Convert The Following Expression To The Indicated Base

Converting Number Bases: A Comprehensive Guide

Converting numbers between different bases is a fundamental concept in mathematics and computer science. Understanding this process is crucial for working with binary (base-2), octal (base-8), decimal (base-10), hexadecimal (base-16), and other number systems. This article provides a comprehensive guide to converting expressions to indicated bases, covering various methods and providing ample examples to solidify your understanding. We will explore both integer and fractional part conversions, tackling common challenges and offering practical tips.

Understanding Number Bases

Before diving into the conversion process, let's establish a clear understanding of number bases. A base, also known as a radix, defines the number of digits used to represent numbers in a particular system. The familiar decimal system (base-10) uses ten digits (0-9). Binary (base-2) uses only two digits (0 and 1), octal (base-8) uses eight digits (0-7), and hexadecimal (base-16) uses sixteen digits (0-9 and A-F, where A represents 10, B represents 11, and so on).

Each digit in a number represents a power of the base. For example, the decimal number 123 can be expressed as:

1 × 10² + 2 × 10¹ + 3 × 10⁰ = 100 + 20 + 3 = 123

Converting from Decimal to Other Bases

Converting a decimal number to another base involves repeatedly dividing the decimal number by the target base and recording the remainders. The remainders, read in reverse order, form the equivalent number in the target base.

Steps for Conversion (Decimal to Other Bases):

1. Divide: Divide the decimal number by the target base.
2. Record Remainder: Note the remainder.
3. Repeat: Repeat steps 1 and 2 using the quotient from the previous division, until the quotient becomes 0.
4. Reverse Remainders: The remainders, read from bottom to top (last remainder to first remainder), form the equivalent number in the target base.

Example 1: Converting Decimal 25 to Binary (Base-2)

Reading the remainders from bottom to top (11001), we get the binary equivalent: 25₁₀ = 11001₂

Example 2: Converting Decimal 47 to Hexadecimal (Base-16)

Reading the remainders from bottom to top (2F), we get the hexadecimal equivalent: 47₁₀ = 2F₁₆

Converting from Other Bases to Decimal

Converting a number from any base to decimal involves multiplying each digit by the corresponding power of the base and summing the results.

Steps for Conversion (Other Bases to Decimal):

1. Identify Digits and Powers: Identify each digit in the number and its corresponding power of the base (starting from the rightmost digit with power 0).
2. Multiply and Sum: Multiply each digit by the corresponding power of the base. Sum the results.

Example 3: Converting Binary 101101₂ to Decimal

101101₂ = (1 × 2⁵) + (0 × 2⁴) + (1 × 2³) + (1 × 2²) + (0 × 2¹) + (1 × 2⁰) = 32 + 0 + 8 + 4 + 0 + 1 = 45₁₀

Example 4: Converting Hexadecimal 3A₁₆ to Decimal

3A₁₆ = (3 × 16¹) + (10 × 16⁰) = 48 + 10 = 58₁₀

Converting Between Bases (Non-Decimal)

Converting directly between bases other than decimal is possible, but often less intuitive. It's generally more efficient to convert first to decimal, then to the target base.

Example 5: Converting Octal 57₈ to Binary

1. Octal to Decimal: 57₈ = (5 × 8¹) + (7 × 8⁰) = 40 + 7 = 47₁₀
2. Decimal to Binary: (See Example 1 for the method) 47₁₀ = 101111₂

Therefore, 57₈ = 101111₂

Fractional Part Conversion

The methods described above apply to the integer part of a number. Converting the fractional part requires a slightly different approach.

Converting Decimal Fractional Part to Other Bases:

1. Multiply: Multiply the fractional part by the target base.
2. Record Integer Part: Record the integer part of the result.
3. Repeat: Repeat steps 1 and 2 using the fractional part of the previous result, until the fractional part becomes 0 or a repeating pattern emerges.
4. Combine: The integer parts, read from top to bottom (first integer to last integer), form the fractional part in the target base.

Example 6: Converting Decimal 0.625 to Binary

Therefore, 0.625₁₀ = 0.101₂

Converting Fractional Part from Other Bases to Decimal:

This process is similar to converting the integer part, but with negative powers of the base.

Example 7: Converting Binary 0.11₂ to Decimal

0.11₂ = (1 × 2⁻¹) + (1 × 2⁻²) = 0.5 + 0.25 = 0.75₁₀

Handling Negative Numbers

Converting negative numbers requires handling the sign separately. Convert the magnitude of the number using the methods described above, then add the negative sign to the result.

Common Errors and Troubleshooting

• Incorrect Remainders: Double-check your remainders during decimal-to-other-base conversions.
• Incorrect Power of the Base: Carefully track the powers of the base during other-base-to-decimal conversions.
• Mixing Bases: Avoid mixing digits from different bases in a single number.
• Fractional Part Truncation: Be aware that the fractional part conversion may produce an infinite repeating pattern in some cases. You might need to truncate the result after a certain number of digits.

Frequently Asked Questions (FAQ)

Q: What is the significance of different number bases in computer science?

A: Binary (base-2) is fundamental because computers represent data using binary digits (bits), 0 and 1. Octal (base-8) and hexadecimal (base-16) are used for representing binary data more compactly, as each octal digit corresponds to three bits and each hexadecimal digit corresponds to four bits.

Q: Can I convert directly from base-8 to base-16 without going through base-10?

A: While possible, it's generally less efficient. Converting to decimal as an intermediate step simplifies the process.

Q: What if I have a very large number to convert?

A: For extremely large numbers, it's best to use computational tools or programming languages that handle arbitrary-precision arithmetic.

Conclusion

Converting numbers between different bases is a valuable skill in mathematics and computer science. By mastering the techniques outlined in this article, you will gain a deeper understanding of number systems and their representation. Remember to practice regularly, paying close attention to detail to avoid common errors. Understanding these conversions is not just about manipulating numbers; it provides a foundational understanding of how information is represented and processed in various systems, particularly in the digital world. With consistent practice and a clear understanding of the underlying principles, you can confidently navigate the world of base conversions and appreciate the elegance and power of different number systems.