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Lecture 1 – Introduction, Numbers and Number System 1.1. Contents Number Systems (Appendix B) .................................................................................................................................. 2 1. Example 1.1 – Converting to Base 10 ......................................................................................................................... 2 1.2. Number Representation ............................................................................................................................................... 2 1.3. Number Conversion .................................................................................................................................................... 3 1. To convert a number N expressed in base 10 to base B .............................................................................................. 3 2. Example 1.2 – Converting Decimal Number to base 8 ............................................................................................... 3 3. Example 1.3 – Converting Decimal Number to base 8 ............................................................................................... 3 4. Example 1.4 – Converting Decimal Number to base 16 ............................................................................................. 3 5. Example 1.5 – Converting Decimal Number to base 2 ............................................................................................... 3 6. Binary-to-hexadecimal (4 bits = 1 hex digit) .............................................................................................................. 4 7. Binary-to-octal (3 bits = 1 octal digit) ......................................................................................................................... 4 8. Hexadecimal-to-binary ................................................................................................................................................ 4 9. Octal-to-binary ............................................................................................................................................................ 4 10. Example 1.6 – Converting Number ........................................................................................................................ 4 1.4. Signed Binary Number ................................................................................................................................................ 4 1.5. Two’s Complement Numbers ..................................................................................................................................... 4 1. Example 1.7 – Two’s Complement Numbers ............................................................................................................. 5 2. Example 1.8 – Two’s Complement Numbers ............................................................................................................. 5 3. Find Decimal Value of Signed Binary Number .......................................................................................................... 5 1.6. Negating two’s Complement Numbers ....................................................................................................................... 6 1. Example 1.9 – Negating in Two’s Complement System............................................................................................. 6 2. Negating Octal and Hexadecimal Numbers ................................................................................................................ 6 a. Example 1.10 – Negating Hexadecimal Numbers in Two’s Complement System ................................................. 7 1.7. Addition in Two’s Complement System ..................................................................................................................... 7 1.8. Subtraction in Two’s Complement System ................................................................................................................. 7 1.9. Examples on Two’s Complements .............................................................................................................................. 8 1. Example 1.11 – Addition and Subtraction in Two’s Complement Systems................................................................ 8 Lecture 1 – Introduction, Numbers, and Number System Page 1 of 8 1.1. Number Systems (Appendix B) Representation of an integer number on base B is done by creating a set of coefficients an … a1a0 with weights specified by powers of the base B. The coefficients in base B are numbers that can take values 0 through B-1. A number N can be written in base B as N an ...a2a1a0 B an B n ... a2 B 2 a1B1 a0 B 0 Fractional numbers can be written by introducing negative powers of the base. A fractional number N can be written in base B as N am ...a2a1a0 .a1a2a3 ...a n B am B m ... a2 B 2 a1B1 a0 B 0 a1B 1 a2 B 2 a3 B 3 ... a n B n There are four common number systems: decimal, hexadecimal, octal, and binary. 1. Example 1.1 – Converting to Base 10 Find the equivalent numbers in the decimal base system of the following numbers: (1204.12)10 = 1204.12 (1204.12)8 = 1*83 + 2*82 + 0*81 + 4*80 + 1*8-1 + 2*8-2 = 644.15625 (1A2F.11)16 = 1*163 + 10*162 + 2*161 + 15*160 + 1*16-1 + 1* 16-2 = 6703.06640 (10000010.10)2= 1*27 + 0*26 + 0*25 + 0*24 + 0*23 + 0*22 + 1*21 + 0*20 + 1*2-1 +0*2-2 = 130.5 1.2. Number Representation Decimal 0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00 9.00 10.00 11.00 12.00 13.00 14.00 15.00 16.00 17.00 18.00 19.00 20.00 Binary 00000000 00000001 00000010 00000011 00000100 00000101 00000110 00000111 00001000 00001001 00001010 00001011 00001100 00001101 00001110 00001111 00010000 00010001 00010010 00010011 00010100 Lecture 1 – Introduction, Numbers, and Number System Octal 00 01 02 03 04 05 06 07 10 11 12 13 14 15 16 17 20 21 22 23 24 Page 2 of 8 Hex 00 01 02 03 04 05 06 07 08 09 0A 0B 0C 0D 0E 0F 10 11 12 13 14 1.3. Number Conversion 1. To convert a number N expressed in base 10 to base B Separate integer part of number from fractional part Divide integer part by B continuously and save the remainder at each step. Multiply fractional part by B continuously and save integer part at each step. N a2a1a0 .a1a2 B a2 B 2 a1B1 a0 B 0 a1B 1 a2 B 2 2. Example 1.2 – Converting Decimal Number to base 8 Converting the following decimal numbers into base 8: (75.32)10 Remainder Fraction 8 75 3 0.32 * 8 8 9 1 0.56 * 8 8 1 1 0.48 * 8 0.84 * 8 0.72 * 8 (75.32)10 ≈ (113.24365)8 Result 2.56 4.48 3.84 6.72 5.76 3. Example 1.3 – Converting Decimal Number to base 8 Converting the following decimal number into base 8: (256.347)10 Remainder Fraction 8 256 0 0.347 * 8 8 32 0 0.776 * 8 84 4 0.208 * 8 0.664 * 8 0.312 * 8 0.496 * 8 (256.347)10 ≈ (400.261523)8 Result 2.776 6.208 1.664 5.312 2.496 3.968 4. Example 1.4 – Converting Decimal Number to base 16 Converting the following decimal number into base 6: (420.625)10 Remainder Fraction 16 420 4 0. 625 * 16 16 26 A 16 1 1 (420.625)10 ≈ (1A4.A)16 5. Example 1.5 – Converting Decimal Number to base 2 Converting the following decimal number into base 6: (53.375)10 Remainder Fraction 0.375 * 2 1 2 53 0.75 * 2 0 2 26 0.50 * 2 1 2 13 0 26 1 23 1 21 (53.375)10 ≈ (110101.011)2 Lecture 1 – Introduction, Numbers, and Number System Page 3 of 8 Result 10.0 Result 0.75 1.50 1.00 6. Binary-to-hexadecimal (4 bits = 1 hex digit) Group binary digits into groups of four and assign each group a hex digit. 1011 1110 0101 0011 B E 5 3 7. Binary-to-octal (3 bits = 1 octal digit) 101 111 010 5 7 2 001 1 8. Hexadecimal-to-binary C 2 1100 0010 8 1000 E 1110 9. Octal-to-binary 5 101 2 010 1 001 6 110 10. Example 1.6 – Converting Number (5CB.2F)16 (147.62)8 (1101 1110 1010. 1010 1001)2 (101 110 010.101 001 011 100)2 1.4. = (0101 1100 1011.0010 1111)2 = (001 100 111.110 010)2 = (DEA.A9)16 = (562.5134)8 Signed Binary Number For signed numbers, in the binary system, the sign of the number is denoted by the left-most bit: 0 = positive 1 = negative For a n-bit number, the remaining n-1 bits represent the magnitude. For signed binary system, there is a duplicate representation of 0 such as +0 (0000 0000) and – 0 (1000 0000). As a result, there is one less combination of bits to represent signed numbers. 1.5. Two’s Complement Numbers All modern computers use the two’s complement number system to represent negative numbers. 1’s Complement of N: complement individual bits (0 1 and 1 0) 2’s Complement of N: complement individual bit and add 1 to it. In the two’s complement system, positive numbers are identical to binary numbers with the sign bit is 0. Lecture 1 – Introduction, Numbers, and Number System Page 4 of 8 Range of number in a system of n bits: Unsigned number: 0 ≤ N ≤ 2n – 1 1’s complement: –2n – 1 + 1 ≤ N ≤ 2n – 1 – 1 2’s complement: –2n – 1 ≤ N ≤ 2n – 1 – 1 The 2’s complement of N is used to represent –N. It can represent integers in the range between –2n-1 to 2n-1-1, where n is the number of bits available for representing N (for unsigned numbers N, the range would be 0 to 2n-1). 1. Example 1.7 – Two’s Complement Numbers Find the two’s complement of the following numbers in 8-bit system: –10, –8, –4, +7, and –14. 8-bit 1’s Compl. +1 2’s Compl. –10 +10 00001010 11110101 11110110 –8 +8 00001000 11110111 11111000 –4 +4 00000100 11111011 11111100 +7 +7 00000111 00000111 –14 +14 00001110 11110001 11110010 2. Example 1.8 – Two’s Complement Numbers Find the equivalent decimal numbers of the following number (2’s complement) in 6-bit system: 6-bit 1’s Compl. +1 Magnitude Numbers 110110 001001 001010 –10 101110 010001 010010 –18 001011 001011 +11 110101 001010 001011 –11 111010 000101 000110 –6 3. Find Decimal Value of Signed Binary Number Observe that the 2’s complement of a negative number is the corresponding positive number. For example, -510=10112 and 2’s complement of 10112 is 01012 Procedure 1) Find the 2’s complement of the number, i.e. its positive value 2) Convert the positive binary number to decimal 3) Add a negative sign to the decimal value Two’s Complement Numbers in 6-bit system +15 +14 +13 +12 +11 +10 Signed System 001111 001110 001101 001100 001011 001010 Lecture 1 – Introduction, Numbers, and Number System 1’s Complement 001111 001110 001101 001100 001011 001010 Page 5 of 8 2’s Complement 001111 001110 001101 001100 001011 001010 +9 +8 +7 +6 +5 +4 +3 +2 +1 0 –1 –2 –3 –4 –5 –6 –7 –8 –9 –10 –11 –12 –13 –14 –15 1.6. 001001 001000 000111 000110 000101 000100 000011 000010 000001 000000 100001 100010 100011 100100 100101 100110 100111 101000 101001 101010 101011 101100 101101 101110 101111 001001 001000 000111 000110 000101 000100 000011 000010 000001 000000 111110 111101 111100 111011 111010 111001 111000 110111 110110 110101 110100 110011 110010 110001 110000 001001 001000 000111 000110 000101 000100 000011 000010 000001 000000 111111 111110 111101 111100 111011 111010 111001 111000 110111 110110 110101 110100 110011 110010 110001 Negating two’s Complement Numbers To negate a number in the Two’s Complement system: Invert all bits of the number Add 1. 1. Example 1.9 – Negating in Two’s Complement System Find the opposite numbers of the followings numbers (2’s complement) in 6-bit system: –10 –8 –4 +7 –14 2’s Compl. 1’s Compl. 110110 111000 111100 000111 110010 001001 000111 000011 111000 001101 +1 2’s Compl. 001010 001000 000100 111001 001110 Opposite Number +10 +8 +4 –7 +14 2. Negating Octal and Hexadecimal Numbers Convert the octal or hex number to its binary equivalent with an appropriate number of bits (e.g. 8, 16, 32 …) Negate the binary equivalent Convert the two’s complement number to its octal or hex equivalent Lecture 1 – Introduction, Numbers, and Number System Page 6 of 8 a. Example 1.10 – Negating Hexadecimal Numbers in Two’s Complement System Find the opposite number of the following hexadecimal numbers A107 FEB7 A0CF FFCA F0FA 1.7. Binary Equivalent 1’s Complement. 1010000100000111 1111111010110111 1010000011001111 1111111111001010 1111000011111010 0101111011111000 0000000101001000 0101111100110000 0000000000110101 0000111100000101 +1 0101111011111001 0000000101001001 0101111100110001 0000000000110110 0000111100000110 Hexadecimal Equivalent 5EF9 0149 5F31 0036 0F06 +5 +(–2) +3 0101 1110 10011 2’s Complement Addition in Two’s Complement System Any carry-out from the signed bit is ignored +5 +(+2) +7 0101 0010 0111 ignore –5 +(+2) –3 1.8. 1011 0010 1101 –5 +(–2) –7 1011 1110 11001 Ignore Subtraction in Two’s Complement System The easiest way of performing subtraction is to negate the subtrahend and add it to the minuend. First, find the 2’s complement of the subtrahend and then perform addition. +5 – (+2) +3 0101 – 0010 2’s Compl 0101 +1110 10011 +5 – (–2) +7 0101 –1110 2’s Compl 0101 +0010 0111 –5 – (–2) –3 1011 –1110 2’s Compl 1011 +0010 1101 ignore –5 – (+2) –7 1011 – 0010 2’s Compl 1011 +1110 11001 ignore Lecture 1 – Introduction, Numbers, and Number System Page 7 of 8 1.9. Examples on Two’s Complements 1. Example 1.11 – Addition and Subtraction in Two’s Complement Systems Perform arithmetic operations of these numbers in 8-bit system: +14 +(+6) +20 –17 –(+11) –28 00001110 00000110 00010100 11101111 – 00001011 +24 – (+12) +12 2’s Compl 11101111 + 11110101 1 11100100 Lecture 1 – Introduction, Numbers, and Number System –23 – (–11) –12 00011000 – 00001100 2’s Compl 00011000 +11110100 1 00001100 11101001 – 11110101 2’s Compl 11101001 + 00001011 11110100 Page 8 of 8