When we type some letters or words, the computer translates them in numbers as computers can understand only numbers. A computer can understand the positional number system where there are only a few symbols called digits and these symbols represent different values depending on the position they occupy in the number.

The value of each digit in a number can be determined using −

- The digit
- The position of the digit in the number
- The base of the number system (where the base is defined as the total number of digits available in the number system)

Decimal Number System

The number system that we use in our day-to-day life is the decimal number system. Decimal number system has base 10 as it uses 10 digits from 0 to 9. In decimal number system, the successive positions to the left of the decimal point represent units, tens, hundreds, thousands, and so on.

Each position represents a specific power of the base (10). For example, the decimal number 1234 consists of the digit 4 in the units position, 3 in the tens position, 2 in the hundreds position, and 1 in the thousands position. Its value can be written as

(1 x 1000)+ (2 x 100)+ (3 x 10)+ (4 x l)

(1 x 10^{3})+ (2 x 10^{2})+ (3 x 10^{1})+ (4 x l0^{0})

1000 + 200 + 30 + 4

1234

As a computer programmer or an IT professional, you should understand the following number systems which are frequently used in computers.

S.No. |
Number System and Description |

1 | Binary Number System
Base 2. Digits used : 0, 1 |

2 | Octal Number System
Base 8. Digits used : 0 to 7 |

3 | Hexa Decimal Number System
Base 16. Digits used: 0 to 9, Letters used : A- F |

Binary Number System

Characteristics of the binary number system are as follows −

- Uses two digits, 0 and 1
- Also called as base 2 number system
- Each position in a binary number represents a
**0**power of the base (2). Example 2^{0} - Last position in a binary number represents a
**x**power of the base (2). Example 2^{x}where**x**represents the last position – 1.

Example

Binary Number: 10101_{2}

Calculating Decimal Equivalent −

Step |
Binary Number |
Decimal Number |

Step 1 | 10101_{2} |
((1 x 2^{4}) + (0 x 2^{3}) + (1 x 2^{2}) + (0 x 2^{1}) + (1 x 2^{0}))_{10} |

Step 2 | 10101_{2} |
(16 + 0 + 4 + 0 + 1)_{10} |

Step 3 | 10101_{2} |
21_{10} |

**Note** − 10101_{2} is normally written as 10101.

Octal Number System

Characteristics of the octal number system are as follows −

- Uses eight digits, 0,1,2,3,4,5,6,7
- Also called as base 8 number system
- Each position in an octal number represents a
**0**power of the base (8). Example 8^{0} - Last position in an octal number represents a
**x**power of the base (8). Example 8^{x}where**x**represents the last position – 1

Example

Octal Number: 12570_{8}

Calculating Decimal Equivalent −

Step |
Octal Number |
Decimal Number |

Step 1 | 12570_{8} |
((1 x 8^{4}) + (2 x 8^{3}) + (5 x 8^{2}) + (7 x 8^{1}) + (0 x 8^{0}))_{10} |

Step 2 | 12570_{8} |
(4096 + 1024 + 320 + 56 + 0)_{10} |

Step 3 | 12570_{8} |
5496_{10} |

**Note** − 12570_{8} is normally written as 12570.

Hexadecimal Number System

Characteristics of hexadecimal number system are as follows −

- Uses 10 digits and 6 letters, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F
- Letters represent the numbers starting from 10. A = 10. B = 11, C = 12, D = 13, E = 14, F = 15
- Also called as base 16 number system
- Each position in a hexadecimal number represents a
**0**power of the base (16). Example, 16^{0} - Last position in a hexadecimal number represents a
**x**power of the base (16). Example 16^{x}where**x**represents the last position – 1

Example

Hexadecimal Number: 19FDE_{16}

Calculating Decimal Equivalent −

Step |
Binary Number |
Decimal Number |

Step 1 | 19FDE_{16} |
((1 x 16^{4}) + (9 x 16^{3}) + (F x 16^{2}) + (D x 16^{1}) + (E x 16^{0}))_{10} |

Step 2 | 19FDE_{16} |
((1 x 16^{4}) + (9 x 16^{3}) + (15 x 16^{2}) + (13 x 16^{1}) + (14 x 16^{0}))_{10} |

Step 3 | 19FDE_{16} |
(65536+ 36864 + 3840 + 208 + 14)_{10} |

Step 4 | 19FDE_{16} |
106462_{10} |

**Note** − 19FDE_{16} is normally written as 19FDE.

There are many methods or techniques which can be used to convert numbers from one base to another. In this chapter, we’ll demonstrate the following −

- Decimal to Other Base System
- Other Base System to Decimal
- Other Base System to Non-Decimal
- Shortcut method – Binary to Octal
- Shortcut method – Octal to Binary
- Shortcut method – Binary to Hexadecimal
- Shortcut method – Hexadecimal to Binary

Decimal to Other Base System

**Step 1** − Divide the decimal number to be converted by the value of the new base.

**Step 2** − Get the remainder from Step 1 as the rightmost digit (least significant digit) of the new base number.

**Step 3** − Divide the quotient of the previous divide by the new base.

**Step 4** − Record the remainder from Step 3 as the next digit (to the left) of the new base number.

Repeat Steps 3 and 4, getting remainders from right to left, until the quotient becomes zero in Step 3.

The last remainder thus obtained will be the Most Significant Digit (MSD) of the new base number.

Example

Decimal Number: 29_{10}

Calculating Binary Equivalent −

Step |
Operation |
Result |
Remainder |

Step 1 | 29 / 2 | 14 | 1 |

Step 2 | 14 / 2 | 7 | 0 |

Step 3 | 7 / 2 | 3 | 1 |

Step 4 | 3 / 2 | 1 | 1 |

Step 5 | 1 / 2 | 0 | 1 |

As mentioned in Steps 2 and 4, the remainders have to be arranged in the reverse order so that the first remainder becomes the Least Significant Digit (LSD) and the last remainder becomes the Most Significant Digit (MSD).

Decimal Number : 29_{10} = Binary Number : 11101_{2.}

Other Base System to Decimal System

**Step 1** − Determine the column (positional) value of each digit (this depends on the position of the digit and the base of the number system).

**Step 2** − Multiply the obtained column values (in Step 1) by the digits in the corresponding columns.

**Step 3** − Sum the products calculated in Step 2. The total is the equivalent value in decimal.

Example

Binary Number: 11101_{2}

Calculating Decimal Equivalent −

Step |
Binary Number |
Decimal Number |

Step 1 | 11101_{2} |
((1 x 2^{4}) + (1 x 2^{3}) + (1 x 2^{2}) + (0 x 2^{1}) + (1 x 2^{0}))_{10} |

Step 2 | 11101_{2} |
(16 + 8 + 4 + 0 + 1)_{10} |

Step 3 | 11101_{2} |
29_{10} |

Binary Number : 11101_{2} = Decimal Number : 29_{10}

Other Base System to Non-Decimal System

**Step 1** − Convert the original number to a decimal number (base 10).

**Step 2** − Convert the decimal number so obtained to the new base number.

Example

Octal Number : 25_{8}

Calculating Binary Equivalent −

Step 1 – Convert to Decimal

Step |
Octal Number |
Decimal Number |

Step 1 | 25_{8} |
((2 x 8^{1}) + (5 x 8^{0}))_{10} |

Step 2 | 25_{8} |
(16 + 5)_{10} |

Step 3 | 25_{8} |
21_{10} |

Octal Number : 25_{8} = Decimal Number : 21_{10}

Step 2 – Convert Decimal to Binary

Step |
Operation |
Result |
Remainder |

Step 1 | 21 / 2 | 10 | 1 |

Step 2 | 10 / 2 | 5 | 0 |

Step 3 | 5 / 2 | 2 | 1 |

Step 4 | 2 / 2 | 1 | 0 |

Step 5 | 1 / 2 | 0 | 1 |

Decimal Number : 21_{10} = Binary Number : 10101_{2}

Octal Number : 25_{8} = Binary Number : 10101_{2}

Shortcut Method ─ Binary to Octal

**Step 1** − Divide the binary digits into groups of three (starting from the right).

**Step 2** − Convert each group of three binary digits to one octal digit.

Example

Binary Number : 10101_{2}

Calculating Octal Equivalent −

Step |
Binary Number |
Octal Number |

Step 1 | 10101_{2} |
010 101 |

Step 2 | 10101_{2} |
2_{8} 5_{8} |

Step 3 | 10101_{2} |
25_{8} |

Binary Number : 10101_{2} = Octal Number : 25_{8}

Shortcut Method ─ Octal to Binary

**Step 1** − Convert each octal digit to a 3-digit binary number (the octal digits may be treated as decimal for this conversion).

**Step 2** − Combine all the resulting binary groups (of 3 digits each) into a single binary number.

Example

Octal Number : 25_{8}

Calculating Binary Equivalent −

Step |
Octal Number |
Binary Number |

Step 1 | 25_{8} |
2_{10} 5_{10} |

Step 2 | 25_{8} |
010_{2} 101_{2} |

Step 3 | 25_{8} |
010101_{2} |

Octal Number : 25_{8} = Binary Number : 10101_{2}

Shortcut Method ─ Binary to Hexadecimal

**Step 1** − Divide the binary digits into groups of four (starting from the right).

**Step 2** − Convert each group of four binary digits to one hexadecimal symbol.

Example

Binary Number : 10101_{2}

Calculating hexadecimal Equivalent −

Step |
Binary Number |
Hexadecimal Number |

Step 1 | 10101_{2} |
0001 0101 |

Step 2 | 10101_{2} |
1_{10} 5_{10} |

Step 3 | 10101_{2} |
15_{16} |

Binary Number : 10101_{2} = Hexadecimal Number : 15_{16}

Shortcut Method – Hexadecimal to Binary

**Step 1** − Convert each hexadecimal digit to a 4-digit binary number (the hexadecimal digits may be treated as decimal for this conversion).

**Step 2** − Combine all the resulting binary groups (of 4 digits each) into a single binary number.

Example

Hexadecimal Number : 15_{16}

Calculating Binary Equivalent −

Step |
Hexadecimal Number |
Binary Number |

Step 1 | 15_{16} |
1_{10} 5_{10} |

Step 2 | 15_{16} |
0001_{2} 0101_{2} |

Step 3 | 15_{16} |
00010101_{2} |

Hexadecimal Number : 15_{16} = Binary Number : 10101_{2}

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