A BINARY NUMBERS CuuDuongThanCong.com https://fb.com/tailieudientucntt dn 100's place 10's place 1's place d2 d1 d0 … 1's place 01's place 001's place d–1 d–2 d–3 … n Number = Σ di × 10i i = –k Figure A-1 The general form of a decimal number CuuDuongThanCong.com https://fb.com/tailieudientucntt d–k Binary Octal 1 1× 1024 +1× + 512 +1× + 256 +1× + 128 210 29 28 27 +1× + 64 26 +0× +0 25 +1× + 16 24 +0× +0 23 +0× +0 22 +0× +0 21 + × 20 +1 × + × + × + × 80 1536 + 448 + 16 + Decimal 2 1 × 103 + × 102 + × 101 + × 100 +0 +1 2000 + Hexadecimal D × 162 + 13 × 161 + × 160 1792 + 208 +1 Figure A-2 The number 2001 in binary, octal, and hexadecimal CuuDuongThanCong.com https://fb.com/tailieudientucntt Decimal 10 11 12 13 14 15 16 20 30 40 50 60 70 80 90 100 1000 2989 Binary 10 11 100 101 110 111 1000 1001 1010 1011 1100 1101 1110 1111 10000 10100 11110 101000 110010 111100 1000110 1010000 1011010 11001000 1111101000 101110101101 Octal 3 10 11 12 13 14 15 16 17 20 24 36 50 62 74 106 120 132 144 1750 5655 Hex 3 A B C D E F 10 14 1E 28 32 3C 46 50 5A 64 3E8 BA Figure A-3 Decimal numbers and their binary, octal, and hexadecimal equivalents CuuDuongThanCong.com https://fb.com/tailieudientucntt Example Hexadecimal Binary Octal B 0 1 0 1 0 0 1 1 0 5 Example Hexadecimal Binary Octal C B 1 1 1 1 0 1 1 1 0 0 7 B A Figure A-4 Examples of octal-to-binary and hexadecimal-tobinary conversion CuuDuongThanCong.com https://fb.com/tailieudientucntt Quotients Remainders 1492 746 373 186 93 46 23 11 1 0 1 1 1 0 = 149210 Figure A-5 Conversion of the decimal number 1492 to binary by successive halving, starting at the top and working downward For example, 93 divided by yields a quotient of 46 and a remainder of 1, written on the line below it CuuDuongThanCong.com https://fb.com/tailieudientucntt 1 1 1 1 1 + × 1499 = 2999 Result + × 749 = 1499 + × 374 = 749 + × 187 = 374 + × 93 = 187 + × 46 = 93 + × 23 = 46 + × 11 = 23 + × = 11 1+2×2=5 0+2×1=2 1+2×0=1 Start here Figure A-6 Conversion of the binary number 101110110111 to decimal by successive doubling, starting at the bottom Each line is formed by doubling the one below it and adding the corresponding bit For example, 749 is twice 374 plus the bit on the same line as 749 CuuDuongThanCong.com https://fb.com/tailieudientucntt N decimal 10 20 30 40 50 60 70 80 90 100 127 128 N binary 00000001 00000010 00000011 00000100 00000101 00000110 00000111 00001000 00001001 00001010 00010100 00011110 00101000 00110010 00111100 01000110 01010000 01011010 01100100 01111111 Nonexistent −N signed mag 10000001 10000010 10000011 10000100 10000101 10000110 10000111 10001000 10001001 10001010 10010100 10011110 10101000 10110010 10111100 11000110 11010000 11011010 11011010 11111111 Nonexistent −N 1’s compl 11111110 11111101 11111100 11111011 11111010 11111001 11111000 11110111 11110110 11110101 11101011 11100001 11010111 11001101 11000011 10111001 10101111 10100101 10011011 10000000 Nonexistent −N 2’s compl 11111111 11111110 11111101 11111100 11111011 11111010 11111001 11111000 11110111 11110110 11101100 11100010 11011000 11001110 11000100 10111010 10110000 10100110 10011100 10000001 10000000 Figure A-7 Negative 8-bit numbers in four systems CuuDuongThanCong.com https://fb.com/tailieudientucntt −N excess 128 01111111 01111110 01111101 01111100 01111011 01111010 01111001 01111000 01110111 01110110 01101100 01100010 01011000 01001110 01000100 00111010 00110000 00100110 00011100 00000001 00000000 Addend Augend Sum Carry +0 0 +1 1 +0 1 +1 Figure A-8 The addition table in binary CuuDuongThanCong.com https://fb.com/tailieudientucntt Decimal 1's complement 2's complement 10 + (−3) 00001010 11111100 00001010 11111101 +7 00000110 00000111 carry discarded 00000111 Figure A-9 Addition in one’s complement and two’s complement CuuDuongThanCong.com https://fb.com/tailieudientucntt ... hexadecimal equivalents CuuDuongThanCong .com https://fb .com/ tailieudientucntt Example Hexadecimal Binary Octal B 0 1 0 1 0 0 1 1 0 5 Example Hexadecimal Binary Octal C B 1 1 1 1 0 1 1 1 0 0 7 B A. .. 1 0 1 1 1 0 0 7 B A Figure A- 4 Examples of octal-to -binary and hexadecimal-tobinary conversion CuuDuongThanCong .com https://fb .com/ tailieudientucntt Quotients Remainders 1492 746 373 186 93... Figure A- 5 Conversion of the decimal number 1492 to binary by successive halving, starting at the top and working downward For example, 93 divided by yields a quotient of 46 and a remainder of