number system

NUMBER SYSTEM

Types

• Binary Number System

• Decimal Number System

• Octal Number System

• Hexadecimal Number System

Binary Number System• It uses only two digits. 0 & 1• These digits (o & 1) are called binary Digits or

binary numbers.• This is positional number system like Decimal

number system.• Each position has a weight that is power of 2

• 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 Number System• These are Base 10 numbers.• It is also positional number system.• We can also write numbers with fractional parts

in the system.• These numbers are from 0 to 9

Position 4 3 2 1 0 -1 -2

Face Value 5 7 2 3 1 . 2 1

Weights 104 103 102 101 100 10-1 10-2

Octal Number System• These numbers have Base 8.• These numbers are from 0 to 7.

• 751(8) is a valid Octal number but 821 can not be a member of this number system.

• 630.4(8) = 6x82 + 3x81 + 0x80 + 4x8-1 =408.5(10)

Position 2 1 0 -1

Face Value 6 3 0 . 4

Weight 82 81 80 8-1

Hexadecimal Number System• This number system uses Base 16.• Numbers are from 0 to 9 and A to F• 758(16) is different from 758(10)

• 758(10) will be called as Seven hundred and fifty eight

• But 758(16) will be called Seven Five Eight Base Sixteen.

• 758.D1(16) = 7x162 + 5x161 + 8x160 + Dx16-1 + 1x16-2 = 1880.8164(10)

Position 2 1 0 -1 -2

Face Value 7 5 8 . D 1

Weight 162 161 160 16-1 16-2

Number System Conversion

Decimal to Binary

• Convert 27 into binary

Number Remainder

2 27

2 13 1

2 6 1

2 3 0

2 1 1

0 1

= 011011(2)

Fractional Decimal to Binary

• Convert 0 . 56 into binary.

Result Fractional Part Integral Part

2 X 0.56 1.12 12 1

2 X 0.12 0.24 24 0

2 X 0.24 0.48 48 0

2 X 0.48 0.96 96 0

2 X 0.96 1.92 92 1

2 X 0.92 1.84 84 1

2 X 0.84 1.68 68 1

2 X 0.68 1.36 36 1

= 10001111(2)

Real Number into Binary• Convert 56 . 25(10) = 0111000 . 01(2)

Number Remainder

2 56

2 28 0

2 14 0

2 7 0

2 3 1

2 1 1

0 1

56=0111000(2)

Result Fractional Part Integral Part

2x0.25 0.5 5 0

2x0.5 1.0 0 1

0 . 25=01

Binary to Decimal

• Convert 011011(2) into Decimal

011011(2) = 0x25 + 1x24 + 1x23 + 0x20 + 1x21 + 1x20 = 27(10)

• Convert 1110 . 11(2) into Decimal

1110 . 11(2) = 1x23 + 1x22 + 1x21 + 0x20 + 1x2-1 + 1x2-2

8 + 4 + 2 + 0 + ½ + ¼ = 14 . 75

Decimal into Hexadecimal

• Convert 185(10) into hexadecimal

Number Remainder

16 185

16 11 9

0 B

185(10) = 0B9 (16)

Hexadecimal into Decimal

• Convert 0B9 (16) into Decimal

0B9(16) = 0x162 + Bx161 + 9x160 = 0x162 + 11x161 + 9x160 = 185(10)

• Convert 0B9.4C (16) into Decimal

0B9 . 4C(16) = 0x162 + Bx161 + 9x160 + 4x16-1 + Cx16-2

0x162 + 11x161 + 9x160 + 4x16-1 + 12x16-2

0 + 176 + 9 + 4/16 + 12/256

0 + 176 + 9 + ¼ + 3/64 = 185 . 296275(10)

Hexadecimal into Binary

• Convert 10A8(16) into Binary

• Convert each digit into Binary separately and write in 4 bits.• Step 1

– 1 = 0001(2)

– 0 = 0000(2)

– A = 1010(2)

– 8 = 1000(2)

• Step 2 : Replace each digit of Hexadecimal number with four bits obtained• 10A8(16) = 0001 0000 1010 1000 (2)

Binary to Hexadecimal

• Convert 10010011(2) into Hexadecimal

Step 1: Divide your number into groups of 4 bits starting from right side.

10010011(2) is divided into 1001 0011

Step 2: Convert each group into hexadecimal

1001 = 9(16) and 0011= 3(16)

Step 3: Replace each group by its hexadecimal equivalent

1001 0011(2) = 93(16)

Decimal into Octal

• Convert 185(10) into Octal

• Convert 0.3 (10) into Octal

R

8 185

8 23 1

8 2 7

8 0 2

185(10) = 0271(8)

8x0.3 = 2.4 0.4 2

8x0.4 = 3.2 0.2 3

8x0.2 = 1.6 0.6 1

8x0.6 = 4.8 0.8 4

8x0.8 = 6.4 0.4 6

0.3(10) = 0.23146(8)

Octal into Decimal• Convert 0271(8) into Decimal

0271(8) = 0x83 + 2x82 + 7x81 + 1x80 = 185(10)

• Convert 107(8) into Binary

Convert each digit independently into Binary

1 = 001(2)

0 = 000(2)

7 = 111(2)

107(8) = 001 000 111 (2)

Binary into Octal

• Convert 10010011(2) into Octal

Step 1: First divide the number into groups of 3 bits starting from right side.

010 , 010 and 011

Step 2: Convert each group into Octal

010(2) = 2(8) 010(2) = 2(8) 011 = 2(8)

Step 3: Replace each group by its Octal equivalent.

010 010 011(2) = 223(8)

1’s Complement Method

• Method 1: 1’s complement of an 8-bit binary number is obtained by subtracting the number from 11111111(2)

11111111

- 10011001

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1’s Complement 01100110

• Method 2: It can directly be obtained by changing all 0’s to 1’s and all 1’s to 0’s.

Original Number 01100110

1’s Complement 10011001

Representation of negative numbers using 1’s Complement

• To represent the negative number in 1’s complement form, we perform following steps.

– Determine the number of bits to represent the number– Convert the modules of the given number in Binary– Place a 0 in MSB

– Take 1’s complement of the result.