how to convert from base 10 to base 8

Daniel Parker

2 min read

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07-02-2025

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Converting numbers between different bases is a fundamental concept in computer science and mathematics. This guide will walk you through the process of converting a base 10 (decimal) number to base 8 (octal). We'll cover the method, provide examples, and even show you how to check your work.

Understanding Base 10 and Base 8

Before diving into the conversion process, let's briefly review the basics of number systems.

• Base 10 (Decimal): This is the number system we use daily. It uses ten digits (0-9) and each position represents a power of 10 (ones, tens, hundreds, thousands, etc.). For example, the number 1234 in base 10 means (1 x 10³)+(2 x 10²)+(3 x 10¹)+(4 x 10⁰).

• Base 8 (Octal): This system uses eight digits (0-7). Each position represents a power of 8 (ones, eights, sixty-fours, etc.).

Method: Repeated Division by 8

The most efficient method for converting base 10 to base 8 involves repeated division by 8. Here's a step-by-step process:

1. Divide: Divide the base 10 number by 8.
2. Record the Remainder: Note the remainder. This will be the rightmost digit (least significant digit) of your octal number.
3. Repeat: Divide the quotient (the result of the division) by 8. Again, note the remainder. This is the next digit to the left.
4. Continue: Keep repeating steps 2 and 3 until the quotient becomes 0.
5. Read the Result: The remainders, read from bottom to top, form the octal equivalent.

Examples

Let's illustrate the method with a few examples:

Example 1: Converting 37 (Base 10) to Base 8

Reading the remainders from bottom to top (4, 5), we get 45₈. Therefore, 37₁₀ = 45₈.

Example 2: Converting 150 (Base 10) to Base 8

Reading the remainders from bottom to top (2, 2, 6), we get 226₈. Thus, 150₁₀ = 226₈.

Example 3: Converting a Larger Number

Let's convert 1234 (Base 10) to Base 8:

The remainders, read from bottom to top, are 2322. Therefore, 1234₁₀ = 2322₈.

Checking Your Work

You can check your conversion by expanding the octal number back to base 10 using the place value method described earlier. For example, let's check our conversion of 2322₈:

(2 x 8³) + (3 x 8²) + (2 x 8¹) + (2 x 8⁰) = 1024 + 192 + 16 + 2 = 1234₁₀

This confirms our conversion is correct.

Conclusion

Converting base 10 numbers to base 8 (octal) is a straightforward process using repeated division. By following the steps outlined above and practicing with a few examples, you'll quickly master this essential skill. Remember to always double-check your work to ensure accuracy. This understanding is crucial in various fields, particularly in computer programming and digital systems where octal representation is sometimes used.