an × 2n + an–1 × 2n–1 + an–2 × 2n–2 + ……… + a0 × 20
For example, binary number 11102 can be shown as:
11102 = 1 × 23 + 1 × 22 + 1 × 21 + 0 × 20
Similarly, binary number 100011102 can be shown as:
100011102 = 1 × 27 + 0 × 26 + 0 × 25 + 0 × 24 + 1 × 23 + 1 × 22 + 1 × 21 + 0 × 20
1.5.3 Octal Number System
In the octal number system, the valid numbers are 0, 1, 2, 3, 4, 5, 6, 7. A subscript 8 indicates that a number is in octal format. For example, the octal number 23 appears as 238.
In general, an octal number is represented as:
an × 8n + an–1 × 8n–1 + an–2 × 8n–2 + ……… + a0 × 80
For example, octal number 2378 can be shown as:
2378 = 2 × 82 + 3 × 81 + 7 × 80
Similarly, octal number 17778 can be shown as:
17778 = 1 × 83 + 7 × 82 + 7 × 81 + 7 × 80
1.5.4 Hexadecimal Number System
In the hexadecimal number system, the valid numbers are: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F. A subscript 16 or subscript H indicates that a number is in hexadecimal format. For example, hexadecimal number 1F can be written as 1F16 or as 1FH. In general, a hexadecimal number is represented as:
an × 16n + an–1 × 16n–1 + an–2 × 16n–2 + ……… + a0 × 160
For example, hexadecimal number 2AC16 can be shown as:
2AC16 = 2 × 162 + 10 × 161 + 12 × 160
Similarly, hexadecimal number 3FFE16 can be shown as:
3FFE16 = 3 × 163 + 15 × 162 + 15 × 161 + 14 × 160
1.6 Converting Binary Numbers into Decimal
To convert a binary number into decimal, write the number as the sum of the powers of 2.
Convert binary number 10112 into decimal.
Write the number as the sum of the powers of 2:
10112 = 1 × 23 + 0 × 22 + 1 × 21 + 1 × 20
= 8 + 0 + 2 = 1
= 11
or, 10112 = 1110
Convert binary number 110011102 into decimal.
Write the number as the sum of the powers of 2:
110011102 = 1 × 27 + 1 × 26 + 0 × 25 + 0 × 24 + 1 × 23 + 1 × 22 + 1 × 21 + 0 × 20
= 128 + 64 + 0 + 0 + 8 + 4 + 2 + 0
= 206
or, 110011102 = 20610
Table 1.1 shows the decimal equivalent of numbers from 0 to 31.
Table 1.1: Decimal equivalent of binary numbers
| Binary | Decimal | Binary | Decimal |
|---|---|---|---|
| 00000000 | 0 | 00010000 | 16 |
| 00000001 | 1 | 00010001 | 17 |
| 00000010 | 2 | 00010010 | 18 |
| 00000011 | 3 | 00010011 | 19 |
| 00000100 | 4 | 00010100 | 20 |
| 00000101 | 5 | 00010101 | 21 |
| 00000110 | 6 | 00010110 | 22 |
| 00000111 | 7 | 00010111 | 23 |
| 00001000 | 8 | 00011000 | 24 |
| 00001001 | 9 | 00011001 | 25 |
| 00001010 | 10 | 00011010 | 26 |
| 00001011 | 11 | 00011011 | 27 |
| 00001100 | 12 | 00011100 | 28 |
| 00001101 | 13 | 00011101 | 29 |
| 00001110 | 14 | 00011110 | 30 |
| 00001111 | 15 | 00011111 | 31 |
1.7 Converting Decimal Numbers into Binary
To convert a decimal number into binary, divide the number repeatedly by 2 and take the remainders. The first remainder is the least significant digit (LSD), and the last remainder is the most significant digit (MSD).
Convert decimal number 2810 into binary.