# Number System Conversions

There are many methods or techniques which can be used to convert numbers from one base to another. We’ll demonstrate here 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

Contents

## Decimal to Other Base System

Steps

**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 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

Steps

**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

Steps

**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

Steps

**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

Steps

**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

Steps

**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

Steps

**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}