 =1 (0,513) 10 ≅ (0,40651) 8

He/She can convert number bases each other 11  Convert a decimal number (214) 10 into a hex number OperationDivisionRemainderResult 214/16136O 0 =6 13/16013O 1 =D (214) 10 = (D6) 16  Convert a decimal number (423) 10 into a hex number OperationDivisionRemainderResult 423/16267O 0 =7 26/16110O 1 =A 1  1O 2 =1 (423) 10 = (1A7) 16

He/She can convert number bases each other 12  Convert a binary number (100.01) 2 into a decimal number = 1* * *2 0, 0* *2 -2 = 1* , 0 + 1* ¼ = , = (4,25) 10  Convert a binary number (100.01) 2 into a decimal number = 1* * *2 0, 0* *2 -2 = 1* , 0 + 1* ¼ = , = (4,25) 10

He/She can convert number bases each other 13  Converting a binary number ( ) 2 into an octal number For integer part; start from just after the dot towards to the leftmost digit and combine 3-bits as a group. Fill zeros if the group is less than 3-bit For real part; start from just after the dot towards to the rightmost digit and combine 3-bits as a group. Fill zeros if the group is less than 3-bit  Converting a binary number ( ) 2 into an octal number For integer part; start from just after the dot towards to the leftmost digit and combine 3-bits as a group. Fill zeros if the group is less than 3-bit For real part; start from just after the dot towards to the rightmost digit and combine 3-bits as a group. Fill zeros if the group is less than 3-bit

He/She can convert number bases each other 14  Converting a binary number ( ) 2 into a hex number For integer part; start from just after the dot towards to the leftmost digit and combine 4-bit as a group. Fill zeros if the group is less than 4-bit For real part; start from just after the dot towards to the rightmost digit and combine 4-bit as a group. Fill zeros if the group is less than 4-bit.C E  Converting a binary number ( ) 2 into a hex number For integer part; start from just after the dot towards to the leftmost digit and combine 4-bit as a group. Fill zeros if the group is less than 4-bit For real part; start from just after the dot towards to the rightmost digit and combine 4-bit as a group. Fill zeros if the group is less than 4-bit.C E

He/She can convert number bases each other 15  Converting an Octal number into a binary number Convert each digit into its 3-bit binary counterpart.  Example ( ) 8 =( ) 2  Converting an octal number into a decimal number Multiply each digit’s value by radix and sum all terms  Example (372.2) 8 = 3x x x x8 -1 = 3x64 + 7x8 + 2x1 +2x0.125 =(250.25) 10  Converting an Octal number into a binary number Convert each digit into its 3-bit binary counterpart.  Example ( ) 8 =( ) 2  Converting an octal number into a decimal number Multiply each digit’s value by radix and sum all terms  Example (372.2) 8 = 3x x x x8 -1 = 3x64 + 7x8 + 2x1 +2x0.125 =(250.25) 10

He/She can convert number bases each other 16  Converting an Octal number into a hex number Step1. Convert each digit into its 3-bit binary counterpart. Step2. Compose 4-bit groups Step3. Convert 4-bit into hex counterpart Integer part group direction. Real part group direction  Example (5431) 8 = ( ? ) 16 Step1. ( ) 2 Step Step3. B 1 9 (5431) 8 = (B19) 16  Converting an Octal number into a hex number Step1. Convert each digit into its 3-bit binary counterpart. Step2. Compose 4-bit groups Step3. Convert 4-bit into hex counterpart Integer part group direction. Real part group direction  Example (5431) 8 = ( ? ) 16 Step1. ( ) 2 Step Step3. B 1 9 (5431) 8 = (B19) 16

He/She can convert number bases each other 17  Converting an hex number into a binary number Convert each digit into its 4-bit binary counterpart.  Example ( ) 16 =( ) 2  Converting an hex number into a decimal number Multiply each digit’s value by radix and sum all terms  Example (372) 16 = 3x x x16 0 = 3x256+ 7x16 + 2x1 = = (881) 10  Converting an hex number into a binary number Convert each digit into its 4-bit binary counterpart.  Example ( ) 16 =( ) 2  Converting an hex number into a decimal number Multiply each digit’s value by radix and sum all terms  Example (372) 16 = 3x x x16 0 = 3x256+ 7x16 + 2x1 = = (881) 10

He/She can convert number bases each other 18  Converting an Hex number into a Octal number Step1. Convert each digit into its 4-bit binary counterpart. Step2. Compose 3-bit groups Step3. Convert 3-bit into octal counterpart Integer part group direction. Real part group direction  Example (E0CA) 16 = ( ? ) 8 Step1. ( ) 2 Step2. ( ) 2 Step3. ( ) 8 (E0CA) 16 = (160312) 8  Converting an Hex number into a Octal number Step1. Convert each digit into its 4-bit binary counterpart. Step2. Compose 3-bit groups Step3. Convert 3-bit into octal counterpart Integer part group direction. Real part group direction  Example (E0CA) 16 = ( ? ) 8 Step1. ( ) 2 Step2. ( ) 2 Step3. ( ) 8 (E0CA) 16 = (160312) 8

He/She can do arithmetic operations on various radix 19  0+0 =0  1+0 =1  1+1 =0 Carry=  0+0 =0  1+0 =1  1+1 =0 Carry= Carry

He/She can do arithmetic operations on various radix 20  0-0 =0  1-0 =1  1-1 =0  0-1 =1 borrow=  0-0 =0  1-0 =1  1-1 =0  0-1 =1 borrow=

He/She can do arithmetic operations on various radix 21 In digital electronics, it is easier to create adder than subtractor unit. M-N can be redefined as M+( r complement of N) N can be negated by 1’s complement but 1’s complement contain both +0 and -0. STEP-1: Convert subtraction operation into addition by using r complement STEP-2: a. If an extra carry is obtained at the end, discard it and the number is assumed as positive. b. If no carry is obtained, apply r complement to result and add 1 In digital electronics, it is easier to create adder than subtractor unit. M-N can be redefined as M+( r complement of N) N can be negated by 1’s complement but 1’s complement contain both +0 and -0. STEP-1: Convert subtraction operation into addition by using r complement STEP-2: a. If an extra carry is obtained at the end, discard it and the number is assumed as positive. b. If no carry is obtained, apply r complement to result and add 1

He/She can do arithmetic operations on various radix 22 Example with extra carry: N can also be negated by 2’s complement (1’s complement+1) M= , N= M-N=? M-N = M+(-N) = M+ (2’s complement of N) = M+ (1’s complement+1) 2 1’s Complement of N= ( ) 2 2s complement of N (1’s complement of N + 1) = ( ) MSB is discarded, the result= ( ) 2

He/She can do arithmetic operations on various radix 23 Example without extra carry: M= (68) 10 N = (84) 10 M-N =? for 2’s complement 1’s complement of N= ( ) 2’s complement of N (1’s complement of N + 1) = ( ) No carry (-16) 10 1s comp ( ) 2 (-16) 10

He/She can do arithmetic operations on various radix

He/She can do arithmetic operations on various radix 25