Converting decimal numbers to octal is a straightforward process that involves converting a base-10 number into a base-8 number. Octal numbers use digits from 0 to 7. This article will detail the methods for converting both whole decimal numbers and decimal fractions to octal, with thorough explanations and examples.

## Converting Whole Decimal Numbers to Octal

To convert a whole decimal number to octal, we use the method of repeated division by 8. Here’s the step-by-step process:

**Divide the decimal number by 8.****Record the remainder (0-7).****Update the decimal number to the quotient obtained from the division.****Repeat steps 1-3 until the quotient is 0.****The octal number is the sequence of remainders read from bottom to top.**

### Example 1: Converting 156 to Octal

**Divide 156 by 8:**- Quotient: 19
- Remainder: 4

**Divide 19 by 8:**- Quotient: 2
- Remainder: 3

**Divide 2 by 8:**- Quotient: 0
- Remainder: 2

Reading the remainders from bottom to top, we get 234.

So, $156_{10} = 234_8$.

## Converting Decimal Fractions to Octal

To convert a decimal fraction to octal, we use the method of repeated multiplication by 8. Here’s how it works:

**Multiply the decimal fraction by 8.****Record the integer part (0-7).****Update the decimal fraction to the fractional part obtained from the multiplication.****Repeat steps 1-3 until the fractional part is 0 or you reach the desired level of precision.****The octal fraction is the sequence of integer parts read in order.**

### Example 2: Converting 0.6875 to Octal

**Multiply 0.6875 by 8:**- Result: 5.5
- Integer part: 5
- Fractional part: 0.5

**Multiply 0.5 by 8:**- Result: 4.0
- Integer part: 4
- Fractional part: 0

The octal fraction is the sequence of integer parts, which gives us 0.54.

So, $0.6875_{10} = 0.54_8$.

### General Conversion Method for Mixed Numbers

For numbers with both whole and fractional parts, convert each part separately and then combine the results.

### Example 3: Converting 156.6875 to Octal

**Convert the whole number part (156) to octal:**- $156_{10} = 234_8$

**Convert the fractional part (0.6875) to octal:**- $0.6875_{10} = 0.54_8$

Combine the two parts to get the octal representation: $156.6875_{10} = 234.54_8$

## Conclusion

Converting decimal numbers to octal involves different methods for whole numbers and fractions. Whole numbers are converted using repeated division by 8, while fractions are converted using repeated multiplication by 8. Understanding these processes is crucial for working with systems that use base-8 numbering, such as some digital and computing applications. With practice, you can easily convert any decimal number to its octal equivalent.