Converting Decimal Numbers to Base 4: A Comprehensive Guide

If you've ever pondered how to convert a decimal number such as 124.68 to base 4, you are not alone. Decimal numbers are the basis of our everyday numerical system, but they can be converted into other bases to serve different purposes. Understanding how to do this is a fundamental skill in various fields, from computer science to cryptography. In this guide, we will explore the process of converting the number 124.68 from base 10 to base 4, providing you with detailed, step-by-step instructions.

Understanding Decimal and Base 4 Systems

The decimal system (base 10) is the most widely used numerical system in everyday life, consisting of ten digits (0-9). Base 4, however, is a numeral system with four symbols: 0, 1, 2, and 3. Converting a decimal number to base 4 involves breaking down the number into its constituent parts and expressing it in base 4.

Converting the Integer Part

Let's start with the integer part of the decimal number 124.68. The process involves repeatedly dividing the number by the base (4) and noting the remainders. These remainders, when read in reverse order, give you the base 4 equivalent.

Method 1: Division Method

1. Divide the number by 4:

124 div 4 31 R 0 31 div 4 7 R 3 7 div 4 1 R 3 1 div 4 0 R 1

Reading the remainders in reverse order gives us the base 4 number: 1330.

Verification: We can verify this by expressing the base 4 number in base 10:

1 × 4^3 3 × 4^2 3 × 4^1 0 × 4^0 64 48 12 0 124

Method 2: Powers of 4

Another method to convert the integer part is to break down the decimal number into multiples of the powers of 4. For the number 124, we have:

124 1 × 4^3 3 × 4^2 3 × 4^1 0 × 4^0 1330 in base 4

Converting the Decimal Part

The decimal part of the number (0.68) can be converted to base 4 by repeatedly multiplying by 4 and noting the integer parts. This process is continued until the decimal part becomes zero or a pattern repeats.

Method 1: Multiplication Method

1. Multiply the decimal part by 4:

0.68 × 4 2.72 → integer part is 2 0.72 × 4 2.88 → integer part is 2 0.88 × 4 3.52 → integer part is 3 0.52 × 4 2.08 → integer part is 2 0.08 × 4 0.32 → integer part is 0 0.32 × 4 1.28 → integer part is 1 0.28 × 4 1.12 → integer part is 1 0.12 × 4 0.48 → integer part is 0 0.48 × 4 1.92 → integer part is 1 0.92 × 4 3.68 → integer part is 3

Notice that the process repeats with 0.68 coming up again. Thus, the base 4 representation of the decimal part is:

0.68 0.2232011013

Combining the integer and decimal parts, we get:

124.68 in base 4 1330.2232011013

Step-by-Step Conversion

Let's break down the entire conversion process into simple steps for clarity:

Conversion of 124.68 to Base 4

Step 1: Integer Part.

Divide the number by 4 and note the remainders: 124 / 4 31 R 0 31 / 4 7 R 3 7 / 4 1 R 3 1 / 4 0 R 1

Reading the remainders upward, we get: 1330 (base 4).

Step 2: Decimal Part.

Multiply the decimal part by 4 and note the integer parts: 0.68 × 4 2.72 → integer part is 2 0.72 × 4 2.88 → integer part is 2 0.88 × 4 3.52 → integer part is 3 0.52 × 4 2.08 → integer part is 2 0.08 × 4 0.32 → integer part is 0 0.32 × 4 1.28 → integer part is 1 0.28 × 4 1.12 → integer part is 1 0.12 × 4 0.48 → integer part is 0 0.48 × 4 1.92 → integer part is 1 0.92 × 4 3.68 → integer part is 3

The process repeats with 0.68 coming up again. Thus, the base 4 representation of the decimal part is: 0.2232011013. Combining the integer and decimal parts, we get: 1330.2232011013.

Therefore, 124.68 in base 10 is equivalent to 1330.2232011013 in base 4.

Conclusion

Understanding how to convert decimal numbers to base 4 is a valuable skill that can be applied in various fields. The methods outlined in this guide—division and powers of 4 for the integer part, and multiplication for the decimal part—provide a comprehensive approach to achieving accurate results.

Further Reading

Explore related topics such as converting between different numerical systems, understanding hexadecimal and binary systems, and applications in computer science and data processing.

References

Algorithms and Data Structures: The Basic Toolbox by Kurt Mehlhorn and Peter Sanders