Understanding Base Systems and Converting Decimal Numbers

When dealing with different numerical systems, it is crucial to understand the base (or radix) of the number system. Each number in a specific base system represents a different value based on its position. For instance, in a base-4 number system, the number 100 represents (1 times 4^2 0 times 4^1 0 times 4^0 16). Understanding this concept allows us to efficiently convert decimal numbers (base-10) to other bases.

Identifying the Number System Base

Let's start by determining the base of the number system where 16 is represented as 100. We can represent this mathematically as:

(1 times b^2 0 times b^1 0 times b^0 16)

This simplifies to:

(b^2 16)

Taking the square root of both sides, we get:

(b 4)

Thus, the base of the number system is 4.

Converting Decimal Numbers to Base 4

Now that we know the base is 4, let's convert the decimal number 56 to its equivalent representation in base 4. We do this by repeatedly dividing the number by 4 and keeping track of the remainders:

56 ÷ 4 14 remainder 0 14 ÷ 4 3 remainder 2 3 ÷ 4 0 remainder 3

Reading the remainders from bottom to top, we get the base 4 representation of 56 as 320.

56 in base 4  320

Generalizing the Method

In the general case, a number system in which 16 is represented as 100 can be defined as having a base (n). Mathematically, this can be written as:

(1 times n^2 0 times n^1 0 times n^0 16)

Which simplifies to:

(n^2 16)

Thus:

(n 4)

For any decimal number, the process of converting it to base 4 involves the same division method outlined above. Let's convert a decimal number to base 4 using an additional example.

For instance, if we have the decimal number 68:

68 ÷ 4 17 remainder 0 17 ÷ 4 4 remainder 1 4 ÷ 4 1 remainder 0 1 ÷ 4 0 remainder 1

Reading the remainders from bottom to top, we get the base 4 representation of 68 as 1010.

68 in base 4  1010

Conclusion

Understanding and converting numbers between different bases is essential for various applications in computer science and mathematics. In this article, we have seen how the base of a number system can be determined and how to convert decimal numbers to a specific base, in this case, base 4. This knowledge can be applied to a wide range of problems and is a fundamental skill in working with number systems.