1. Assoc. Prof. Dr. Ahmet Turan ÖZCERİT

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

3. 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

4. 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

5. 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 2 2 2 + d 1 2 1 + d 0 2 0, d 1 2 -1 +d 2 2 -2 +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 2 2 2 + d 1 2 1 + d 0 2 0, d 1 2 -1 +d 2 2 -2 +d n 2 -n Binary SystemDecimal System 000 011 102 113

6. He/She can make operation on binary numbers 6  What is binary value of following binary number: 1 0110 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 B= 256 + 0 + 64 + 32 + 0 + 8 + 4 + 0 + 1 B= 365 10  What is binary value of following binary number: 1 0110 1101, 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 -1 +1*2 -2+ 0*2 - 3+ 1*2 -4 B= 256 + 0 + 64 + 32 + 0 + 8 + 4 + 0 + 1 + 0.5 +0.25 + 0+ 0.0625 B= 365,8125 10  What is binary value of following binary number: 1 0110 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 B= 256 + 0 + 64 + 32 + 0 + 8 + 4 + 0 + 1 B= 365 10  What is binary value of following binary number: 1 0110 1101, 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 -1 +1*2 -2+ 0*2 - 3+ 1*2 -4 B= 256 + 0 + 64 + 32 + 0 + 8 + 4 + 0 + 1 + 0.5 +0.25 + 0+ 0.0625 B= 365,8125 10

7. 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 2 8 2 + d 1 8 1 + d 0 8 0, d 1 8 -1 +d 2 8 -2 +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 2 8 2 + d 1 8 1 + d 0 8 0, d 1 8 -1 +d 2 8 -2 +d n 8 -n BinaryOctal 0000 0011 0102 0113 1004 1015 1106 1117

8. 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 2 16 2 + d 1 16 1 + d 0 16 0, d 1 16 -1 +d 2 16 -2 +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 2 16 2 + d 1 16 1 + d 0 16 0, d 1 16 -1 +d 2 16 -2 +d n 16 -n BinaryHex 00000 00011 00102 00113 01004 01015 01106 01117 BinaryHex 10008 10019 1010A 1011B 1100C 1101D 1110E 1111F

9. 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 = (1110101,1101110….) 2

10. He/She can convert number bases each other 10  Convert a real decimal number (0,513) 10 into an octal number OperationMultiplyIntegerResult 0.513*84.1044o 0 =1 0.104*80.8320o 1 =0 0.832*86.6566o 2 =6 0.656*85.2485o 3 =5 0.248*81.9841o 4 =1 (0,513) 10 ≅ (0,40651) 8

11. 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

12. He/She can convert number bases each other 12  Convert a binary number (100.01) 2 into a decimal number 100.01= 1*2 2 + 0*2 1 + 0*2 0, 0*2 -1 + 1*2 -2 = 1*4 + 0 + 0, 0 + 1* ¼ = 4 + 0 +0, 0+ 0.25 = (4,25) 10  Convert a binary number (100.01) 2 into a decimal number 100.01= 1*2 2 + 0*2 1 + 0*2 0, 0*2 -1 + 1*2 -2 = 1*4 + 0 + 0, 0 + 1* ¼ = 4 + 0 +0, 0+ 0.25 = (4,25) 10

13. He/She can convert number bases each other 13  Converting a binary number (110111011.1100111) 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 1 5 6 6. 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.6 3 4  Converting a binary number (110111011.1100111) 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 1 5 6 6. 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.6 3 4 1101110110.. 1100111

14. He/She can convert number bases each other 14  Converting a binary number (110111011.1100111) 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 3 7 6. 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 (110111011.1100111) 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 3 7 6. 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 1101110110.. 1100111

15. 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 (673.124) 8 =(110 111 011. 001 010 100) 2  Converting an octal number into a decimal number Multiply each digit’s value by radix and sum all terms  Example (372.2) 8 = 3x8 2 + 7x8 1 + 2x8 0 + 2x8 -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 (673.124) 8 =(110 111 011. 001 010 100) 2  Converting an octal number into a decimal number Multiply each digit’s value by radix and sum all terms  Example (372.2) 8 = 3x8 2 + 7x8 1 + 2x8 0 + 2x8 -1 = 3x64 + 7x8 + 2x1 +2x0.125 =(250.25) 10

16. 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. (101 100 011 001) 2 Step2. 1011 0001 1001 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. (101 100 011 001) 2 Step2. 1011 0001 1001 Step3. B 1 9 (5431) 8 = (B19) 16

17. 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 (673.124) 16 =(0110 0111 0011. 0001 0010 0100) 2  Converting an hex number into a decimal number Multiply each digit’s value by radix and sum all terms  Example (372) 16 = 3x16 2 + 7x16 1 + 2x16 0 = 3x256+ 7x16 + 2x1 = 768 + 112 + 1 = (881) 10  Converting an hex number into a binary number Convert each digit into its 4-bit binary counterpart.  Example (673.124) 16 =(0110 0111 0011. 0001 0010 0100) 2  Converting an hex number into a decimal number Multiply each digit’s value by radix and sum all terms  Example (372) 16 = 3x16 2 + 7x16 1 + 2x16 0 = 3x256+ 7x16 + 2x1 = 768 + 112 + 1 = (881) 10

18. 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. (1110 0000 1100 1010) 2 Step2. (001 110 000 011 001 010) 2 Step3. ( 1 6 0 3 1 2) 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. (1110 0000 1100 1010) 2 Step2. (001 110 000 011 001 010) 2 Step3. ( 1 6 0 3 1 2) 8 (E0CA) 16 = (160312) 8

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

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

21. 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

22. 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=1010100, N=1000100 M-N=? M-N = M+(-N) = M+ (2’s complement of N) = M+ (1’s complement+1) 2 1’s Complement of N= (0111011) 2 2s complement of N (1’s complement of N + 1) = (0111100) 2 1010100 0111100 1 0010000 MSB is discarded, the result= (0010000) 2

23. He/She can do arithmetic operations on various radix 23 Example without extra carry: M=1000100 2 (68) 10 N =1010100 2 (84) 10 M-N =? for 2’s complement 1’s complement of N= (0101011) 2’s complement of N (1’s complement of N + 1) = (0101100) 1000100 0101100 1110000 No carry (-16) 10 1s comp 0001111 0000001 -(0010000) 2 (-16) 10

24. He/She can do arithmetic operations on various radix 24 1101 13 0101 5 1101 65 0000 1101 100001 110010 101 101 1010 00101 101 0000

25. He/She can do arithmetic operations on various radix 25