Writer:-Rashedul Hasan. Editor:- Jasim Uddin Number system Writer:-Rashedul Hasan. Editor:- Jasim Uddin

Commonly used Number System Base Number system Digits/symbol used 2 Binary 0,1 8 Octal 0,1,2,3,4,5,6,7 10 Decimal 0,1,2,3,4,5,6,7,8,9 16 Hexadecimal 0,1,2,3,4,5,6,7,8,9, A,B,C,D,E,F

Decimal Number System The decimal numeral system has ten as its base. It is the most widely used numeral system, perhaps because humans have ten digits over both hands. uses various symbols (called digits) for no more than ten distinct values (0, 1, 2, 3, 4, 5, 6, 7, 8 and 9) to represent any numbers,

Decimal Number System The decimal system is a positional numeral system; it has positions for units, tens, hundreds, etc. The position of each digit conveys the multiplier (a power of ten) to be used with that digit—each position has a value ten times that of the position to its right.

For example 848 8 Hundreds or 8*100 or 8*102 4 tens or 4*10 or 4*101 8 units or 8*1 or 8*100

Form the example, Both the 8 are not equal. Left most 8 occupies the hundred or 102 position is called MSD that is Most Significant digit. Right most 8 occupies the units or 100 position is called LSD that is Least Significant digit. The total value=8*102 + 4*101 +8*100 = 800+40+8 848.

Another example 1492.76 1 thousand or 1*1000 or 1* 103 4 Hundreds or 4*100 or 4*102 9 tens or 9*10 or 9*101 2 units or 2*1 or 2*100 7 tenths or 7*0.1 or 7*10-1 6 hundredths or 6*0.01 or 6*10-2

Total value = 1* 103 + 4*102 + 9*101 + 2*100 + 7*10-1 + 6*10-2 = 1000+400+90+2+0.7+0.06 =1492.76

Binary Number System The binary numeral system, or base-2 number system, is a numeral system that represents numeric values using two symbols, 0 and 1. The digits in binary system are called bits. In binary number system, the value of each digit is based on 2, and powers of 2.

Since binary is a base-2 system, each digit represents an increasing power of 2, with the rightmost digit representing 20, the next representing 21, then 22, and so on. 25 24 23 22 21 20 In binary system, if the bit is zero (0), its value is zero.

Bit: O or 1 Byte: a group of 8 bits is called Byte. Word: a group of 16 bits is called Word. Kilobytes KB 210 Megabyte MB 220 Gigabyte GB 230 Terabyte TB 240

Binary to Decimal If the digit is one (1), its value is determined by its position from the right. For example, the binary number 100101 is converted to decimal form by, [(1) × 25] + [(0) × 24] + [(0) × 23] + [(1) × 22] + [(0) × 21] + [(1) × 20] = [1 × 32] + [0 × 16] + [0 × 8] + [1 × 4] + [0 × 2] + [1 × 1] = 37

Decimal to Binary Divide the number by 2, the remainder is either 0 or 1. Place the remainder to the right of partial quotient obtained in step 1. Divide the partial quotient of step 1 by 2, placing the remainder to the right of new partial quotient Repeat the step 1,2,3 until a quotient of zero is obtained. The binary number is equal to the remainders arranged so that first remainder is the LSB and the last remainder is MSB of binary number.

Operation Remainder 118 ÷ 2 = 59 0 59 ÷ 2 = 24 1 29 ÷ 2 = 14 1 14 ÷ 2 = 7 0 7 ÷ 2 = 3 1 3 ÷ 2 = 1 1 1 ÷ 2 = 0 1 Reading the sequence of remainders from the bottom up gives the binary numeral 11101102

Decimal to Binary 47 Dividers number remainders 2 47 2 23 1 (LSB) 2 47 2 23 1 (LSB) 2 11 1 2 5 1 2 2 1 2 1 0 0 1 (MSB) 4710 = 101111

Decimal to Binary 76 Dividers number remainders 2 76 2 38 0 (LSD) 2 76 2 38 0 (LSD) 2 19 0 2 9 1 2 4 1 2 2 0 2 1 0 0 1(MSD) 7610 = 1001100

Incase of Fraction 0.625 Multiplier decimal fraction 2 * 0.625 1.25 1 2 * 0.625 1.25 1 2 * 0.25 .5 0 2 * .5 1.0 1

Incase of Fraction 0.86 Multiplier decimal fraction 2 * 0.86 1.72 1 2 * 0.86 1.72 1 2 * 0.72 1.44 1 2 * .44 .88 0 2 * .88 1.76 1

87.125 87 [Integral part] Dividers number remainders 2 87 2 43 1 (LSD) 2 87 2 43 1 (LSD) 2 21 1 2 10 1 2 5 0 2 2 1 2 1 0 0 1(MSD) 8710 = 1010111

87.125 .125 [Fraction part] Multiplier decimal fraction 2 * 0.125 .25 0 2 * 0.25 .5 0 2 * .5 1.0 1

Binary to Decimal 10101 1 0 1 0 1 20*1 = 1 21*0 = 0 22*1 = 4 23*0 = 0 1 0 1 0 1 20*1 = 1 21*0 = 0 22*1 = 4 23*0 = 0 24*1 = 16 21 101012 = 21

In case of fraction 0.1011 0. 1 0 1 1 2-1*1 = 0.5 2-2*0 = 0 0. 1 0 1 1 2-1*1 = 0.5 2-2*0 = 0 2-3*1 =0.125 2-4*1 = 0.0625 0.6875 0.1011 = 0.6875

Convert binary to Decimal 1101.1101

Octal Number system The octal number system has a base of eight. And they are, 0,1,2,3,4,5,6 and 7. The digit position of an octal number can have only value for 0 to 7. the digit positions in an octal number have weights as follows, 84 83 82 81 80 8-1 8-2 8-3 8-4

Decimal to Octal conversion Divide the Decimal number by 8 Place reminder to the right of partial quotient obtained in step 1. Divide the partial quotient of step 1 by 8, placing the remainder to the right of new partial quotient Repeat the step 1,2,3 until a quotient of zero is obtained. The binary number is equal to the remainders arranged so that first remainder is the LSD and the last remainder is MSD of octal number.

Decimal to Octal conversion 573 Dividers number remainders 8 573 8 71 5 (LSD) 8 8 7 8 1 0 0 1 (MSD) 57310 = 1075

Decimal to Octal conversion 2536 Dividers number remainders 8 2536 8 317 0 (LSD) 8 39 5 8 4 7 0 4 (MSD) 253610 = 4750

Octal to Decimal The extreme right hand digit is multiplied by 80 the second from the right by 81 and So on. Then add all this products to get decimal equivalent of the octal number. In case of octal fraction, multiply the first digit after octal point by 8-1, second digit from octal point by 8-2

Octal to Decimal 1075 1 0 7 5 80 * 5 = 5 81 * 7 = 56 82 * 0 = 0 1 0 7 5 80 * 5 = 5 81 * 7 = 56 82 * 0 = 0 83* 1 = 512 573

Octal to Decimal 0.44 0. 4 4 8-1 * 4 = 0.5 8-2 * 4 = 0.0625 0.5625

Octal to Decimal Assignment 4750 0.6256

Octal to Binary 576 5 7 6 111 101 110 576 = 101111110

Octal to Binary In case of Fraction 0.216 0. 2 1 6 010 001 110 0. 2 1 6 010 001 110 0.216 = 0.010001110

Octal to Binary 27.12

Binary to Octal Group the binary bits in three. For grouping the bits in three, move towards left from binary point. In case of even number, add zero or zeros at appropriate place. Replace each group of threes bits by equivalent octal numbers.

110111101 110 111 101 6 5 7

11000110 011 000 110 3 6

Assignment 1111000 Ans. 170

Hexadecimal The base is 16, it has 16 possible digit symbol. 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E and F The digit positions in an octal numbers have weights as follows, 164 163 162 161 160 16-1 16-2 16-3 16-4 Each hexadecimal digit represent group of four (4) binary digits. hexadecimal digit A-F are equivalent to decimal values 10 – 15.

Relationship between Hexadecimal, Decimal and Binary digits. 0000 1 0001 2 0010 3 0011 4 0100 5 0101 6 0110 7 0111 8 1000 9 1001 A 10 1010 B 11 1011 C 12 1100 D 13 1101 E 14 1110 F 15 1111

Decimal to Hexadecimal Divide the number by 16 Place reminder to the right of partial quotient obtained in step 1. Divide the partial quotient of step 1 by 16, placing the remainder to the right of new partial quotient Repeat the step 1,2,3 until a quotient of zero is obtained. The binary number is equal to the remainders arranged so that first remainder is the LSD and the last remainder is MSD of octal number.

Decimal to Hexadecimal 741 Dividers number remainders 16 741 16 46 5 (LSD) 16 2 14 i.e E 0 2(MSD) 74110 = 2E5

Decimal to Hexadecimal 2536

Decimal to Hexadecimal In case of Fraction 0.256 16 * 0.256 4.096 4 16 * 0.096 1.536 1 16 * 0.536 8.576 8 16 * 0.576 9.216 9

Decimal to Hexadecimal 0.3942

Decimal to Hexadecimal 97.236 Integral Part Dividers number remainders 16 97 16 6 1 (LSD) 16 0 6 (MSD)

Decimal to Hexadecimal Fraction Part Multiplier fraction & Partial product 16 * 0.236 3.776 3 16 * 0.776 12.416 12/C 16 * 0.416 6.656 6 16 * 0.656 10.496 10/A

Hexadecimal to Decimal 1F95 1 F 9 5 160 * 5 = 5 161 * 9 = 144 162 * F = 3840 163* 1 = 4096 8085

Hexadecimal to Decimal

Hexadecimal to Decimal 0.48 0. 4 8 16-1 * 4 = 0.25 16-2 * 8 = 0.03125 0.28125

Assignment 0.D2F 61.3C6A

Hexadecimal to Binary 59C 5 9 C 0101 1001 1100

Hexadecimal to Binary 0.2D6 0. 2 D 6 0010 1101 0110

Hexadecimal to Binary ABCD A B C D 1010 1011 1100 1101

Hexadecimal to Binary 11011110 1101 1110 D E

Hexadecimal to Binary 110000110 0001 1000 0110 1 8 6