The document discusses different number systems including binary, decimal, octal, and hexadecimal. It provides details on how to convert between these number systems, including how to convert fractional numbers between bases. Conversion methods covered include dividing numbers into place values to determine the digit values in the target base. The document also discusses representing negative numbers using 1's complement notation.

1. NUMBER SYSTEM

2. Types
• Binary Number System
• Decimal Number System
• Octal Number System
• Hexadecimal Number System

3. 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

4. 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

5. 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

6. 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

7. Number System Conversion

8. 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)

9. 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)

10. 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

11. 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

12. Decimal into Hexadecimal
• Convert 185(10) into hexadecimal
Number Remainder
16 185
16 11 9
0 B
185(10) = 0B9 (16)

13. 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)

14. 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)

15. 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)

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)

17. 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)

18. 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)

19. 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
---------------------
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

20. 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.