How to Convert Decimal 0.5625 to Base Six

Converting between number bases can sometimes be a cumbersome task, especially for non-integer values. In this article, we will explore the process of converting the decimal 0.5625 to base six, also known as the base-6 number system. We will break down each step and provide a detailed explanation to ensure a thorough understanding.

Understanding Number Bases

The base of a number system determines how many unique digits can be used. The decimal system, which we commonly use, is a base-10 system, meaning it uses ten digits: 0 through 9. In contrast, the base-six system uses only six digits: 0, 1, 2, 3, 4, and 5. This means that each digit in a base-six number can only be one of these values.

Converting a Decimal to Base Six

The process of converting a decimal to base six involves a series of multiplications and divisions by six. Here is a step-by-step guide:

Step 1: Multiply the Decimal Fraction by Six

To convert the fractional part of the number, we repeatedly multiply the fraction by six. After each multiplication, we record the integer part as the next digit in the base-six representation and discard the decimal part for the next iteration.

Let's convert 0.5625 from base ten to base six:

0.5625 * 6  3.375 - Record 3, use 0.375 as the next fraction0.375 * 6  2.25 - Record 2, use 0.25 as the next fraction0.25 * 6  1.5 - Record 1, use 0.5 as the next fraction0.5 * 6  3.0 - Record 3, use 0.0 as the next fraction

Step 2: Combine the Recorded Digits

After performing the multiplications, we combine the recorded digits to form the base-six equivalent of the original decimal.

Therefore, 0.562510 is 0.32136.

Verification

To ensure the accuracy of our conversion, we can verify it by performing the reverse conversion. We multiply each digit of the base-six number by the corresponding power of six and sum the results:

3/6 2/62 1/63 3/64 0.562510

This process confirms that our conversion is correct.

Conclusion

Converting a decimal to a different base can be done systematically by repeatedly multiplying the fraction by the new base and recording the integer parts. This method works well for both whole numbers and decimal fractions. Understanding this process is vital when working with different number systems.

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